2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 11doi: 10.28924/ada/ma.2.11 on the α−ψ− contractive mappings in c∗-algebra valued b-rectangular metric spaces and fixed point theorems mohamed rossafi1,∗, abdelkarim kari2, hafida massit3 1lasma laboratory department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, b. p. 1796 fes atlas, morocco mohamed.rossafi@usmba.ac.ma 2ams laboratory faculty of sciences ben m’sik, hassan ii university, casablanca, morocco abdkrimkariprofes@gmail.com 3laboratory of partial differential equations, spectral algebra and geometry department of mathematics, faculty of sciences, university ibn tofail, kenitra, morocco massithafida@yahoo.fr ∗correspondence: mohamed.rossafi@usmba.ac.ma abstract. this present paper extends a version of α−ψ−contraction in c∗-algebra valued rectangu-lar b-metric spaces and establishing the existence and uniqueness of fixed point for them. non-trivialexamples are further provided to support the hypotheses of our results. 1. introduction a c∗-algebra valued metric spaces were introduced by ma et al. [6] as a generalization of metricspaces they proved certain fixed point theorems, by giving the definition of c∗-algebra valuedcontractive mapping analogous to banach contraction principle. many mathematicians worked onthis interesting space.various fixed point results were established on such spaces, see [1–3] and references therein.combining conditions used for definitions of c∗-algebra valued metric and generalized metricspaces, g kalapana and tasneem [4] announced the notions of c∗-algebra valued metric space andestablish nice results of fixed point on such space.in this paper, inspired by the work done in [9], we introduce the notion of α − ψ−contractionand establish some new fixed point theorems for mappings in the setting of complete c∗-algebravalued rectangular bmetric spaces.moreover, an illustrative examples is presented to support the obtained results. received: 18 dec 2021. key words and phrases. fixed point; c∗-algebra valued metric spaces; α − ψ− contraction; α − ψ − c∗ valuedcontraction. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 2 2. preliminaries throughout this paper, we denote a by an unital (i.e ,unity element i) c∗-algebra with linearinvolution ∗, such that for all x,y ∈a, (xy)∗ = y∗x∗,and x∗∗ = x.we call an element x ∈a a positive element, denote it by x � θif x ∈ ah = {x ∈ a : x = x∗} and σ(x) ⊂ r+,where σ(x) is the spectrum of x.using positiveelement ,we can define a partial ordering � on ah as follows : x � y if and only if y −x � θwhere θ means the zero element in a. we denote the set x ∈a : x � θ by a+ and |x| = (x∗x) 12 .and a′ will denote the set {a ∈a+; ab = ba,∀b ∈a} lemma 2.1. [8] suppose that a is a unital c∗-algebra with a unit i,(1) for any x ∈a+ we have x � i ⇐⇒‖x‖≤ 1(2) if a ∈a+ with ‖a‖ < 1 2 then i −a is unvertible and ‖a(1 −a)−1‖ < 1(3) suppose that a,b ∈a+ and ab = ba, then ab � θ(4) let a ∈ a′, if b,c ∈ a, with b � c � θ, and i − a ∈ a′+ is invertible operator, then (i −a)−1b � (i −a)−1c definition 2.2. [4] let x be a non-empty set and b ∈ a such that b � i. supposa the mapping d : x ×x →a+ satisfies:(i) d(x,y) = θ if and only if x = y;(ii) d(x,y) = d(y,x) for all distinct points x,y ∈ x;(iii) d(x,y) � b[d(x,u) + d(u,v) + d(v,y)] for all x,y ∈ x and for all distinct points u,v ∈ x −{x,y}.then (x,a+,d) is called a c∗-algebra valued rectangular b−metric space. example 2.3. let x = r and a = m2(r). define d(x,y) = diag(|x − y|, 2|x − y|) where x,y ∈r. it is easy to verify d is a c∗− algebra-valued rectangular b− metric and (x,m2(r),d)is a copmlete c∗-algebra valued rectangular b−metric space. definition 2.4. [9] if ψ : a → b is a linear mapping in c∗-algebra, it is said to be positive if ψ(a+) ⊆ b+. in this case ψ(ah) ⊆ bh, and the restriction map ψ : ah → bh is increasing. definition 2.5. [9] suppose that a and b are c∗-algebra .a mapping ψ : a → b is said to be c∗homomorphism if :(i) ψ(ax + by) = aψ(x) + bψ(y) for all a,b ∈c and x,y ∈ a(ii) ψ(xy) = ψ(x)ψ(y) for all x,y ∈ a https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 3 (iii) ψ(x∗) = ψ(x)∗ for all x ∈ a(iv) ψ maps the unit in a to the unit in b. definition 2.6. [9] let a and b be c∗-algebra spaces and let ψ : a → b be a homomorphismthen ψ is called an ∗− homomorphism if it is one to one ∗− homomorphism.a c∗-algebra a is ∗−isomorphic to a c∗-algebra b if there exists ∗− isomorphism of a onto b. definition 2.7. [9] let ψ be the set of positive functions ψ : a+ → a+ satisfying the followingconditions : (i) ψ is continous and nondecrasing(ii) ψ(a) = θ if and only if a = θ(iii) limn−→∞ψn(a) = θ, (a � θ) ,∑∞n=1ψn(a) < ∞(iv) the series ∑∞k=1bkψk(a) < ∞ for a � θ is increasing and continuous at θ. corollary 2.8. [9] every c∗− homomorphism is contractive and hence bounded. lemma 2.9. every ∗− homomorphism is positive. definition 2.10. [9] let x be a nonempty set and α : x ×x →a′+ be a function, wesay that theself map t is α− admissible if (x,y) ∈ x ×x,α(x,y) � i ⇒ α(tx,ty) � i,where i the unit of a. definition 2.11. [9] let (x,a,d) be a c∗-algebra valued b− metric space and t : x → x ismapping, we say that t is an α−ψ− contractive mapping if there exist two functions α : x×x → a+ and ψ ∈ ψ such that α(x,y)d(tx,ty) � ψ(d(x,y)), for all x,y ∈ x 3. main result in [9] introduced the concept of α−ψ− contractive mappings in a unital c∗-algebra valued b−metric space. in this paper we will develop the definitions in case of unital c∗-algebra valuedrectangular b− metric space and give some banach fixed point theorems. definition 3.1. let (x,a,d) be a c∗-algebra valued b− rectangular metric space and t : x → xis mapping, we say that t is an α − ψ− contractive mapping if there exist two functions α : x ×x →a+ and ψ ∈ ψ such that α(x,y)d(tx,ty) � ψ(d(x,y)), f orallx,y ∈ x (3.1) theorem 3.2. let (x,a,d) be a complete c∗-algebra valued rectangular b− metric space and let t : x → x be a α,ψ− contractive mapping satisfying the following conditions: (i) t is α− admissible https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 4 (ii) there exists x0 ∈ x such that α(x0,tx0) � i(iii) for all x,y ∈ x ,there exists z ∈ x such that α(x,z) � i and α(y,z) � i(iv) t is continuous then, t has a unique fixed point in x. proof. let x0 ∈ x such that α(x0,tx0) � i and define a sequence {xn}∈ x such that xn+1 = txn, ∀n ∈ n. suppose that there exists n ∈ n such that xn = txn. then xn is a fixed point of t andthe proof is finished.hence, we assume that xn 6= txn+1, ∀n ∈n, since t is α−admissible, we get α(x0,x1) = α(x0,tx0) � i ⇒ α(tx0,t2x0) = α(x1,x2) � i. continuing this process, we have α(xn,xn+1) � i ∀n ∈n. (3.2) by 3.1 and 3.2, we get d(xn,xn+1) = d(txn−1,txn) � α(xn−1,xn)d(txn−1,txn) � ψ(d(xn−1,xn)) � � ψn(d(x0,x1)). for m ≥ 1 and p ≥ 1, it follows that d(xm+p,xm) � b[d(xm+p,xm+p−1) + d(xm+p−1,xm+p−2) + d(xm+p−2,xm)] � bd(xm+p,xm+p−1) + bd(xm+p−1,xm+p−2) + b[b[d(xm+p−2,xm+p−3) + d(xm+p−3,xm+p−4) + d(xm+p−4,xm)]] = bd(xm+p,xm+p−1) + bd(xm+p−1,xm+p−2) + b 2d(xm+p−2,xm+p−3) + b 2d(xm+p−3,xm+p−4) + b2d(xm+p−4,xm) � bd(xm+p,xm+p−1) + bd(xm+p−1,xm+p−2) + b2d(xm+p−2,xm+p−3) + b2d(xm+p−3,xm+p−4) + .... + b p−1 2 d(xm+3,xm+2) + b p−1 2 d(xm+2,xm+1) + b p−1 2 d(xm+1,xm) � bψm+p−1(d(x0,x1)) + bψm+p−2(d(x0,x1)) + ... + b p−1 2 d(x0,x1)since b � i, using definition 2.6 we have d(xm,xm+p) � bψm+p−1(d(x0,x1)) +bψm+p−2(d(x0,x1)) +...+b p−1 2 d(x0,x1) → θ as n → +∞therefore {xn} is a cauchy sequence in x. by the completeness of (x,a,d) there exists an x ∈ x such that limn→∞xn = limn→∞txn−1 = x. from continuity of t and by uniqueness of the limit, we get tx = x, ie. x is a fixed point of t .now suppose that y 6= x is another fixed point of t . https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 5 from (iii), there exists z ∈ x such that α(x,z) � i and α(y,z) � i.since t is α− admissible, we have α(x,tnz) � i and α(y,tnz) � i for all n ∈n using (1), we obtain d(x,tnz) = d(tx,t (tn−1z)) � α(x,tn−1z)d(tx,t (tn−1z)) � ψn(d(x,z)) → θ as n →∞. thus, tnz = x. similary tnz = y as n →∞ so, the uniqueness of the limit we obtain x = y . � example 3.3. let x = r and a = m2(r) as given in example 2.3, define t : x → x, by tx = x 3and α : x ×x → m2(r) such that α(x,y) = ( |x −y| 0 0 0 ) thus, t is α− admissible, and ψ : m2(r)+ → m2(r)+ , ψ(a) = ( a2 0 0 a2 ) ∀a ∈ (r)+. this is clear that t is α−ψ− contractive mapping and satisfies α(x,y)d(tx,ty) � ψ(d(x,y)), for all x,y ∈ x theorem 3.4. let (x,a,d) be a complete c∗-algebra valued rectangular b− metric space and let t : x → x be a α,ψ− contractive mapping of kannan type ie, α(x,y)d(tx,ty) � ψ(d(tx,x) + d(ty,y)) (3.3) for all x,y ∈ x where ψ ∈ ψ and α : x ×x →a+ and the following conditions holds: (i) t is α− admissible(ii) there exists x0 ∈ x such that α(x0,tx0) � i(iii) t is continuous then, t has a fixed point in x. proof. by (3.3), we obtain d(xn,xn+1) = d(txn−1,txn) � α(xn−1,xn)d(txn−1,txn) � ψ(d(txn−1,xn−1) + d(txn,xn)) = ψ(d(xn,xn−1) + d(xn+1,xn)) = ψ(d(xn,xn−1)) + ψ(d(xn+1,xn)) (i −ψ)(d(xn,xn−1)) � ψ(d(xn,xn−1)) from lemma 2.1 and definition 2.6, we obtain https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 6 d(xn,xn+1) � (i −ψ)−1ψ(d(xn,xn−1)) = φ(d(xn,xn−1)) where φ = (i −ψ)−1ψ therefore d(xn,xn+1) � φn(d(x0,x1))∀n ∈n for any m ≥ 1 and p ≥ 1 similary in theorem 3.1 we have d(xm,xm+p) � bψm+p−1(d(x0,x1)) +bψm+p−2(d(x0,x1)) +...+b p−1 2 d(x0,x1) → θ as n → +∞.thus {xn} is a cauchy sequence in x. by the completeness of (x,a,d), there exists x ∈ xsuch that limn→∞xn = limn→∞txn−1 = x. the continuity of t gives that x is a fixed point of t .to prove that x is the unique fixed point, we suppose that y ∈ x is another fixed point of t .then θ � d(x,y) = d(tx,ty) � α(x,y)d(tx,ty) � ψ(d(tx,x) + d(ty,y)) = ψ(d(x,x) + d(y,y)) = θ hence x = y .therefore the fixed point is unique. � theorem 3.5. let (x,a,d) be a complete c∗-algebra valued rectangular b− metric space and let t : x → x be a α,ψ− contractive mapping of banach-kannan type ie, α(x,y)d(tx,ty) � ψ(d(x,y) + d(tx,x) + d(ty,y)) (3.4) for all x,y ∈ x where ψ ∈ ψ and α : x ×x →a+ such that ψ(1 −ψ)−1 � 1 2i , and the following conditions holds: (i) t is α− admissible(ii) there exists x0 ∈ x such that α(x0,tx0) � i(iii) t is continuous then, t has a fixed point in x proof. using (3.4), we get d(xn,xn+1) = d(txn−1,txn) � α(xn−1,xn)d(txn−1,txn) � ψ(d(xn−1,xn) + d(txn−1,xn−1) + d(txn,xn)) = ψ(d(xn−1,xn)2i + d(xn,xn+1)) ⇒ (i −ψ)(d(xn,xn+1)) � 2iψ(d(xn,xn−1)) ⇒ d(xn,xn+1) � 2i(i −ψ)−1ψ(d(xn,xn−1)) � φ(d(xn,xn−1)).where https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 7 ϕ = 2i(i −ψ)−1ψ. then d(xn,xn+1) � φn(d(x0,x1). we refer to the proof of the theorem 3.1 we get that x is a fixed point of t . now, if y 6= x isanother fixed point of t , we have θ � d(x,y) = d(tx,ty) � α(x,y)d(tx,ty) � ψ(d(x,y) + d(tx,x) + d(ty,y)) = ψ(d(x,y) + d(x,x) + d(y,y)) = ψ(d(x,y). so d(x,y) = θ ; ie x = y . � 4. applications as application of α−ψ contractive in unital c∗-algebra valued rectangular b− metric spaces,existence and uniqueness results for a type of operator equation is given. example 4.1. suppose that h is a hilbert space, b(h) is the set of linear bounded operators on h. let a1,a2, ...,an, ... ∈ b(h)which satisfy ∑∞n=1‖an‖ < 1 and q ∈ b(h)+.then the operator equation x −∑∞n=1a∗nxan = q has a unique solution in b(h). proof. set a = (∑∞n=1‖an‖)p with p ≥ 1, then ‖a‖ < 1.without loss of generality, one can supposethat a > 0.choose a positive operator m ∈ b(h).for x,y ∈ b(h) and p ≥ 1, set d(x,y ) = ‖x −y‖pm.then d(x,y ) is a c∗-algebra valued rectangular b− metric.suppose that x,y,z,w ∈ b(h) we have ‖x −y‖p � 2p (‖x −z‖p + ‖z −w‖p + ‖w −y‖p).which implies that d(x,y ) � a[d(x,z) + d(z,w ) + d(w,y )]where a = 2pi. consider the map t : b(h) → b(h) such that t (x) = ∑∞n=1a∗nxan + q.then d(t (x),t (y )) = ‖t (x),t (y )‖pm = ‖ ∑∞ n=1a ∗ n(x −y )an‖pm � ∑∞ n=1‖an‖ 2p‖x −y‖pm � a2d(x,y ) https://doi.org/10.28924/ada/ma.2.11 eur. j. math. anal. 10.28924/ada/ma.2.11 8 let α : b(h) ×b(h) → b(h)+ defined by α(x,y ) = (x,y )iand ψ : b(h)+ → b(h)+ defined by ψ(x) = x.we get α(x,y )d(tx,ty ) � ψ(d(x,y )).using theorem 3.1, there exists a unique fixed point x in b(h). � 5. acknowledgments it is our great pleasure to thank the referee for his careful reading of the paper and for severalhelpful suggestions. references [1] h.h. alsulami, r.p. agarwal, e. karapınar, f. khojasteh, a short note on c∗-valued contraction mappings, j. inequal.appl. 2016 (2016) 50. https://doi.org/10.1186/s13660-016-0992-5.[2] s. chandok, d. kumar, c. park, c∗−algebra-valued partial metric space and fixed point theorems. proc. math. sci.129 (2019) 37. https://doi.org/10.1007/s12044-019-0481-0.[3] m. jleli, b. samet, a new generalization of the banach contraction principle. j. inequal. appl. 2014 (2014), 38. https://doi.org/10.1186/1029-242x-2014-38.[4] g. kalapana, z.s. tasneem c∗−algebra-valued rectangular b-metric spaces and some fixed point theorems, commun.fac. sci. univ. ank. ser. a1 math. stat. 68 (2019) 2198-2208. https://doi.org/10.31801/cfsuasmas.598146.[5] w. a. kirk, n. shahzad, generalized metrics and caristi’s theorem, fixed point theory appl. 2013 (2013) 129. https://doi.org/10.1186/1687-1812-2013-129.[6] z. ma, l. jiang, h. sun, c∗-algebra-valued metric spaces and related fixed point theorems, fixed point theoryappl. (2014) 2014, 206. https://doi.org/10.1186/1687-1812-2014-206.[7] h. massit, m. rossafi, fixed point for ψ− contractive mapping in c∗− algebra valued rectangular b-metric, j. math.comput. sci. 11(2021) 6507-6521. https://doi.org/10.28919/jmcs/6363.[8] g.j. murphy, c∗-algebras and operator theory, academic press, london, uk, 1990.[9] s. omran, i. masmali, on the (α−ψ)-contractive mappings in c∗-algebra valued b-metric spaces and fixed pointtheorems, j. math. 2021 (2021) 7865976. https://doi.org/10.1155/2021/7865976.[10] b. samet, c. vetro, p. vetro, fixed point theorems for α−ψ−contractive type mappings, nonlinear anal.: theorymethods appl. 75 (2012) 2154–2165. https://doi.org/10.1016/j.na.2011.10.014. https://doi.org/10.28924/ada/ma.2.11 https://doi.org/10.1186/s13660-016-0992-5 https://doi.org/10.1007/s12044-019-0481-0 https://doi.org/10.1186/1029-242x-2014-38 https://doi.org/10.31801/cfsuasmas.598146 https://doi.org/10.1186/1687-1812-2013-129 https://doi.org/10.1186/1687-1812-2014-206 https://doi.org/10.28919/jmcs/6363 https://doi.org/10.1155/2021/7865976 https://doi.org/10.1016/j.na.2011.10.014 1. introduction 2. preliminaries 3. main result 4. applications 5. acknowledgments references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 8doi: 10.28924/ada/ma.2.8 stability of positive weak solution for generalized weighted p-fisher-kolmogoroff nonlinear stationary-state problem salah a. khafagy∗, hassan m. serag department of mathematics, faculty of science, al-azhar university, nasr city (11884), cairo, egypt salahabdelnaby.211@azhar.edu.eg, serraghm@yahoo.com ∗correspondence: salahabdelnaby.211@azhar.edu.eg abstract. in the present paper, we investigate the stability results of positive weak solution for thegeneralized fisher–kolmogoroff nonlinear stationary-state problem involving weighted p-laplacianoperator −d∆p,pu = ka(x)u[ν−υu] in ω, bu = 0 on ∂ω, where ∆p,p with p > 1 and p = p (x)is a weight function, denotes the weighted p-laplacian defined by ∆p,pu ≡ div[p (x)|∇u|p−2∇u],the continuous function a(x) : ω → r satisfies either a(x) > 0 or a(x) < 0 for all x ∈ ω,d,k,νand υ are positive parameters and ω ⊂ rn is a bounded domain with smooth boundary bu = δh(x)u + (1 −δ) ∂u ∂n where δ ∈ [0, 1], h : ∂ω → r+ with h = 1 when δ = 1. 1. introduction: in this paper we study the stability results of positive weak solution for the generalized weighted p-fisher–kolmogoroff nonlinear stationary-state problem −d∆p,pu = ka(x)f (u) = ka(x)u[ν −υu] in ω, bu = 0 on ∂ω, } (1.1) where ∆p,p with p > 1 and p = p (x) is a weight function, denotes the weighted p-laplaciandefined by ∆p,pu ≡ div[p (x)|∇u|p−2∇u] (see for details [6]), the continuous function a(x) : ω → rsatisfies either a(x) > 0 or a(x) < 0 for all x ∈ ω,d,k,ν,ν and υ are positive parameter and ω ⊂ rn is a bounded domain with smooth boundary bu = δh(x)u + (1 − δ)∂u ∂n where δ ∈ [0, 1], h : ∂ω → r+ with h = 1 when δ = 1. system (1.1) is the generalized weighted p-fisher–kolmogoroff nonlinear stationary-state problem [21], where d is the diffusion coefficient, k is theis the linear reproduction rate and u is the population density. situations where d is space-dependent are arising in more and more modelling situations of biomedical importance from diffusionof genetically engineered organisms in heterogeneous environments to the effect of white and grey received: 12 sep 2021. key words and phrases. stability; weak solution; p-laplacian.1 https://adac.ee https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 2 matter in the growth and spread of brain tumours. problem (1.1) arises from the population biologyof one species.systems of type (1.1) have received considerable attention in the last decade (see, e.g., [18,19,24]and the references therein). it has been shown that for some certian values of ν,υ, system (1.1)has a rich mathematical structure. in [8, 23] the system (1.1) is considered under the hypothesis p (x) = (k/d) = 1,p = 2 and f (u) = u.this corresponds to the emden-fowler stationary-stateproblem of polytropic index of order one. while in [9, 19], system (1.1) is considered under thehypothesis p (x) = (k/d) = 1,p = 2 and f (u) = u − u2,where u is the population denistyof degree two.this corresponds to the logestic nonlinear stationary-state problem. due to theappearance of weighted p-laplacian operator in (1.1) and the particular cases; the extensions arechallenging and nontrivial.many authors are interested in the study of stability and instability of nonnegative solutions oflinear [2] , semilinear (see [10,26]), semiposiotne (see [3,25]), nonlinear (see [1,16]) and singular (see[17]) systems, due to the great number of applications in reaction-diffusion problems, in autocatalyticreaction, in temperature on plasma, population dynamics, etc.; see [4, 23] and references therein.also, in the recent past, many authors devoted their attention to study the weighted p-laplaciannonlinear systems (see [11, 12, 14, 15]).tertikas in [25] have been proved the stability and instability results of positive solutions for thesemilinear system −∆u = λf (u) in ω, bu = 0 on ∂ω, under various choices of the function f . in [3], the authors have been studied the uniqueness andstability of nonnegative solutions for classes of nonlinear elliptic dirichlet problems in a ball, whenthe nonlinearity is monotone, negative at the origin, and either concave or convex. in the case p (x) = a(x) = 1, p = 2 and a function λf (u) instead of λuα + uβ, system (1.1) have been studiedby several authors (see [5, 7, 20]).khafagy in [13] have been studied the stability and instability of positive weak solution for thenonlinear system −∆p,pu + a(x)|u|p−2u = λb(x)uα in ω, bu = 0 on ∂ω. } (1.2) where 0 < α < p − 1. he proved that if 0 < α < p − 1 and b(x) > 0(< 0) for all x ∈ ω, thenevery positive weak solution u of (1.2) is linearly stable (unstable) respectively. definition 1.1. we recall that, if u be any positive weak solution of (1.1), then the linearizedequation of (1.1) about u is given by −(p− 1)div[p (x)|∇u|p−2∇φ] − (k/d)a(x)[ν − 2υu]φ = µφ,x ∈ ω, bφ = 0, x ∈ ∂ω, } (1.3) where µ is the eigenvalue corresponding to the eigenfunction φ. https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 3 definition 1.2. [3] a solution u of (1.1) is called stable solution if all eigenvalues of (1.3) arestrictly positive, which can be implied if the principal eigenvalue µ1 > 0. otherwise u unstable. 2. main results the main goal of this section is to prove the stability and instability of the positive weak solution u of (1.1). our main results are formulate in the following theorems. theorem 2.1. if α + 1 < p < β + 1 and a(x) > 0 for all x ∈ ω, then every positive weak solution of ( 1.1) is linearly stable. proof. let u0 be any positive weak solution of (1.1), then the linearized equation bout u0 is −(p− 1)div[p (x)|∇u0|p−2∇φ] − (k/d)a(x)[ν − 2υu0]φ = µφ, x ∈ ω bφ = 0, x ∈ ∂ω. } (2.1) let µ1 be the first eigenvalue of (2.1) and let ψ(x) ≥ 0 be the corresponding eigenfunction.multiplying (1.1) by ψ and integrating over ω, we have − ∫ ω ψdiv[p (x)|∇u0|p−2∇u0]dx = (k/d) ∫ ω a(x)[νu0 −υu20 ]ψdx. (2.2) the first term of the l.h.s. of (2.2) may be written in the form∫ ω ψdiv[p (x)|∇u0|p−2∇u0]dx = ∫ ω ψ∇u0∇[p (x)|∇u0|p−2]dx + ∫ ω ψ[p (x)|∇u0|p−2]div(∇u0)dx. applying green’s first identity, we have∫ ω ψdiv[p (x)|∇u0|p−2∇u0]dx = ∫ ω ψ∇u0∇[p (x)|∇u0|p−2]dx − ∫ ω ∇[ψ(p (x)|∇u0|p−2)∇u0dx + ∫ ∂ω ψ[p (x)|∇u0|p−2] ∂u0 ∂n ds, = − ∫ ω ∇ψ[p (x)|∇u0|p−2]∇u0dx + ∫ ∂ω ψ[p (x)|∇u0|p−2] ∂u0 ∂n ds. (2.3) https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 4 from (2.3) in (2.2), we have (k/d) ∫ ω a(x)[νu0 −υu20 ]ψdx = ∫ ω ∇ψ[p (x)|∇u0|p−2]∇u0dx − ∫ ∂ω ψ[p (x)|∇u0|p−2] ∂u0 ∂n ds + ∫ ω a(x)ψ|u0|p−2u0]dx. (2.4) also, multiplying (2.1) by (−u0) and integrating over ω, we have −µ1 ∫ ω u0ψdx = (p− 1) ∫ ω u0div[p (x)|∇u0|p−2∇ψ]dx −(p− 1) ∫ ω u0a(x)|u0|p−2ψ +λ ∫ ω a(x)[ν − 2υu0]ψdx. (2.5) the first term of the l.h.s. of (2.5) may be written in the form ∫ ω u0div[p (x)|∇u0|p−2∇ψ]dx = ∫ ω u0[p (x)|∇u0|p−2]∇·∇ψdx + ∫ ω u0∇ψ∇[p (x)|∇u0|p−2]dx. using green’s first identity, one have ∫ ω u0div[p (x)|∇u0|p−2∇ψ]dx = − ∫ ω ∇[u0p (x)|∇u0|p−2]∇ψ + ∫ ω u0∇[p (x)|∇u0|p−2]∇ψdx + ∫ ∂ω u0[p (x)|∇u0|p−2] ∂ψ ∂n ds, = − ∫ ω [p (x)|∇u0|p−2]∇u0∇ψ + ∫ ∂ω u0[p (x)|∇u0|p−2] ∂ψ ∂n ds. (2.6) https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 5 from (2.6) in (2.5) we have −µ1 ∫ ω u0ψdx = (p− 1)[ ∫ ∂ω u0[p (x)|∇u0|p−2] ∂ψ ∂n ds − ∫ ω [p (x)|∇u0|p−2]∇u0∇ψ] +(k/d) ∫ ω a(x)[νu0 − 2υu20 ]ψdx. (2.7) multiplying (2.4) by (p− 1) and adding with (2.7), we have −µ1 ∫ ω u0ψdx = (p− 1)[ ∫ ∂ω u0[p (x)|∇u0|p−2] ∂ψ ∂n ds − ∫ ∂ω ψ[p (x)|∇u0|p−2] ∂u0 ∂n ds] +(k/d) ∫ ω a(x)[νu0 − 2υu20 ]ψdx −(p− 1)(k/d) ∫ ω a(x)[νu0 −υu20 ]ψdx. hence −µ1 ∫ ω u0ψdx = (p− 1) ∫ ∂ω [p (x)|∇u0|p−2][u0 ∂ψ ∂n −ψ ∂u0 ∂n ]ds +(k/d) ∫ ω a(x)νu0[1 − (p− 1)]ψdx +(k/d) ∫ ω a(x)υu20 [(p− 1) − 2]ψdx. (2.8) now, when δ = 1, we have bu0 = u0 = 0 for s ∈ ∂ω and also we have ψ = 0 for s ∈ ∂ω. then∫ ∂ω [p (x)|∇u0|p−2][u0 ∂ψ ∂n −ψ ∂u0 ∂n ]ds = 0. (2.9) also, when δ 6= 1, we have ∂u0 ∂n = − δhu0 1 −δ and ∂ψ ∂n = − δhψ 1 −δ , which implies again the result given by (2.9).hence −µ1 ∫ ω u0ψdx = (k/d) ∫ ω a(x)[νu0[2 −p] + υu20 [p− 3]]ψdx. (2.10) https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 6 since 2 < p < 3 and a(x) > 0 for all x, then (2.10) becomes −µ1 ∫ ω u0ψdx < 0, (2.11) so µ1 > 0 and the result follows. � theorem 2.2. if 2 < p < 3 and a(x) < 0 for all x ∈ ω, then every positive weak solution of (1.1) is unstable. proof. as in the proof of theorem 1., we have −µ1 ∫ ω u0ψdx > 0, (2.12) so µ1 < 0 and the result follows. � 3. applications and related results here we introduce some examples to demonstrate the effectiveness of our results. example 3.1. consider the emden-fowler steady-state problem of polytropic index of order one [8], −∆u = λa(x)u in ω, bu = 0 on ∂ω, } (3.1) with a(x) > 0 for all x ∈ ω.here p (x) = 1, (k/d) = λ,p = 2. then according to theorem 1., every positive weak solution of (3.1) is unstable. example 3.2. consider the population denisty steady-state problem of degree two [19], −∆pu = λa(x)[u −u2] in ω, bu = 0 on ∂ω, } (3.2) with a(x) > 0 for all x ∈ ω.hence, according to theorem 1., every positive weak solution of (3.1) is stable. example 3.3. consider the chemotaxis steady-state problem of degree two [9, 19], −∆pu = λa(x)[−u + u2] in ω, bu = 0 on ∂ω, } (3.3) with a(x) > 0 for all x ∈ ω.hence, according to theorem 1., every positive weak solution of (3.1) is unstable. https://doi.org/10.28924/ada/ma.2.8 eur. j. math. anal. 10.28924/ada/ma.2.8 7 references [1] g. afrouzi, s. rasouli, stability properties of non-negative solutions to a non-autonomous p-laplacian equation,chaos solitons fractals. 29 (2006) 1095-1099. https://doi.org/10.1016/j.chaos.2005.08.165.[2] g. afrouzi, z. sadeeghi, stability results for a class of elliptic problems, int. j. nonlinear sci. 6 (2008) 114-117. http://www.internonlinearscience.org/upload/papers/20110307063941556.pdf.[3] i. ali, a. castro, r, shivaji, uniqueness and stability of nonnegative solutions for semipositone problems in a ball,proc. amer. math. soc. 117 (1993) 775-782. 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z. zhitao, lectures on solution set of semilinear elliptic equations (in tokyo metropolitan university), 2005.[24] s. shabani, rokn-e-vafa and t. h. tehrani, diffusive logistic equations with harvesting and heterogeneity understrong growth rate, adv. nonlinear anal. 8 (2019) 455-467. https://doi.org/10.1515/anona-2016-0208.[25] a. tertikas, stability and instability of positive solutions of semilinear problems, proc. amer. math. soc. 114 (1992)1035-1040. https://doi.org/10.1090/s0002-9939-1992-1092928-2.[26] i. voros, stability properties of nonnegative solutions of semilinear symmetric cooperative systems, electronic j. diff.eqn. 105 (2004) 1-6. https://ejde.math.txstate.edu/volumes/2004/105/voros.pdf. https://doi.org/10.28924/ada/ma.2.8 https://doi.org/10.1090/s0002-9947-02-03005-2 https://doi.org/10.1515/anona-2016-0208 https://doi.org/10.1090/s0002-9939-1992-1092928-2 https://ejde.math.txstate.edu/volumes/2004/105/voros.pdf 1. introduction: 2. main results 3. applications and related results references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 12doi: 10.28924/ada/ma.2.12 some investigations on a class of analytic and univalent functions involving q-differentiation ayotunde olajide lasode∗ , timothy oloyede opoola department of mathematics, faculty of physical sciences, university of ilorin, ilorin, nigeria lasode_ayo@yahoo.com, opoola.to@unilorin.edu.ng ∗correspondence: lasode_ayo@yahoo.com abstract. we use the concept of q-differentiation to define a class eq(β,δ) of analytic and univalentfunctions. the investigations thereafter includes coefficient estimates, inclusion property and someconditions for membership of some analytic functions to be in the class eq(β,δ). our results generalizesome known and new ones. 1. introduction and definitions we let ud = {z : z ∈ c, |z| < 1} represent the unit disk and a represent the class ofnormalized analytic functions of the form f (z)= z + ∞∑ m=2 amz m, z ∈ud (1) where f (0) = 0 = f ′(0)−1. also, let s represent a subset of a containing functions univalentin ud. a function f in s is a member of class bt (δ) of bounded turning functions of order δ if itsatisfies the geometric condition ref ′(z) > δ ∈ [0,1), z ∈ud. let bt (0)=bt represent the class of bounded turning functions. it is known (see [1]) that f ∈btare univalent functions. also, a function f in s is a member of class cv(δ) of convex functions oforder δ if it satisfies the geometric condition re ( z f ′′(z) f ′(z) +1 ) > δ ∈ [0,1), z ∈ud. let cv(0)= cv represent the class of convex functions.the importance of operators in geometric function theory cannot be underrated. for instancesee [2, 13, 15] for some known ones.in 1908, jackson [7] (see also [3, 4, 8–11]) initiated the concept of q-calculus as follows. received: 21 jan 2022. key words and phrases. analytic functions; carathéodory functions; univalent functions; bounded turning function;coefficient bound; inclusion property and q-calculus. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.12 https://orcid.org/0000-0002-2657-7698 eur. j. math. anal. 10.28924/ada/ma.2.12 2 definition 1.1. for q ∈ (0,1), the q-differentiation of function f ∈a is defined by dqf (0)= f ′(0), dqf (z)= f (z)− f (qz) z(1−q) (z 6=0) and d2qf (z)=dq(dqf (z)). (2) obviously, applying (2) in (1) gives us dqf (z)=1+ ∞∑ m=2 [m]qamz m−1 and zd2qf (z)= ∞∑ m=2 [m−1]q[m]qamzm−1 (3) where [m]q = 1−qm1−q and lim q↑1 [m]q = m.for example if f (z)= zm, then by using (2), dqf (z)=dq(zm)= 1−qm 1−q zm−1 = [m]qz m−1 and observe that lim q↑1 dqf (z)= lim q↑1 ( [m]qz m−1) = mzm−1 = f ′(z) where f ′(z) is the classical differentiation.in this work, the q-differential operator was used to define a class of analytic functions andgeneralize some results. 2. relevant lemmas we represent by p the well-known class of analytic functions of the form p(z)=1+ ∞∑ m=1 cmz m, re p(z) > 0, z ∈ud (4) and by p(δ)⊆p(0)=p the class whose members are of the form pδ(z)=1+ ∞∑ m=1 (1−δ)cmzm, re p(z) > δ ∈ [0,1), z ∈ud. (5) the following lemmas shall be required to proof our results. lemma 2.1 ( [14]). let g(z) = ∞∑ m=1 amz m ≺ g(z) = ∞∑ m=1 bmz m, z ∈ ud where g(z) is univalent in ud and g(ud) is a convex domain, then |am| ≤ |b1|, m ∈ n. equality holds for the function g(z)= g(τzm), |τ|=1. the lemmas that follow are the q-analogous versions of the original ones as referenced. lemma 2.2 ( [6]). let p(z) be analytic in ud such that p(0)=1. if re ( zdq(p(z)) p(z) +1 ) > 3δ −1 2δ , z ∈ud, then for α =(δ −1)/δ (δ ∈ [1/2,1)), re p(z) > 2α. the constant 2α is the best possible. lemma 2.3 ( [5]). let u = u1+u2i and v = v1+v2i such that γ(u,v) :c2 −→c is a complex-valued function such that https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 3 (1) γ(u,v) is continuous in π ⊂c2,(2) (1,0)∈ π and re(γ(1,0)) > 0 and(3) re(γ(ξ+(1−ξ)u2i,v1))≤ ξ (0≤ ξ < 1)) if (ξ+(1−ξ)u2i,v1)∈ π and v1 ≤−12(1−ξ)(1+u 2 2) and re(γ(ξ+(1−ξ)u2i,v1))≥ ξ (ξ > 1) if (ξ+(1−ξ)u2i,v1)∈ π and v1 ≥ 12(1−ξ)(1+u 2 2). if p(z)∈p for (p(z),zdqp(z))∈ π and re(γ(p(z), zdqp(z))) > ξ, z ∈ud, then rep(z) > ξ in ud. 3. main results the definition of the investigated class is as follows.a function f (z)∈a is a member of the class eq(β,δ) if the condition re ( dqf (z)+ 1+eiβ 2 zd2qf (z) ) > δ, δ ∈ [0,1), β ∈ (−π,π], z ∈ud (6) holds.when parameters in (6) are varied, the class eq(β,δ) reduces to some well-known classes ofanalytic functions that have been studied by some authors. these are cited in our corollaries andremarks.the following are the proved results. theorem 3.1. let β ∈ (−π,π] and δ ∈ [0,1), if condition (6) holds, then eq(β,δ)⊂bt q(δ). bt q(δ) is the class of q-bounded turning function of order δ. proof. let p(z) = dqf (z) so that dqp(z) = d2qf (z) and for κ = (1+ eiβ)/2, then (6) can beexpressed as re(p(z)+κzdqp(z)) > δ. (7) in view of the conditions in lemma 2.3 and for p(z) in (7), we define the function γ(u,ν)= u +κν on the domain π of c2, then (i) clearly, γ(u,ν) satisfies the condition (1) in lemma 2.3,(ii) for (1,0)∈ π, γ(1,0)=1 =⇒ re(γ(1,0)) > 0 and(iii) γ(δ +(1−δ)u2i,ν1)= δ + 1+cosδ2 ν1 +((1−δ)u2 + sinδ2 ν1) i, thus, re(γ(δ +(1−δ)u2i,ν1))= δ + 1+cosβ 2 ν1 ≤ δ for ν1 ≤−12(1−δ)(1+u22). https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 4 now since γ(u,ν) satisfies all the conditions (1−3) in lemma 2.3, then it implies that rep(z)=re(dqf (z)) > δ, z ∈ud hence the proof is complete. � corollary 3.2 ( [1]). since class bt q(δ) is well-known to consist of univalent functions, then eq(β,δ)⊂bt q(δ) consists of univalent functions. corollary 3.3. lim q↑1 eq(β,δ)⊂bt (δ), z ∈ud. theorem 3.4. if f ∈a is such that re ( zdq(dqf (z)+κzd2qf (z)) dqf (z)+κzd2qf (z) ) > δ −1 2δ , (8) then re(dqf (z)+κzd2qf (z)) > 2 (δ−1)/δ, δ ∈ [1/2,1), z ∈ud and κ =(1+eiβ)/2. proof. from (6), let p(z)=dqf (z)+κzd2qf (z), then by logarithmic q-differentiation we obtain zdqp(z) p(z) +1= zdq(dqf (z)+κzd2qf (z)) dqf (z)+κzd2qf (z) +1. now applying lemma 2.2 gives re ( zdqp(z) p(z) +1 ) =re ( zdq(dqf (z)+κzd2qf (z)) dqf (z)+κzd2qf (z) +1 ) > 3δ −1 2δ implies that re ( zdq(dqf (z)+κzd2qf (z)) dqf (z)+κzd2qf (z) ) > δ −1 2δ and by the same lemma 2.2 the proof in complete. � corollary 3.5. if f ∈a satisfies condition (8), then f ∈eq(β,2(δ−1)/δ). corollary 3.6. if f ∈ lim q↑1 eq(β,1/2) is such that re ( z(1+κ)f ′′(z)+κz2f ′′′(z) f ′(z)+κzf ′′(z) ) > − 1 2 , then re(f ′(z)+κzf ′′(z)) > 1/2, z ∈ud. https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 5 corollary 3.7. if f ∈eq(π,1/2) is such that re ( zdq(dqf (z)) dqf (z) ) > − 1 2 , (9) then re(dqf (z)) > 1 2 . this means that if condition (9) holds, then f is a q-bounded turning function of order 1/2. now if q ↑ 1, then re ( zf ′′(z) f ′(z) ) > − 1 2 , (10) implies re(f ′(z)) > 1 2 z ∈ud. this means that if condition (10) holds, then f is a bounded turning function of order 1/2. corollary 3.8. if f ∈eq(0,1/2) is such that re ( zdq(dqf (z)+zd2qf (z)) dqf (z)+zd2qf (z) ) > − 1 2 , (11) then re(dqf (z)+zd2qf (z)) > 1 2 and if q ↑ 1, re ( 2zf ′′(z)+z2f ′′′(z) f ′(z)+zf ′′(z) ) > − 1 2 implies that re(f ′(z)+zf ′′(z)) > 1/2, z ∈ud. theorem 3.9. let β ∈ (−π,π] and δ ∈ [0,1), then the function f (z)= z +amz m ∈eq(β,δ), m = {2,3, . . .} (12) if |am| ≤ 2 [m]q { |xm|− ((2+[m−1]q)cosθ+[m−1]q cos(β +θ0)) } (13) where xm =2+[m−1]q(1+eiβ) |xm|= √ 2 { 2+[m−1]q(2+[m−1]q)(1+cosβ) } ≥ 2  (14) and θ0 attains minimum at θ0 = π +arctan ( −[m−1]q sinβ 2+[m−1]q(1+cosβ) ) . (15) https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 6 proof. firstly, applying (2) in (12) gives dqf (z)=1+[m]qamzm−1 zd2qf (z)= [m−1]q[m]qamzm−1 } . (16) note that it suffices to study the condition that for |z|=1,∣∣∣∣dqf (z)+ 1+eiβ2 zd2qf (z)−1 ∣∣∣∣ < re {dqf (z)+ 1+eiβ2 zd2qf (z) } (17) so that by putting (16) into (17) we obtain∣∣∣∣[m]qamzm−1 + 12[m−1]q[m]q(1+eiβ)amzm−1 ∣∣∣∣ < re { 1+[m]qamz m−1 + 1 2 [m−1]q[m]q(1+eiβ)amzm−1 } . now letting |am|= r , amzm−1 = reiθ and using (14) we obtain∣∣∣∣12[m]qreiθxm ∣∣∣∣ ≤re {1+[m]qreiθ + 12[m−1]q[m]q(1+eiβ)reiθ } (18) so that 1 2 [m]qr|xm| ≤ref (19) where f =1+[m]qreiθ + 1 2 [m−1]q[m]q(1+eiβ)reiθ in (18). further simplification gives f =1+[m]qr cosθ+ 1 2 [m−1]q[m]qr cosθ+ 1 2 [m−1]q[m]qr cos(β +θ)+im(f) so that ref =1+ 1 2 [m]qr{2cosθ+[m−1]q cosθ+[m−1]q cos(β +θ)}= ψ. (20) now (19) becomes 1 2 [m]qr|xm| ≤ 1+ 1 2 [m]qr{(2+[m−1]q)cosθ+[m−1]q cos(β +θ)} and by simplification we obtain (13).to know the values of θ where (20) attains minimum implies that ∂ψ ∂θ =− r[m]q 2 { (2+[m−1]q)sinθ+[m−1]q sin(β +θ) } implies that (2+[m−1]q)sinθ+[m−1]q sin(β +θ)=0 so that tanθ = −[m−1]q sinβ 2+[m−1]q(1+cosβ)which simplifies to (15). � https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 7 corollary 3.10. let f (z)= z +amzm ∈eq(0,δ) and m = {2,3, . . .}, then |am| ≤ 1 [m]q {√ 1+2[m−1]q +[m−1]2q +1+[m−1]q } and if q ↑ 1, then |am| ≤ 1 2m2 . corollary 3.11. let f (z)= z +amzm ∈eq(π,δ) and m = {2,3, . . .}, then |am|5 1 2[m]q and if q ↑ 1, then |am| ≤ 1 2m . remark 3.12. let q ↑ 1, then theorem 3.9 becomes the result in [18]. theorem 3.13 (coefficient estimates). let β ∈ (−π,π], δ ∈ [0,1) and let g(z) = 1+ b1z + b2z 2 + · · · ∈ cv(δ). if f ∈a belongs to eq(β,δ), then |am| ≤ 2(1−δ)|b1| [m]q|xm| , m = {2,3, . . .} (21) where |xm| is defined in (14). proof. let f (z)∈eq(β,δ), therefore from (6) and using (5), dqf (z)+ 1+eiβ 2 zd2qf (z)= δ +(1−δ)p(z), z ∈ud. (22) now putting (3) and (4) into (22) and simplifying gives 1+ ∞∑ m=2 { 1+[m−1]q ( 1+eiβ 2 )} [m]qamz m−1 =1+ ∞∑ m=2 (1−δ)cm−1zm−1 which implies that {2+[m−1]q(1+eiβ)} [m]q 2 am =(1−δ)cm−1, m = {2,3, . . .} where by applying (14) we obtain xm [m]q 2(1−δ) am = cm−1, m = {2,3, . . .}. (23) since g(ud) is a convex domain, then from lemma 2.1, (23) becomes∣∣∣∣xm [m]q2(1−δ)am ∣∣∣∣ = |cm−1| ≤ |b1| and simplifying further we obtain (21). � https://doi.org/10.28924/ada/ma.2.12 eur. j. math. anal. 10.28924/ada/ma.2.12 8 corollary 3.14. let f (z)∈eq(0,δ), then |am| ≤ (1−δ)|b1|√ 1+2[m−1]q +[m−1]2q and if q ↑ 1, then |am| ≤ (1−δ)|b1| m , m = {2,3, . . .}. corollary 3.15. let f ∈eq(π,δ), then |am| ≤ (1−δ)|b1| [m]q and if q ↑ 1, then |am| ≤ (1−δ)|b1| m , m = {2,3, . . .} remark 3.16. let p(z)∈p and φ(z)=1+ 2 π2 ( ln 1+ √ z 1− √ z )2. if q ↑ 1, (1) β = π and g(z)= p(z), then theorem 3.13 becomes the result in [12].(2) and g(z)= p(z), then theorem 3.13 becomes the result in [16].(3) and g(z)= φ(z), then theorem 3.13 becomes the result in [18].(4) and β =0, then theorem 3.13 becomes the result in [17]. acknowledgment. the authors would like to thank the referees for their careful reading of thismanuscript and their valuable suggestions. references [1] j.w. alexander, functions 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https://doi.org/10.1007/bfb0066543 https://www.jams.jp/notice/mj/50-1.html https://doi.org/10.2298/fil1614743s https://doi.org/10.1515/dema-2005-0106 https://doi.org/10.1515/dema-2005-0106 1. introduction and definitions 2. relevant lemmas 3. main results references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 6doi: 10.28924/ada/ma.2.6 solving equilibrium problem and fixed point problem by normal s-iteration process in hilbert space shamshad husain, mohd asad∗ department of applied mathematics, faculty of engineering and technology, aligarh muslim university, aligarh, india s_husain68@yahoo.com, masad19932015@gmail.com ∗correspondence: masad19932015@gmail.com abstract. the main purpose of this paper is to find a common element in the solution set of equilibriumproblem and fixed point problem of non-expansive mappings in the real hilbert space with the helpof normal s-iteration process. also, under some acceptable assumptions, we prove the sequencesinduced by above stated process converge weakly to a point in the solution set of above statedproblems. at the end, we give a numerical example to justify our work. the results studied in thiswork philosophize and boost some contemporary and known results in this direction. 1. introduction and auxiliary results everywhere in this paper except stated otherwise, let h be a real hilbert space equipped withinner product 〈·, ·〉 and induced norm ‖ · ‖. let c be a non-empty closed and convex subset of h. we denote strong and weak convergence of a sequence {xn} ∈ h by the symbols → and ⇀respectively.let t : c → h be a nonexpansive mapping. the so called fixed point problem for mapping t is tofind an element p ∈ c such that tp = p. (1) denote the set of solution of the problem (1) by fix(t ) = {p ∈ c : tp = p}. t is said to benonexpansive iff ‖tp−tq‖2 ≤‖p−q‖2, ∀p,q ∈ c. received: 10 dec 2021. key words and phrases. equilibrium problem; normal s-iteration; fixed point problem; hilbert space; non-expansivemapping. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 2 in 2011, d.r. sahu [4] studied problem (1) and proposed an iterative method known as normals-iteration process which is defied as follows: let x1 ∈ c be chosen arbitrarily, yn = (1 −αn)xn + αntxn, xn+1 = tyn, ∀n ≥ 1, (2) where {αn} ⊂ (0 , 1). under some acceptable conditions of {αn}, sahu proved that the sequence {xn} induced by the algorithm (2) converges weakly to an element of solution set of problem (1).the performance of normal s-iteration process is much better than mann and picard iterationprocess for nonexpansive mappings(see [4], [5]).elsewhere, let f : c ×c → r be a bifunction such that for all p ∈ c, f (p,p) = 0. then the socalled equilibrium problem is to find p ∈ c such that f (p,q) ≥ 0, ∀q ∈ c. (3) denote the solution set of problem (3) by ep (f ). problem (3) contains nash equilibrium prob-lems, fixed point problems, variational inequality problems, minimization problems and optimizationproblems as its special cases(see [7, 16]).in this paper, we consider a problem which is formulated as follows: find p ∈ c, such that p ∈ ω := fix(t ) ∩ep (f ). (4) in past few years, many researchers have found a common solution of problem (4) by varioustechniques(see [4], [3], [2], [12], [1], [14], [10]). impelled and inspired by these approaches, the mainobjective of this paper is to find a common element in the solution set of problem (4) with the helpof normal s-iteration process in the framework of real hilbert space. also we prove some weakconvergence theorem under some acceptable conditions.now we define some basic auxiliary results which are very helpful throughout this work.the metric projection pc from h into c is defined as: for any p ∈ c, ‖p−pc(p)‖≤‖p−q‖, ∀q ∈ c. it is to be noted that the metric projection is nonexpansive. further for any p ∈ h and s ∈ c, s = pc(p) ⇐⇒ 〈p− s,s −q〉≥ 0, ∀q ∈ c. a mapping t is said to be monotone iff for all p,q ∈ h 〈tp−tq,p−q〉≥ 0. lemma 1.1. [9] let h be a hilbert space. then for all p,q ∈ h and α ∈ [0, 1] the followings hold: (i) ‖p−q‖2 = ‖p‖2 −‖q‖2 − 2〈p−q,q〉;(ii) ‖p + q‖2 ≤‖p‖2 + 2〈q,p + q〉;(iii) ‖αp + (1 −α)q‖2 = α‖p‖2 + (1 −α)‖q‖2 −α(1 −α)‖p−q‖2. https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 3 assumption 1.1. [6] let f : c ×c →r be a bi-function satisfying the subsequent conditions:(i) f (p,p) ≥ 0, ∀p ∈ c;(ii) f is monotone, i.e. f (p,q) + f (q,p) ≤ 0, ∀p,q ∈ c;(iii) f is upper semi continuous, i.e. for each p,q,s ∈ c, lim t→0 supf (λs + (1 −λ)p,q) ≤ f (p,q); (5) (iv) for each fixed p ∈ c, the function q 7→ f (p,q) is convex and lower semi continuous; lemma 1.2. [7] assume that the bi-function f : c ×c → r satisfy the conditions of assumption1.1. then for fixed r > 0 and p ∈ h, there exists s ∈ c such that f (q,p) + 1 r 〈q −p,p− s〉≥ 0, ∀q ∈ c. (6) lemma 1.3. [12] assume that the bi-function f : c ×c →r satisfy the conditions of assumption1.1. if for r > 0 and p ∈ h, defined a mapping tfr : h → c as follows: tfr (p) = { s ∈ c : f (s,q) + 1 r 〈q − s,s −p〉≥ 0, ∀q ∈ c } . (7) then the followings hold: (i) tfr is non-empty and single valued.(ii) tfr is firmly non-expansive, i.e., ‖tfr (p) −t f r (q)‖ 2 ≤〈tfr (p) −t f r (q),p−q〉 ∀p,q ∈ h. (iii) fix(tfr ) = ep(f ).(iv) ep(f ) is closed and convex. lemma 1.4. [11] let {an} be a sequence of non negetive real numbers such that an+1 ≤ (1 −αn)an + αnδn + γn, ∀n ≥ 0, where αn ∈ (0, 1) and δn ⊂r satisfies the following conditions:(i) ∑∞n=0αn = ∞;(ii) lim n→∞ supδn ≤ 0.(iii) γn ≥ 0 (n ≥ 1),∑γn < ∞.then lim n→∞ an = 0. lemma 1.5. [13] let c be a closed and convex subset of h and t : c → c be a non-expansivemapping. then(i) fix(t ) is a closed and convex subset of c;(ii) i −t is demiclosed at 0. https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 4 lemma 1.6. [8] let f : c ×c →r be a non linear bi-function satisfying the assumption 1.1 andlet tfr be defined as above in lemma 1.3. if for r > 0, let p,q ∈ h and r1, r2 > 0, then ‖tfr2 (q) −t f r1 (p)‖≤‖q −p‖ + ∣∣∣∣r2 − r1r2 ∣∣∣∣‖tfr2 (q) −q‖. lemma 1.7. [15] let xn and yn be two bounded sequences in a banach space x and let βn be asequence in [0, 1] which satisfy the following conditions: 0 < lim n→∞ inf βn ≤ lim n→∞ sup βn < 1. suppose xn+1 = (1 −βn)zn + βnxn for all integers n ≥ 0, and lim n→∞ sup (‖zn+1 − zn‖−‖xn+1 − xn‖) ≤ 0, then lim n→∞ ‖xn − zn‖ = 0. 2. main result in this section we study and analyze normal s-iteration process for solving equilibrium problemand fixed point problem for nonexpansive mapping and its convergence analysis. theorem 2.1. let c ⊂ h be a nonempty closed and convex subsets of h. let f : c ×c → r bea nonlinear bifunction satisfying assumption 1.1. let t : c → h be a nonexpansive mapping suchthat fix(t ) 6= . assume that ω := fix(t ) ∩ep (f ) 6= . let {xn}be a sequence defined as follows:choose x1 ∈ h arbitrarily, yn = t f rn (xn), zn = (1 −αn)yn + αntyn, xn+1 = tzn, ∀n ≥ 1, (8) where {αn}⊂ [0 , 1] and {rn}⊂ (0 ,∞) satisfying the following conditions: c1: lim n→∞ αn = 0, ∑∞ n=1αn(1 −αn) = ∞, ∑∞ n=1 |αn −αn−1| < ∞; c2: lim n→∞ inf rn > 0, ∑∞ n=0 |rn+1 − rn| < ∞;then the sequence {xn} induced by process (8) converges weakly to an element in ω. proof. take p ∈ ω. then by process (8), we obtain ‖xn+1 −p‖ = ‖tzn −p‖≤‖zn −p‖, ≤‖(1 −αn)yn + αntyn −p‖, ≤ (1 −αn)‖yn −p‖ + αn‖tyn −p‖, ≤ (1 −αn)‖yn −p‖ + αn‖yn −p‖, ≤‖yn −p‖≤‖tfrn (xn) −p‖, ≤‖xn −p‖. https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 5 by using mathematical induction, we have ‖xn+1 −p‖≤‖xn −p‖≤‖x1 −p‖, ∀n ≥ 1. hence the sequence {xn} is bounded and so are the sequences {yn},{zn},{tyn} and {tzn} arealso bounded.let m = supn≥0{‖yn −xn‖ + ‖xn −q‖2 + ‖tyn‖ + ‖tzn‖}.since yn = tfrn (xn) and yn−1 = tfrn−1(xn−1), then we obtain f (yn,q) + 1 rn 〈q −yn,yn −xn〉≥ 0, ∀q ∈ c, (9) f (yn−1,q) + 1 rn−1 〈q −yn−1,yn−1 −xn−1〉≥ 0, ∀q ∈ c. (10) replace q by yn in (10) and q by qn−1 in (9) and adding them with the assumption 1.1(ii),we obtain 〈yn −yn−1, yn−1 −xn−1 rn−1 − yn −xn rn 〉≥ 0, and hence 〈yn −yn−1,yn−1 −yn −xn−1 − rn−1 rn (yn −xn)〉≥ 0. this implies that by using lemma 1.6 ‖yn −yn−1‖2 ≤〈yn −yn−1,xn −xn−1 + ( 1 − rn−1 rn ) (yn −xn)〉, ≤‖yn −yn−1‖ { ‖xn −xn−1‖ + ∣∣∣∣rn − rn−1rn ∣∣∣∣‖yn −xn‖}, ‖yn −yn−1‖≤‖xn −xn−1‖ + ∣∣∣∣rn − rn−1rn ∣∣∣∣‖yn −xn‖, from process (8)(c2), we have lim n→∞ inf rn > 0. therefore there exists r > 0 such that rn > r forlarge enough n ∈n. then for n ≥ 1, ‖yn −yn−1‖≤‖xn −xn−1‖ + 1 r |rn − rn−1|m. (11) consider ‖xn+1 −xn‖ = ‖tzn −tzn−1‖≤‖zn −zn−1‖, ≤‖(1 −αn)yn + αntyn − (1 −αn−1)yn−1 −αn−1tyn−1‖, ≤‖(1 −αn)yn − (1 −αn)yn−1 + (1 −αn)yn−1 − (1 −αn−1)yn−1 + αntyn −αntyn−1 + −αntyn−1 −αn−1tyn−1‖, ≤ (1 −αn)‖yn −yn−1‖ + 2|αn −αn−1|m + αn‖yn −yn−1‖, ≤‖yn −yn−1‖ + 2|αn −αn−1|m. (12) https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 6 using (11) and (12), we obtain ‖xn+1 −xn‖≤‖xn −xn−1‖ + 1 r |rn − rn−1|m + 2|αn −αn−1|m. (13) by applying lemma 1.4, we obtain lim n→∞ ‖xn+1 −xn‖ = 0. (14) by using process (8)(c1)(c2) along with lemma 1.7 and (13), we obtain lim n→∞ ‖xn −zn‖ = 0. (15) furthermore, for any p ∈ ω, we have from process (8) ‖yn −p‖2 = ‖tfrn (xn) −p‖ 2, ≤〈tfrn (xn) −t f rn (p),xn −p〉, ≤〈yn −p,xn −p〉, ≤ 1 2 { ‖yn −p‖2 + ‖xn −p‖2 −‖xn −yn‖2 } , ≤‖xn −p‖2 −‖xn −yn‖2. (16) from convaxity of function x 7→ ‖x‖2 and (16), we obtain ‖xn+1 −p‖2 = ‖tzn −p‖2, ≤‖zn −p‖2, ≤‖(1 −αn)yn + αntyn −p‖2, ≤ (1 −αn)‖yn −p‖2 + αn‖tyn −p‖2, ≤‖yn −p‖2, ≤‖xn −p‖2 −‖xn −yn‖2. and so, ‖xn −yn‖2 ≤‖xn −p‖2 −‖xn+1 −p‖2, ≤ (‖xn −p‖−‖xn+1 −p‖)(‖xn −p‖ + ‖xn+1 −p‖), ≤‖xn −xn+1‖(‖xn −p‖ + ‖xn+1 −p‖). since the sequence {xn} is bounded and lim n→∞ ‖xn+1 −xn‖ = 0. we have lim n→∞ ‖xn −yn‖ = 0. (17) https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 7 further, ‖xn+1 −p‖2 = ‖tzn −p‖2, ≤‖zn −p‖2, ≤‖(1 −αn)yn + αntyn −p‖2, ≤ (1 −αn)‖yn −p‖2 + αn‖tyn −p‖2 −αn(1 −αn)‖yn −tyn‖2, ≤‖yn −p‖2 −αn(1 −αn)‖yn −tyn‖2, ≤‖xn −p‖2 −‖xn −yn‖2 −αn(1 −αn)‖yn −tyn‖2, and so, αn(1 −αn)‖yn −tyn‖2 ≤‖xn −p‖2 −‖xn+1 −p‖2 −‖xn −yn‖2, ≤‖xn −xn+1‖(‖xn −p‖ + ‖xn+1 −p‖) −‖xn −yn‖2, using process (8)(c1), (14) and (17), we obtain lim n→∞ ‖yn −tyn‖ = 0. (18) consider ‖zn −tzn‖≤‖zn −yn‖ + ‖yn −tyn‖ + ‖tyn −tzn‖, (19) ≤‖zn −yn‖ + ‖yn −tyn‖ + ‖yn −zn‖. (20) by using (15) and (18), we obtain lim n→∞ ‖zn −tzn‖ = 0. (21) since {xn} is bounded. there exists a subsequence {xni} ⊂ {xn} such that xn ⇀ p̂. since lim n→∞ ‖xn −yn‖ = 0 and {yn} is bounded, this implies that yni ⇀ p̂ ∈ c. now by (18) we have ‖tyni −yni‖→ 0. (22) from (22) and lemma 1.5, we conclude that p̂ ∈ fix(t ).next we prove that p̂ ∈ ep (f ). since yn = tfrn (xn), we have f (yn,q) + 1 rn 〈q −yn,yn −xn〉≥ 0, ∀q ∈ c.by using assumption 1.1(ii), we obtain 1 rn 〈q −yn,yn −xn〉≥ f (q,yn), and so, 〈q −yni, yni −xni rni 〉≥ f (q,yni ). (23) https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 8 since ‖yni−xni‖ rni ≤ ‖yni−xni‖ r → 0 and yni ⇀ p̂, therefore by assumption 1.1(iv), we obtain lim ni→∞ inf f (q,yni ) ≤ lim ni→∞ 〈q −yni, yni −xni rni 〉 = 0. that is, f (q, p̂) ≤ 0, ∀q ∈ c. (24) further for any λ ∈ (0 , 1) and q ∈ c, let qλ = λq + (1 − λ)p̂, then qλ ∈ c and so we have f (qλ, p̂) ≤ 0. it follows from the assumption 1.1 and (24), that 0 = f (qλ,qλ), ≤ λf (qλ,q) + (1 −λ)f (qλ, p̂), ≤ λf (qλ,q). this implies that f (qλ,q) ≥ 0, ∀λ ∈ (0 , 1). letting λ → 0+ by assumption 1.1, we have f (p̂,q) ≥ 0, ∀q ∈ c. this implies that p̂ ∈ ep (f ) and hence p̂ ∈ ω. this completes theproof. � 3. numerical example here we give numerical examples for supporting our main results. all codes are done by matlab2021a. example 3.1. set h = r. let c = [0 + ∞). suppose t : c → h, is defined by t (p) = p 3 . it can be easily seen that, here fix(t ) = {0}. also, we define f (s,q) = 3q2 + 2sq − 5s2, it is easy to check that f satisfy the conditions of assumption 1.1. so, for rn = r > 0, tfr (p) is non-empty and single-valued for each p ∈ c. hence for r > 0, there exists s ∈ c such that f (s,q) + 1 r 〈q − s,s −p〉≥ 0 ∀ q ∈ c, which is equivalent to 3rq2 + (s −p + 2rs)q + (ps − 5rs2 − s2) ≥ 0, ∀ q ∈ c. after solving the above inequality, we get s = p 1+8r for each r > 0 i.e. tfr (p) = p 1+8r for each r > 0. it can be easily seen that here ep (f ) = {0}. this implies that ω := fix(t )∩ep (f ) = {0}. now, let us choose r = 1 8 , and {αn} = 1(n+6). {αn} satisfy the conditions of main result. table. for different initial value, we present a table of iterations here. https://doi.org/10.28924/ada/ma.2.6 eur. j. math. anal. 10.28924/ada/ma.2.6 9 no. of iterations x0 = 1 x0 = −1 1 1.000000 -1.0000002 0.150794 -0.1419233 0.023038 -0.0204074 0.003555 -0.0029645 0.000553 -0.0004346 0.000087 -0.0000647 0.000014 -0.0000098 0.000002 -0.0000019 0.000000 0.000000 0 2 4 6 8 10 12 14 16 18 20 number of iterations -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x n x 0 = 1 x 0 = -1 figure 1. graphical representation of sequence {xn} for different choices of initialvalue x0. references [1] a. moudafi, m. théra, proximal and dynamical approaches to equilibrium problems, in: m. théra, r. tichatschke(eds.), ill-posed variational problems and regularization techniques, springer berlin heidelberg, berlin, heidel-berg, 1999: pp. 187–201. https://doi.org/10.1007/978-3-642-45780-7_12.[2] a. moudafi, viscosity approximation methods for fixed-points problems, j. math. anal. appl. 241 (2000) 46–55. https://doi.org/10.1006/jmaa.1999.6615. https://doi.org/10.28924/ada/ma.2.6 https://doi.org/10.1007/978-3-642-45780-7_12 https://doi.org/10.1006/jmaa.1999.6615 eur. j. math. anal. 10.28924/ada/ma.2.6 10 [3] s.d. flam, a.s. antipin, equilibrium programming using proximal-like algorithms, math. program. 78 (1996) 29–41. https://doi.org/10.1007/bf02614504.[4] d. r. sahu, applications of the s-iteration process to constrained minimization problems and split feasibilityproblems, fixed point theory, 12 (2011) 187–204.[5] d.r. sahu, a. pitea, m. verma, a new iteration technique for nonlinear operators as concerns convex programmingand feasibility problems, numer. algor. 83 (2020) 421–449. 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279–291. https://doi.org/10.1016/j.jmaa.2004.04.059.[11] h.-k. xu, iterative algorithms for nonlinear operators, j. lond. math. soc. 66 (2002) 240–256. https://doi.org/ 10.1112/s0024610702003332.[12] p.l. combettes, s.a. hirstoaga, equilibrium programming in hilbert spaces, j. nonlinear convex anal. 6 (2005)117–136.[13] q. zhang, c. cheng, strong convergence theorem for a family of lipschitz pseudocontractive mappings in a hilbertspace, math. computer model. 48 (2008) 480–485. https://doi.org/10.1016/j.mcm.2007.09.014.[14] s. takahashi, w. takahashi, viscosity approximation methods for equilibrium problems and fixed point problems inhilbert spaces, j. math. anal. appl. 331 (2007) 506–515. https://doi.org/10.1016/j.jmaa.2006.08.036.[15] t. suzuki, strong convergence of krasnoselskii and mann’s type sequences for one-parameter nonexpansive semi-groups without bochner integrals, j. math. anal. appl. 305 (2005) 227–239. https://doi.org/10.1016/j.jmaa. 2004.11.017.[16] s. husain, n. singh, ∆-convergence for proximal point algorithm and fixed point problem in cat(0) spaces, fixedpoint theory appl. 2019 (2019), 8. https://doi.org/10.1186/s13663-019-0658-3. https://doi.org/10.28924/ada/ma.2.6 https://doi.org/10.1007/bf02614504 https://doi.org/10.1007/s11075-019-00688-9 https://doi.org/10.1155/2012/843486 https://doi.org/10.1155/2012/843486 https://doi.org/10.1007/978-3-319-48311-5 https://doi.org/10.1007/978-3-319-48311-5 https://doi.org/10.1016/j.jmaa.2004.04.059 https://doi.org/10.1112/s0024610702003332 https://doi.org/10.1112/s0024610702003332 https://doi.org/10.1016/j.mcm.2007.09.014 https://doi.org/10.1016/j.jmaa.2006.08.036 https://doi.org/10.1016/j.jmaa.2004.11.017 https://doi.org/10.1016/j.jmaa.2004.11.017 https://doi.org/10.1186/s13663-019-0658-3 1. introduction and auxiliary results 2. main result 3. numerical example references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 4doi: 10.28924/ada/ma.2.4 ∗-k-operator frame for hom∗a(x) mohamed rossafi1,∗, roumaissae el jazzar2 and ali kacha2 1lasma laboratory department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, b. p. 1796 fes atlas, morocco mohamed.rossafi@usmba.ac.ma 2laboratory of partial differential equations, spectral algebra and geometry department of mathematics, faculty of sciences, university ibn tofail, kenitra, morocco roumaissae.eljazzar@uit.ac.ma, ali.kacha@yahoo.fr ∗correspondence: rossafimohamed@gmail.com abstract. in this work, we introduce the concept of ∗-k-operator frames in hilbert pro-c∗-modules,which is a generalization of k-operator frame. we present the analysis operator, the synthesisoperator and the frame operator. we also give some properties and we study the tensor product of ∗-k-operator frame for hilbert pro-c∗-modules. 1. introduction duffin and schaeffer introduced the notion of frame in nonharmonic fourier analysis in 1952 [3].in 1986 the work of duffin and schaeffer were reintroduced and developed by grossman andmeyer [7]. the concept of frame on hilbert space has already been successfully extended to proc∗-algebras and hilbert modules. many properties of frames in hilbert c∗-modules are valid forframes of multipliers in hilbert modules over pro-c∗-algebras [9].operator frames for b(h) is a new notion of frames that li and cio introduced in [11] andgeneralized by rossafi in [16]. in this work we introduce the notion of ∗-k-operator frame for thespace hom∗a(x) of all adjointable operators on a hilbert pro-c∗-module for x .this paper is divided into three sections. in section 2 we recall some fundamental definitionsand notations of hilbert pro-c∗-modules. in section 3 we introduce the ∗-k-operator frame andwe give some of its properties. lastly we investigate tensor product of hilbert pro-c∗-modules, weshow that tensor product of ∗-k-operator frames for hilbert pro-c∗-modules x and y, present an ∗-k-operator frames for x ⊗y, and tensor product of their frame operators is the frame operatorof their tensor product of ∗-k-operator frames. received: 18 nov 2021. key words and phrases. frame; ∗-k-operator frame; k-operator frame pro-c∗-algebra; hilbert pro-c∗-modules; tensorproduct. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 2 2. preliminaries the basic information about pro-c∗-algebras can be found in the works [4–6, 8, 12, 14, 15]. c∗-algebra whose topology is induced by a family of continuous c∗-seminorms instead of a c∗-norm is called pro-c∗-algebra. hilbert pro-c∗-modules are generalizations of hilbert spacesby allowing the inner product to take values in a pro-c∗-algebra rather than in the field of complexnumbers.pro-c∗-algebra is defined as a complete hausdorff complex topological ∗-algebra a whosetopology is determined by its continuous c∗-seminorms in the sens that a net {aα} converges to 0if and only if p(aα) converges to 0 for all continuous c∗-seminorm p on a [8, 10, 15], and we have:1) p(ab) ≤ p(a)p(b)2) p(a∗a) = p(a)2 for all a,b ∈aif the topology of pro-c∗-algebra is determined by only countably many c∗-seminorms, then it iscalled a σ-c∗-algebra.we denote by sp(a) the spectrum of a such that: sp(a) = {λ ∈c : λ1a −a is not invertible } forall a ∈a. where a is unital pro-c∗-algebra with unite 1a.the set of all continuous c∗-seminorms on a is denoted by s(a). if a+ denotes the set of allpositive elements of a, then a+ is a closed convex c∗-seminorms on a. example 2.1. every c∗-algebra is a pro-c∗-algebra. proposition 2.2. [8] let a be a unital pro-c∗-algebra with an identity 1a. then for any p ∈ s(a), we have: (1) p(a) = p(a∗) for all a ∈ a(2) p (1a) = 1(3) if a,b ∈a+ and a ≤ b, then p(a) ≤ p(b)(4) if 1a ≤ b, then b is invertible and b−1 ≤ 1a(5) if a,b ∈a+ are invertible and 0 ≤ a ≤ b, then 0 ≤ b−1 ≤ a−1(6) if a,b,c ∈a and a ≤ b then c∗ac ≤ c∗bc(7) if a,b ∈a+ and a2 ≤ b2, then 0 ≤ a ≤ b definition 2.3. [15] a pre-hilbert module over pro-c∗-algebra a, is a complex vector space ewhich is also a left a-module compatible with the complex algebra structure, equipped with an a-valued inner product 〈., .〉 e×e →a which is c-and a-linear in its first variable and satisfiesthe following conditions: 1) 〈ξ,η〉∗ = 〈η,ξ〉 for every ξ,η ∈ e2) 〈ξ,ξ〉≥ 0 for every ξ ∈ e https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 3 3) 〈ξ,ξ〉 = 0 if and only if ξ = 0 for every ξ,η ∈ e. we say e is a hilbert a-module (or hilbert pro-c∗-module over a ). if e iscomplete with respect to the topology determined by the family of seminorms p̄e(ξ) = √ p(〈ξ,ξ〉) ξ ∈ e,p ∈ s(a) let a be a pro-c∗-algebra and let x and y be hilbert a-modules and assume that i and j becountable index sets. a bounded a-module map from x to y is called an operators from x to y.we denote the set of all operator from x to y by homa(x ,y). definition 2.4. [1] an a-module map t : x −→y is adjointable if there is a map t∗ : y −→xsuch that 〈tξ,η〉 = 〈ξ,t∗η〉 for all ξ ∈x ,η ∈y, and is called bounded if for all p ∈ s(a), thereis mp > 0 such that p̄y(tξ) ≤ mpp̄x (ξ) for all ξ ∈x .we denote by hom∗a(x ,y), the set of all adjointable operator from x to y and hom∗a(x) = hom∗a(x ,x) definition 2.5. [1] let a be a pro-c∗-algebra and x ,y be two hilbert a-modules. the operator t : x →y is called uniformly bounded below, if there exists c > 0 such that for each p ∈ s(a), p̄y(tξ) 6 cp̄x (ξ), for all ξ ∈x and is called uniformly bounded above if there exists c′ > 0 such that for each p ∈ s(a), p̄y(tξ) > c ′p̄x (ξ), for all ξ ∈x ‖t‖∞ = inf{m : m is an upper bound for t} p̂y(t ) = sup{p̄y(t (x)) : ξ ∈x , p̄x (ξ) 6 1} it’s clear to see that, p̂(t ) 6 ‖t‖∞ for all p ∈ s(a). proposition 2.6. [2]. let x be a hilbert module over pro-c∗-algebra a and t be an invertible element in hom∗a(x) such that both are uniformly bounded. then for each ξ ∈x ,∥∥t−1∥∥−2∞ 〈ξ,ξ〉≤ 〈tξ,tξ〉≤ ‖t‖2∞〈ξ,ξ〉. 3. ∗-k-operator frame for hom∗a(x) we begin this section with the definition of a k-operator frame. definition 3.1. let {ti}i∈i be a family of adjointable operators on a hilbert a-module x over aunital pro-c∗-algebra, and let k ∈ hom∗a(x). {ti}i∈i is called a k-operator frame for hom∗a(x),if there exist two positive constants a,b > 0 such that a〈k∗ξ,k∗ξ〉≤ ∑ i∈i 〈tiξ,tiξ〉≤ b〈ξ,ξ〉,∀ξ ∈x . (3.1) https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 4 the numbers a and b are called lower and upper bound of the k-operator frame, respectively. if a〈k∗ξ,k∗ξ〉 = ∑ i∈i 〈tiξ,tiξ〉, the k-operator frame is an a-tight. if a = 1, it is called a normalized tight k-operator frame or aparseval k-operator frame. we will now move to define the ∗-k-operator frame for hom∗a(x). definition 3.2. let {ti}i∈i be a family of adjointable operators on a hilbert a-module x overa unital pro-c∗-algebra, and let k ∈ hom∗a(x). {ti}i∈i is called a ∗-k-operator frame for hom∗a(h), if there exists two nonzero elements a and b in a such that a〈k∗ξ,k∗ξ〉a∗ ≤ ∑ i∈i 〈tiξ,tiξ〉≤ b〈ξ,ξ〉b∗,∀ξ ∈x . (3.2) the elements a and b are called lower and upper bounds of the ∗-k-operator frame, respectively.if a〈k∗ξ,k∗ξ〉∗ = ∑ i∈i 〈tiξ,tiξ〉, the ∗-k-operator frame is an a-tight. if a = 1, it is called a normalized tight ∗-k-operator frameor a parseval ∗-k-operator frame. example 3.3. let l∞ be the set of all bounded complex-valued sequences. for any u = {uj}j∈n,v = {vj}j∈n ∈ l∞, we define uv = {ujvj}j∈n,u∗ = {ūj}j∈n,‖u‖ = sup j∈n |uj|. then a = {l∞,‖.‖} is a c∗-algebra. then a is pro-c∗-algebra.let x = c0 be the set of all null sequences. for any u,v ∈x we define 〈u,v〉 = uv∗ = {ujūj}j∈n. therefore x is a hilbert a-module.define fj = {f ji }i∈n∗ by f ji = 12 + 1i if i = j and f ji = 0 if i 6= j ∀j ∈n∗.now define the adjointable operator tj : x →x , tj{(ξi )i} = (ξif ji )i .then for every x ∈x we have∑ j∈n 〈tjξ,tjξ〉 = { 1 2 + 1 i }i∈n∗〈ξ,ξ〉{ 1 2 + 1 i }i∈n∗. so {tj}j is a {12 + 1i }i∈n∗-tight ∗-operator frame.let k : h→h defined by kξ = {ξi i }i∈n∗.then for every ξ ∈x we have 〈k∗ξ,k∗ξ〉≤ ∑ j∈n 〈tjξ,tjξ〉 = { 1 2 + 1 i }i∈n∗〈ξ,ξ〉{ 1 2 + 1 i }i∈n∗. https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 5 this shows that {tj}j∈n is an ∗-k-operator frame with bounds 1,{12 + 1i }i∈n∗. remark 3.4. (1) every ∗-operator frame for hom∗a(x) is an ∗-k-operator frame, for any k ∈ hom∗a(x): k 6= 0.(2) if k ∈ hom∗a(x) is a surjective operator, then every ∗-k-operator frame for hom∗a(x) isan ∗-operator frame. example 3.5. let x be a finitely or countably generated hilbert a-module. hom∗a(x). let k ∈ hom∗a(x) an invertible element such that both are uniformly bounded and k 6= 0. let {ti}i∈i be an ∗-operator frame for x with bounds a and b, respectively. we have a〈ξ,ξ〉a∗ ≤ ∑ i∈i 〈tiξ,tiξ〉≤ b〈ξ,ξ〉b∗,∀ξ ∈x . or 〈k∗ξ,k∗ξ〉≤ ‖k‖2∞〈ξ,ξ〉,∀ξ ∈x .then ‖k‖−1∞ a〈k ∗ξ,k∗ξ〉(‖k‖−1∞ a) ∗ ≤ ∑ i∈i 〈tiξ,tiξ〉≤ b〈ξ,ξ〉b∗,∀ξ ∈x . so {ti}i∈i is ∗-k-operator frame for x with bounds ‖k‖−1∞ a and b, respectively. in what follows, we introduce the analysis, the synthesis and the frame operator. we alsoestablish some properties.let {ti}i∈i be an ∗-k-operator frame for hom∗a(x). define an operator r : x → l2(x) by rξ = {tiξ}i∈i,∀ξ ∈x , then r is called the analysis operator. the adjoint of the analysis operator r, r∗ : l2(x) →x is given by r∗({ξi}i ) = ∑i∈i t∗i ξi,∀{ξi}i ∈ l2(x). the operator r∗ is calledthe synthesis operator. by composing r and r∗, the frame operator s : x → x is given by sξ = r∗rξ = ∑ i∈i t ∗ i tiξ.note that s need not be invertible in general. but under some condition s will be invertible. theorem 3.6. let k be a surjective operators in hom∗a(x). if {ti}i∈i is an ∗-k-operator frame for hom∗a(x), then the frame operator s is positive, invertible and adjointable. in addition we have the reconstruction formula, ξ = ∑ i∈i t ∗ i tis −1ξ, ∀ξ ∈x . proof. we start by showing that, s is a self-adjoint operator. by definition we have ∀ξ,η ∈h 〈sξ,η〉 = 〈∑ i∈i t∗i tiξ,η 〉 = ∑ i∈i 〈t∗i tiξ,η〉 = ∑ i∈i 〈ξ,t∗i tiη〉 https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 6 = 〈 ξ, ∑ i∈i t∗i tiη 〉 = 〈ξ,sη〉. then s is a selfadjoint.the operator s is clearly positive.by (2) in remark 3.4 {ti}i∈i is an ∗-operator frame for hom∗a(x).the definition of an ∗-operator gives a1〈ξ,ξ〉a∗1 ≤ ∑ i∈i 〈tiξ,tiξ〉≤ b〈ξ,ξ〉b∗. thus by the definition of norm in l2(x) p̄x (rξ) 2 = p̄x ( ∑ i∈i 〈tiξ,tiξ〉) ≤ p̄x (b)2p(〈ξ,ξ〉),∀ξ ∈x . (3.3) therefore r is well defined and p̄x (r) ≤ p̄x (b). it’s clear that r is a linear a-module map. wewill then show that the range of r is closed. let {rξn}n∈n be a sequence in the range of r suchthat limn→∞rξn = η. for n,m ∈n, we have p(a〈ξn −ξm,ξn −ξm〉a∗) ≤ p(〈r(ξn −ξm),r(ξn −ξm)〉) = p̄x (r(ξn −ξm))2. seeing that {rξn}n∈n is cauchy sequence in x , then p(a〈ξn −ξm,ξn −ξm〉a∗) → 0, as n,m →∞.note that for n,m ∈n, p(〈ξn −ξm,ξn −ξm〉) = p(a−1a〈ξn −ξm,ξn −ξm〉a∗(a∗)−1) ≤ p(a−1)2p(a〈ξn −ξm,ξn −ξm〉a∗). thus the sequence {ξn}n∈n is cauchy and hence there exists ξ ∈x such that ξn → ξ as n →∞.again by (3.3), we have p̄x (r(ξn −ξm))2 ≤ p̄x (b)2p(〈ξn −ξ,ξn −ξ〉). thus p(rξn − rξ) → 0 as n → ∞ implies that rξ = η. it is therefore concluded that therange of r is closed. we now show that r is injective. let ξ ∈ x and rξ = 0. note that a〈ξ,ξ〉a∗ ≤〈rξ,rξ〉 then 〈ξ,ξ〉 = 0 so ξ = 0 i.e. r is injective.for ξ ∈x and {ξi}i∈i ∈ l2(x) we have 〈rξ,{ξi}i∈i〉 = 〈{tiξ}i∈i,{ξi}i∈i〉 = ∑ i∈i 〈tiξ,ξi〉 = ∑ i∈i 〈ξ,t∗i ξi〉 = 〈ξ, ∑ i∈i t∗i ξi〉. then r∗({ξi}i∈i) = ∑i∈i t∗i ξi . since r is injective, then the operator r∗ has closed range and x = range(r∗), therefore s = r∗r is invertible � https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 7 let k ∈ hom∗a(x), in the following theorem we constructed an ∗-k-operator frame by using an ∗-operator frame. theorem 3.7. let {ti}i∈i be an ∗-k-operator frame in x with bounds a, b and k ∈ hom∗a(x) be an invertible element such that both are uniformly bounded. then {tik}i∈i is an ∗-k∗-operator frame in x with bounds a, ‖k‖∞b. the frame operator of {tik}i∈i is s ′ = k∗sk, where s is the frame operator of {ti}i∈i. proof. from a〈ξ,ξ〉a∗ ≤ ∑ i∈i 〈tiξ,tiξ〉≤ b〈ξ,ξ〉b∗,∀ξ ∈x . we get for all ξ ∈x , a〈kξ,kξ〉a∗ ≤ ∑ i∈i 〈tikξ,tikξ〉≤ b〈kξ,kξ〉b∗ ≤‖k‖∞b〈ξ,ξ〉(‖k‖∞b)∗. then {tik}i∈i is an ∗-k∗-operator frame in x with bounds a, ‖k‖∞b.by definition of s,we have skξ = ∑i∈i t∗i tikξ. then k∗sk = k∗ ∑ i∈i t∗i tikξ = ∑ i∈i k∗t∗i tikξ. hence s′ = k∗sk. � corollary 3.8. let k ∈ hom∗a(x) and {ti}i∈i be an ∗-operator frame. then {tis −1k}i∈i is an ∗-k∗-operator frame, where s is the frame operator of {ti}i∈i. proof. result of the theorem 3.7 for the ∗-operator frame {tis−1}i∈i. � 4. tensor product we denote by a⊗b, the minimal or injective tensor product of the pro-c∗-algebras a and b, itis the completion of the algebraic tensor product a⊗alg b with respect to the topology determinedby a family of c∗-seminorms. suppose that x is a hilbert module over a pro-c∗-algebra a and y is a hilbert module over a pro-c∗-algebra b. the algebraic tensor product x ⊗alg y of x and y is a pre-hilbert a⊗b-module with the action of a⊗b on x ⊗alg y defined by (ξ⊗η)(a⊗b) = ξa⊗ηb for all ξ ∈x ,η ∈y,a ∈a and b ∈b and the inner product 〈·, ·〉 : ( x ⊗alg y)×(x ⊗alg y) →a⊗alg b. defined by 〈ξ1 ⊗η1,ξ2 ⊗η2〉 = 〈ξ1,ξ2〉⊗〈η1,η2〉 and we know that for z = ∑ni=1ξi⊗ηi in x⊗algy we have 〈z,z〉a⊗b = ∑i,j〈ξi,ξj〉a⊗〈ηi,ηj〉b ≥ 0and 〈z,z〉a⊗b = 0 iff z = 0. https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 8 the external tensor product of x and y is the hilbert module x ⊗y over a⊗b obtained by thecompletion of the pre-hilbert a⊗b-module x ⊗alg y.if p ∈ m(x) and q ∈ m(y) then there is a unique adjointable module morphism p ⊗ q : a⊗b →x ⊗y such that (p ⊗q)(a⊗b) = p (a) ⊗q(b) and (p ⊗q)∗(a⊗b) = p∗(a) ⊗q∗(b)for all a ∈ a and for all b ∈ b (see, for example, cite the minimal or injective tensor product ofthe pro-c∗-algebras a and b, denoted by a⊗b, is the completion of the algebraic tensor product a⊗alg b with respect to the topology determined by a family of c∗-seminorms. suppose that xis a hilbert module over a pro-c∗-algebra a and y is a hilbert module over a pro-c∗-algebra b.the algebraic tensor product x ⊗alg y of x and y is a pre-hilbert a⊗b-module with the actionof a⊗b on x ⊗alg y defined by (ξ⊗η)(a⊗b) = ξa⊗ηb for all ξ ∈x ,η ∈y,a ∈a and b ∈b and the inner product 〈·, ·〉 : ( x ⊗alg y)×(x ⊗alg y) →a⊗alg b. defined by 〈ξ1 ⊗η1,ξ2 ⊗η2〉 = 〈ξ1,ξ2〉⊗〈η1,η2〉 we also know that for z = ∑ni=1ξi⊗ηi in x⊗algy we have 〈z,z〉a⊗b = ∑i,j〈ξi,ξj〉a⊗〈ηi,ηj〉b ≥ 0and 〈z,z〉a⊗b = 0 iff z = 0.the external tensor product of x and y is the hilbert module x ⊗y over a⊗b obtained by thecompletion of the pre-hilbert a⊗b-module x ⊗alg y.if p ∈ m(x) and q ∈ m(y) then there is a unique adjointable module morphism p ⊗ q : a⊗b →x ⊗y such that (p ⊗q)(a⊗b) = p (a) ⊗q(b) and (p ⊗q)∗(a⊗b) = p∗(a) ⊗q∗(b)for all a ∈ a and for all b ∈ b (see, for example, [9])let i and j be countable index sets. theorem 4.1. let x and y be two hilbert pro-c∗-modules over unitary pro-c∗-algebras a and b, respectively. let {ti}i∈i ⊂ hom∗a(x) be an ∗-k-operator frame for x with bounds a and b and frame operators st and {pj}j∈j ⊂ hom∗b(y) be an ∗-l-operator frame for k with bounds c and d and frame operators sl. then {ti ⊗ lj}i∈i,j∈j is an ∗-k⊗l-operator frame for hibert a⊗b-module x ⊗y with frame operator st ⊗sp and bounds a⊗c and b ⊗d. proof. the defintion of ∗-k-operator frame {ti}i∈i and ∗-l-operator frame {pj}j∈j gives a〈k∗ξ,k∗ξ〉aa∗ ≤ ∑ i∈i 〈tiξ,tiξ〉a ≤ b〈ξ,ξ〉ab∗,∀ξ ∈x . c〈l∗η,l∗η〉bc∗ ≤ ∑ j∈j 〈pjη,pjη〉b ≤ d〈η,η〉bd∗,∀η ∈y. https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 9 therefore (a〈k∗ξ,k∗ξ〉aa∗) ⊗ (c〈l∗η,l∗η〉bc∗) ≤ ∑ i∈i 〈tiξ,tiξ〉a ⊗ ∑ j∈j 〈pjη,pjη〉b ≤ (b〈ξ,ξ〉ab∗) ⊗ (d〈η,η〉bd∗),∀ξ ∈x ,∀η ∈y.then (a⊗c)(〈k∗ξ,k∗ξ〉a ⊗〈l∗η,l∗η〉b)(a∗ ⊗c∗) ≤ ∑ i∈i,j∈j 〈tiξ,tiξ〉a ⊗〈pjη,pjη〉b ≤ (b ⊗d)(〈ξ,ξ〉a ⊗〈η,η〉b)(b∗ ⊗d∗),∀ξ ∈x ,∀η ∈y.consequently we have (a⊗c)〈k∗ξ⊗l∗η,k∗ξ⊗l∗η〉a⊗b(a⊗c)∗ ≤ ∑ i∈i,j∈j 〈tiξ⊗pjη,tiξ⊗pjη〉a⊗b ≤ (b ⊗d)〈ξ⊗η,ξ⊗η〉a⊗b(b ⊗d)∗,∀ξ ∈x ,∀η ∈y. then for all ξ⊗η in x ⊗y we have (a⊗c)〈(k ⊗l)∗(ξ⊗η), (k ⊗l)∗(ξ⊗η)〉a⊗b(a⊗c)∗ ≤ ∑ i∈i,j∈j 〈(ti ⊗pj)(ξ⊗η), (ti ⊗pj)(ξ⊗η)〉a⊗b ≤ (b ⊗d)〈ξ⊗η,ξ⊗η〉a⊗b(b ⊗d)∗. the last inequality is true for every finite sum of elements in x ⊗alg y and then it’s true for all z ∈x ⊗k. it shows that {ti ⊗pj}i∈i,j∈j is an ∗-k ⊗l-operator frame for hilbert a⊗b-module x ⊗y with lower and upper bounds a⊗c and b ⊗d, respectively.by the definition of frame operator st and sp we have stξ = ∑ i∈i t∗i tiξ,∀ξ ∈x . spη = ∑ j∈j p∗j pjη,∀η ∈y. therefore (st ⊗sp )(ξ⊗η) = stξ⊗spη = ∑ i∈i t∗i tiξ⊗ ∑ j∈j p∗j pjη = ∑ i∈i,j∈j t∗i tiξ⊗p ∗ j pjη = ∑ i∈i,j∈j (t∗i ⊗p ∗ j )(tiξ⊗pjη) https://doi.org/10.28924/ada/ma.2.4 eur. j. math. anal. 10.28924/ada/ma.2.4 10 = ∑ i∈i,j∈j (t∗i ⊗p ∗ j )(ti ⊗pj)(ξ⊗η) = ∑ i∈i,j∈j (ti ⊗pj)∗(ti ⊗pj)(ξ⊗η). then by the uniqueness of frame operator, the last expression is equal to st⊗p (ξ⊗η). consequentlywe have (st ⊗sp )(ξ⊗η) = st⊗p (ξ⊗η). the last equality is true for every finite sum of elementsin x ⊗alg y and then it’s true for all z ∈ x ⊗y. it follows that (st ⊗sp )(z) = st⊗p (z). thus st⊗p = st ⊗sp . � references [1] n. haddadzadeh, g-frames in hilbert modules over pro-c*-algebras, int. j. ind. math. 9(4) (2017) 259-267.[2] m. azhini and n. haddadzadeh, fusion frames in hilbert modules over pro-c∗-algebras, int. j. ind. math. 5(2)(2013) article id ijim-00211.[3] r. j. duffin and a. c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952) 341-366.[4] m. fragoulopoulou, an introduction to the representation theory of topological ∗-algebras, schriftenreihe, univ.münster, 48 (1988) 1-81.[5] m. fragoulopoulou, tensor products of enveloping locally c∗-algebras, schriftenreihe, univ. münster (1997) 1-81.[6] m. fragoulopoulou, topological algebras with involution, north holland, amsterdam, 2005.[7] a. grossman, and y. meyer, painless nonorthogonal expansions, j. math. phys. 27 (1986) 1271-1283.[8] a. inoue, locally c∗-algebra, mem. fac. sci. kyushu univ. ser. a, math. 25(2) (1972) 197-235.[9] m. joita, on frames in hilbert modules over pro-c∗-algebras, topol. appl. 156 (2008) 83-92.[10] e. c. lance, hilbert c∗-modules, a toolkit for operator algebraists, london math. soc. lecture note series 210.cambridge univ. press, cambridge, 1995.[11] c. y. li and h. x. cao, operator frames for b(h), wavelet analysis and applications. birkhäuser basel, 2006.67-82.[12] a. mallios, topological algebras: selected topics, north holland, amsterdam, 1986.[13] m. naroei and a. nazari, some properties of –frames in hilbert modules over pro-c∗-algebras, sahand commun.math. anal. 16 (2019) 105–117.[14] n. c. phillips, inverse limits of c*-algebras, j. oper. theory. 19 (1988) 159-195.[15] n. c. phillips, representable k-theory for σ -c∗-algebras, k-theory, 3 (1989) 441-478.[16] m. rossafi and s. kabbaj, operator frame for end∗a(h), j. linear topol. algebra. 8 (2019) 85-95. https://doi.org/10.28924/ada/ma.2.4 1. introduction 2. preliminaries 3. -k-operator frame for homa(x) 4. tensor product references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 18doi: 10.28924/ada/ma.2.18 developments of newton’s method under hölder conditions samundra regmi1, ioannis k. argyros2,∗, santhosh george3, christopher i. argyros4 1learning commons, university of north texas at dallas, dallas, tx, usa samundra.regmi@untdallas.edu 2department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 3department of mathematical and computational sciences,national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 4department of computing and technology, cameron university, lawton, ok 73505, usa christopher.argyros@cameron.edu ∗correspondence: iargyros@cameron.edu abstract. the semi-local convergence criteria for newton’s method are weakened without new con-ditions. moreover, tighter error distances are provided as well as a more precise information on thelocation of the solution. 1. introduction the computation of a solution x∗ of nonlinear equation f (x) = 0 (1.1) is important in computational sciences, since many applications can be written as (1.1). here f : ω ⊆ x −→ y is fréchet-differentiable operator, x,y are banach spaces and ω 6= ∅ is aconvex and open set. but this can be attained only in special cases. that explains why mostsolution methods for (1.1) are iterative. there is a plethora of methods for solving (1.1) [1–14].among them newton’s method (nm) defined by x0 ∈ ω, xn+1 = xn −f ′(xn)−1f (xn) (1.2) seems to be the most popular [2,4]. but the convergence domain is small, limiting the applicability ofnm. that is why we have developed a technique that determines a subset ω0 of ω also containingthe iterates {xn}. hence, the hölder constants are at least as tight as the ones in ω. this crucial received: 20 mar 2022. key words and phrases. banach space; hölder condition; semi-local convergence; convergence criteria.1 https://adac.ee https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 2 modification leads to: weaker sufficient convergence criteria, the extension of the convergencedomain, tighter error estimates on ‖x∗ −xn‖,‖xn+1 −xn‖ and a more precise information on x∗.it is worth noticing that these advantages are obtained without additional conditions, since inpractice the evolution of the old hölderian constants require that of the new conditions as specialcases. 2. convergence we introduce certain hölder conditions crucial for the semi-local convergence. let p ∈ (0, 1].suppose there exists x0 ∈ ω such that f ′(x0)−1 ∈ l(y,x). definition 2.1. operator f ′ is center hölderian on ω if there exists h0 > 0 such that ‖f ′(x0)−1f ′(w) −f ′(x0)‖≤ h0‖w −x0‖p (2.1) for all w ∈ ω. set ω0 = u(x0, 1 h 1 p 0 ) ∩ ω. (2.2) definition 2.2. operator f ′ is center hölderian on ω0 if there exists h > 0 such that ‖f ′(x0)−1f ′(w) −f ′(u)‖≤ h̃‖w −u‖p, (2.3) where h̃ = { h, w = u −f ′(u)−1f (u),u ∈ d0 k, w,u ∈ ω0. . we present the results with h although k can be used too. but notice h ≤ k. definition 2.3. operator f ′ is center hölderian on ω if there exists h1 > 0 such that ‖f ′(x0)−1f ′(w) −f ′(u)‖≤ h1‖w −u‖p (2.4) for all w,u ∈ ω. remark 2.4. it follows from (2.2), that ω0 ⊆ ω. (2.5) then, by (2.1)-(2.5) the following items hold h0 ≤ h1 (2.6) and h ≤ h1. (2.7) we shall assume that h0 ≤ h. (2.8) https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 3 otherwise the results that follow hold with h0 replacing h. notice that h0 = h0(x0, ω), h1 = h1(x0, ω), h = h(x0, ω0) and h0h1 can be small (arbitrarily) [2–4]. in earlier studies [1, 5–14] the estimate ‖f ′(z)−1f ′(x0)‖≤ 1 1 −h1‖z −x0‖ 1 p (2.9) for all z ∈ u(x0, 1 h 1 p 1 ) was found using (2.4). but, if we use (2.1) to obtain the weaker and more precise estimate ‖f ′(z)−1f ′(x0)‖≤ 1 1 −h0‖z −x0‖ 1 p (2.10) for all z ∈ u(x0, 1 h 1 p 0 ). this modification in the proofs and exchanging h1 by h leads to the advantages as already mentioned in the introduction. that is why we omit the proofs in our results that follow. notice also that in practice the computation of h1 require that of h0 and h as special cases. hence, the applicability of nm is extended without additional conditions. let d ≥ 0 be such that ‖f ′(x0)−1f (x0)‖≤ d. (2.11) we assume that (2.1)-(2.3) hold from now on unless otherwise stated. first we extend the resultsby keller [11] for nm. similarly the results for the chord method can also be extended. we leavethe details to the motivated reader. for brevity we skip the extensions on the radii of convergenceballs, and only mention convergence criteria and error estimates. theorem 2.5. assume: hrλ < 1 + λ 2 + λ , d ≤ [ 1 − 2 + λ 1 + λ hrλ ] λ and ū(x0, r) ⊂ ω. then, limn−→∞xn = x∗ ∈ u(x0, r0) and f (x∗) = 0. furthermore, ‖x∗ −xn‖≤ ( µ 1 λ 2 + λ )(1+λ)p r µ 1 λ , where µ = hr λ 1−h0rλ 1 1+λ < 1. proof. see theorem 2 in [11]. � https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 4 theorem 2.6. assume: hdλ < 1 2 + λ ( λ 1 + λ )λ and ū(x0, r) ⊂ ω. then, limn−→∞xn = x∗ ∈ u(x0, r0), f (x∗) = 0 and ‖x∗ −xn‖≤ ( λ 1 p 1 −λ )(1+p)n d µ 1 p , where λ = hr p 0 1−h0r p 0 ( d r0 )p 1 1+p < 1 and r0 is the minimal positive root of scalar equation (2 + p)ht1+p − (1 + p)(t −d) = 0 provided that r ≥ r0. proof. see theorem 4 in [11]. � theorem 2.7. assume: hdp ≤ 1 − ( p 1 + p )p , r ≥ 1 + p 2 + p− (1 + p)p d and ū(x0, r) ⊂ ω. then, limn−→∞xn = x∗ ∈ u(x0, r),f (x∗) = 0 and ‖x∗ −xn‖≤ ( 1 1 + p )n [(1 + p)h 1 p d](1+p) n h 1 p . proof. see theorem 5 in [11]. �next, we extend a result given in [6] which in turn extended earlier ones [1,7–14]. it is convenientto define function on the interval [0,∞) by g(t) = h 1 + p t1+p − t + d gβ(t) = βh 1 + p t1+p − t + d (β ≥ 0) h(t) = t1+p + (1 + p)t (1 + p)1+p − 1 , v(p) = max t≥0 h(t), δ(p) = min{β ≥ 1 : max h(t) ≤ β, 0 ≤ t ≤ t(β)} https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 5 and scalar sequence {sn} by s0 = 0, sn = sn−1 − gd(sn−1) g′(sn−1) . then, we can show: theorem 2.8. assume: d ≤ 1 v(p) ( p 1 + p )p and u(x0, r̄) ⊆ ω, where r̄ is the minimal solution of equation gv (p) = 0, gv (t) = v(p)h 1 + p t1+p − t + d. proof. see theorem 2.2 in [6]. �next, we present the extensions of the work by rokne in [13] but for the newton-like method(nlm) xn+1 = xn −l−1n f (xn), where ln is a linear operator approximating f ′(xn). theorem 2.9. assume: ‖l(x) −l(x0)‖≤ m0‖x −x0‖p for all x ∈ ω. set ω0 = u(x0, 1 (γ2m0) 1 p ). ‖f ′(x) −f ′(y)‖≤ m̄‖x −y‖p for all x,y ∈ ω0, ‖f ′(x) −l(x)‖≤ γ0 + γ1‖x −x0‖p for all x ∈ ω0, and some γ0 ≥ 0,γ1 ≥ 0. l(x0)−1 ∈ l(y,x) with ‖l(x0)−1‖ ≤ γ2 and ‖l(x0)−1f (x0)‖≤ γ3, function q defined by q(t) = t1+p(γ2γ0 + γ2m0) + t( γ2m̄d p 1 + p + γ2γ0 − 1) −γ2m0γ3tp + γ3 has a smallest positive zero r > γ3, γ2m̄r p < 1, ρ = p 1 −γ2m̄rp [ γ2m̄d p 1 + p + γ2γ0 + γ2γ1r p ] < 1, ū(x0,r) ⊂ ω. then limn−→∞xn = x∗ and f (x∗) = 0. https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 6 proof. see theorem 1 in [13]. �many results on newton’s method were also reported in the elegant book in [9]. next, we showhow to extend one of them. the details of how to extend the result of them are left to the motivatedreader. theorem 2.10. suppose: conditions (2.1), (2.3), (2.8), and (c) h0 = hdp ∈ (0,ρ) where ρ is the only solution of equation (1 + p)p(1 − t)1+p − tp = 0, p ∈ (0, 1] in (0, 1 2 ] and u(x0,s) ⊂ ω, where s = (1+p)(1−h0) (1+p)−(2+p)h0 hold. then, sequence {xn} converges to a solution x∗ of equation f (x) = 0. moreover, {xn},x∗ ∈ u[x0,s] and x∗ is the only solution in ω ∩u(x0, d h 1/p 0 ). moreover, the following error estimates hold ‖xn −x∗‖≤ en, where en = δ (1+p)n−1 p2 a n 1−δ (1+p)n p a d, with δ = h1 h0 , a = 1 −h0, h1 = h0f1(h0)1+pf2(h0)p, f1(t) = 11−t and f2(t) = t1+p. finally, we extend the results by f. cianciaruso and e. de pascale in [6] who in turn extendedearlier ones [1, 5, 7, 11, 12, 14]. define scalar sequence {vn} for h = dph by v0 = 0,v1 = h 1 p , vn+1 = vn + (vn −vn−1)1+p (1 + p)(1 −vpn ) . (2.12) next, we extend theorem 2.1 and theorem 2.3 in [6], respectively. theorem 2.11. let function f : [1,∞) −→ [0,∞),r : [0,∞) −→ [0,∞) be defined by f (t) = (1 − 1 t ) 1 + p ((1 + p) 1 1−p + (t(t − 1)p) 1 1−p )1−p and r(t) = (1 + p) 1 p ((1 + p) 1 1−p + (t(t − 1)p) 1 1−p )1−p . suppose that h ≤ f (m), (2.13) where m is a global maximum for function f , given explicitly by m = 1+ √ 1+4(1+p)pp1−p 2 . then, the following assertion hold vn ≤ r(m)(1 − 1 mn ), (2.14) vn+1 vn ≤ 1 − 1 mn+1 1 − 1 mn , (2.15) https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 7 vn ≤ vn+1 ≤ r(m) < 1 and limn−→∞vn = v∗ ∈ [0,r(m)]. simply use h for h1 in [6]. � theorem 2.12. under condition (2.13) further suppose that r∗ = h− 1 p v∗ ≤ ρ and u(x0,ρ) ⊆ ω. then, sequence {xn} generated by nm is well defined in u(x0,v∗), stays in u(x0,v∗) and converges to the unique solution x∗ ∈ u[x0,v∗] of equation f (x) = 0, so that ‖xn+1 −xn‖≤ vn+1 −vn and ‖x∗ −xn‖≤ v∗ −vn. proof. simply use h for h1 used in [6]. remark 2.13. (1) if k = h1 the last two results coincide with the corresponding ones in [6]. but if k < h1 then the new results constitute an improvement with benefits already stated in the introduction. notice that the majorizing sequence {wn} in [6] was defined for h1 = dph1 by w0 = 0,w1 = h 1 p 1 , wn+1 = wn + (wn −wn−1)1+p (1 + p)(1 −wpn ) , (2.16) and the convergence criterion is h1 ≤ f (m). (2.17) it then follows by (2.7), (2.12), (2.13), (2.16) and (2.17) that h1 ≤ f (m) ⇒ h ≤ f (m) (2.18) but not necessarily vice versa, unless if h = h1, vn ≤ wn, 0 ≤ vn+1 −vn ≤ wn+1 −wn and 0 ≤ v∗ ≤ w∗ = lim n−→∞ wn. (2) in view of (2.9) and (2.10) sequence {un} defined for each n = 0, 1, 2, . . . by https://doi.org/10.28924/ada/ma.2.18 eur. j. math. anal. 10.28924/ada/ma.2.18 8 u0 = 0,u1 = h 1 p 1 , u2 = u1 + h0(u1 −u0)1+p (1 + p)(1 −h0u p 1) , un+1 = un + h(un −un−1)1+p (1 + p)(1 −h0u p n ) is a tighter majorizing sequence than {vn} and can replace it in theorem 2.11 and theorem 2.12. concerning the uniqueness of the solution x∗ we provide a result based only on (2.1). proposition 2.14. suppose: (1) the point x∗ ∈ u(x0,a) ⊂ ω is a simple solution of equation f (x) = 0 for some a > 0. (2) condition (2.1) holds. (3) there exist b ≥ a such that h0 ∫ 1 0 ((1 −τ)a + τb)pdτ < 1. (2.19) let g = u[x0,b] ∩ ω. then, the point x∗ is the only solution of equation f (x) = 0 in the set g. proof. let z∗ ∈ g with f (z∗) = 0. by (2.1) and (2.19), we obtain in turn for q = ∫1 0 f ′(x∗ + τ(z∗ −x∗))dτ ‖f ′(x0)−1(q−f ′(x0))‖ ≤ h0 ∫ 1 0 ‖x∗ + τ(z∗ −x∗) −x0‖pdτ ≤ h0 ∫ 1 0 [(1 −τ)‖x∗ −x0‖ + τ‖z∗ −x0‖]pdτ ≤ h0 ∫ 1 0 ((1 −τ)a + τb)pdτ < 1, showing z∗ = x∗ by the invertibility of q and the approximation q(x∗−z∗) = f (x∗)−f (z∗) = 0. �notice that if k = h1 the results coincide to the ones of theorem 3.4 in [9]. but, if k < h1then they constitute an extension. remark 2.15. (a) we gave the results in affine invariant form. (b)the results in this study can be extended more if we consider the set s = u(x1, 1h1/p − d) provided that h1/pd < 1. moreover, suppose s ⊂ ω. then, s ⊂ ω0, so the hölderian constant corresponding to s is at least as small as k, and can replace it in all previous results. references [1] j. appell, e.d. pascale, j.v. lysenko, p.p. zabrejko, new results on newton-kantorovich approximations with appli-cations to nonlinear integral equations, numer. funct. anal. optim. 18 (1997) 1–17. https://doi.org/10.1080/ 01630569708816744. https://doi.org/10.28924/ada/ma.2.18 https://doi.org/10.1080/01630569708816744 https://doi.org/10.1080/01630569708816744 eur. j. math. anal. 10.28924/ada/ma.2.18 9 [2] i.k. argyros, s. hilout, inexact newton-type methods, j. complex. 26 (2010) 577–590. https://doi.org/10.1016/ j.jco.2010.08.006.[3] i.k. argyros, convergence and applications of newton-type iterations, springer new york, 2008. https://doi. org/10.1007/978-0-387-72743-1.[4] i.k. argyros, s. george, mathematical modeling for the solution of equations and systems of equations with appli-cations, volume-iv, nova publisher, ny, 2021.[5] f. cianciaruso, e. de pascale, newton–kantorovich approximations when the derivative is hölderian: old and newresults, numer. funct. anal. optim. 24 (2003) 713–723. https://doi.org/10.1081/nfa-120026367.[6] f. cianciaruso, e. de pascale, estimates of majorizing sequences in the newton–kantorovich method: a furtherimprovement, j. math. anal. appl. 322 (2006) 329–335. https://doi.org/10.1016/j.jmaa.2005.09.008.[7] n.t. demidovich, p.p. zabreiko, j.v. lysenko, some remarks on the newtonkantorovich method for nonlinear equa-tions with hölder continuous linearizations, izv. akad. nauk, beloruss, 3 (1993) 22-26 (russian).[8] e. de pascale, p.p. zabrejko, convergence of the newton-kantorovich method under vertgeim conditions: a newimprovement, z. anal. anwend. 17 (1998) 271–280. https://doi.org/10.4171/zaa/821.[9] j. a. ezquerro, m. hernandez-veron, mild differentiability conditions for newton’s method in banach spaces, fron-tiers in mathematics, birkhauser cham, switzerland, (2020), https://doi.org/10.1007/978-3-030-48702-7.[10] l.v. kantorovich, g.p. akilov, functional analysis in normed spaces, the macmillan co, new york, (1964).[11] h.b. keller, newton’s method under mild differentiability conditions, j. computer syst. sci. 4 (1970) 15–28. https: //doi.org/10.1016/s0022-0000(70)80009-5.[12] j.v. lysenko, conditions for the convergence of the newton-kantorovich method for nonlinear equations with hölderlinearization, dokl. akad. nauk. bssr, 38 (1994) 20-24. (in russian).[13] j. rokne, newton’s method under mild differentiability conditions with error analysis, numer. math. 18 (1971)401–412. https://doi.org/10.1007/bf01406677.[14] b.a. vertgeim, on some methods of the approximate solution of nonlinear functional equations in banach spaces,uspekhi mat. nauk. 12 (1957) 166-169 (in russian). engl. transl: amer. math. soc. transl. 16 (1960) 378-382. https://doi.org/10.28924/ada/ma.2.18 https://doi.org/10.1016/j.jco.2010.08.006 https://doi.org/10.1016/j.jco.2010.08.006 https://doi.org/10.1007/978-0-387-72743-1 https://doi.org/10.1007/978-0-387-72743-1 https://doi.org/10.1081/nfa-120026367 https://doi.org/10.1016/j.jmaa.2005.09.008 https://doi.org/10.4171/zaa/821 https://doi.org/10.1007/978-3-030-48702-7 https://doi.org/10.1016/s0022-0000(70)80009-5 https://doi.org/10.1016/s0022-0000(70)80009-5 https://doi.org/10.1007/bf01406677 1. introduction 2. convergence references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 2doi: 10.28924/ada/ma.2.2 on geometric constants for discrete morrey spaces adam adam, hendra gunawan∗ analysis and geometry group, faculty of mathematics and natural sciences, bandung institute of technology, bandung 40132, indonesia adam_adam@students.itb.ac.id, hgunawan@math.itb.ac.id ∗correspondence: hgunawan@math.itb.ac.id abstract. in this paper we prove that the n-th von neumann-jordan constant and the n-th jamesconstant for discrete morrey spaces `pq where 1 ≤ p < q < ∞ are both equal to n. this resulttells us that the discrete morrey spaces are not uniformly non-`1, and hence they are not uniformly n-convex. 1. introduction let n ≥ 2 be a non-negative integer and (x,‖·‖) be a banach space. the n-th von neumannjordan constant for x [6] is defined by c (n) nj (x) := sup {∑ ±‖u1 ±u2 ±···±un‖ 2 x 2n−1 ∑n i=1‖ui‖x : ui 6=0, i =1,2, . . . ,n } and the n-th james constant for x [7] is defined by c (n) j (x) := sup{min‖u1 ±u2 ±···±un‖ : ui ∈ sx, i =1,2, . . . ,n}. note that in the definition of c(n) nj (x), the sum ∑± is taken over all possible combinations of ±signs. similarly, in the definition of c(n) j (x), the minimum is taken over all possible combinationsof ± signs, while the supremum is taken over all ui’s in the unit sphere sx := {u ∈ x : ‖u‖=1}.these constants measure some sort of convexity of a banach space.we say that x is uniformly n-convex [2] if for every ε ∈ (0,n] there exists a δ ∈ (0,1) such thatfor every u1,u2, . . . ,un ∈ sx with ‖u1 ±u2 ±···±un‖≥ ε for all combinations of ± signs exceptfor ‖u1 +u2 + · · ·+un‖, we have ‖u1 +u2 + · · ·+un‖≤ n(1−δ). received: 31 aug 2021. key words and phrases. n-th von neumann-jordan constant; n-th james constant; discrete morrey spaces; uniformlynon-`1 spaces; uniformly n-convex spaces. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.2 https://orcid.org/0000-0001-7879-8321 eur. j. math. anal. 10.28924/ada/ma.2.2 2 meanwhile, we say that x is uniformly non-`1n [1, 5, 8] if there exists a δ ∈ (0,1) such that for every u1,u2, . . . ,un ∈ sx we have min‖u1 ±u2 ±···±un‖≤ n(1−δ). note that for n =2, uniformly non-`1n spaces are known as uniformly nonsquare spaces, while for n = 3 they are known as uniformly non-octahedral spaces. one may verify that if x is uniformly n-convex, then x is uniformly non-`1n [2].now a few remarks about the two constants, and their associations with the uniformly non-`1nand uniformly n-convex properties. • 1≤ c(n) nj (x)≤ n and c(n) nj (x)=1 if and only if x is a hilbert space [6]. • 1 ≤ c(n) j (x) ≤ n. if dim(x) = ∞, then √n ≤ c(n) j (x) ≤ n. moreover, if x is a hilbertspace, then c(n) j (x)= √ n [7]. • x is uniformly non-`1n if and only if c(n)nj(x) < n [6]. • x is uniformly non-`1n if and only if c(n)j (x) < n [7]. the last two statements tell us that if c(n) nj (x)= n or c(n) j (x)= n, then x is not uniformly non-`1nand hence not uniformly n-convex.in this paper, we shall compute the value of the two constants for discrete morrey spaces. let ω :=n∪{0} and m =(m1,m2, . . . ,md)∈zd . define sm,n := {k ∈zd : ‖k −m‖∞ ≤ n} where n ∈ ω and ‖m‖∞ = max{|mi| : 1 ≤ i ≤ d}. denote by |sm,n| the cardinality of sm,n for m ∈zd and n ∈ ω. then we have |sm,n|=(2n +1)d .now let 1 ≤ p ≤ q < ∞. define `pq = `pq(zd) to be the discrete morrey space as introducedin [3], which consists of all sequences x :zd →r with ‖x‖`pq := sup m∈zd,n∈ω |sm,n| 1 q −1 p ( ∑ k∈sm,n |xk|p )1 p < ∞, where x := (xk) with k ∈ zd . one may observe that these discrete morrey spaces are banachspaces [3]. note, in particular, that for p = q, we have `pq = `q.from [4] we already know that cnj(`pq) = cj(`pq) = 2 for 1 ≤ p < q < ∞, which impliesthat `pq are not uniformly nonsquares for those p’s and q’s. in this paper, we shall show that c (n) nj (` p q) = c (n) j (` p q) = n for 1 ≤ p < q < ∞, which leads us to the conclusion that `pq arenot uniformly non-`1n for those p’s and q’s, which is sharper than the existing result. (if x is notuniformly non-`1n, then x is not uniformly non-`1n−1, provided that n ≥ 3.) https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 3 2. main results the value of the n-th von neumann-jordan constant and the n-th james constant for discretemorrey spaces are stated in the following theorems. to understand the idea of the proof, we firstpresent the result for n =3. theorem 2.1. for 1≤ p < q < ∞, we have c(3) nj (` p q(zd))= c (3) j (` p q(zd))=3. proof. to prove the theorem, it suffices for us to find x(1),x(2),x(3) ∈ `pq such that∑ ±‖x (1) ±x(2) ±x(3)‖2 ` p q 22 ∑3 i=1‖x(i)‖`pq =3 for the von neumann-jordan constant, and min‖x(1) ±x(2) ±x(3)‖`pq =3 for the james constant. case 1: d =1. let j ∈z be a nonnegative, even integer such that j > 4 qq−p −1, or equivalently (j +1) 1 q −1 p < 4 −1 p . construct x(1),x(2),x(3) ∈ `pq(z) as follows: • x(1) =(x(1) k )k∈z is defined by x (1) k = 1, k =0, j,2j,3j, 0, otherwise; • x(2) =(x(2) k )k∈z is defined by x (2) k =  1, k =0, j, −1, k =2j,3j, 0, otherwise; • x(3) =(x(3) k )k∈z is defined by x (3) k =  1, k =0,2j, −1, k = j,3j, 0, otherwise. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 4 the three sequences are in the unit sphere of `pq(z). indeed, for the first sequence, we have ‖x(1)‖`pq = sup m∈z,n∈ω |sm,n| 1 q −1 p ( ∑ k∈sm,n |x(1) k |p )1 p = sup m∈z∩[0,3j],n∈z∩[0,3j/2] |sm,n| 1 q −1 p ( ∑ k∈sm,n |x(1) k |p )1 p =max{1,(j +1) 1 q −1 p2 1 p ,(2j +1) 1 q −1 p3 1 p ,(3j +1) 1 q −1 p4 1 p}. since (3j +1)1q−1p < (2j +1)1q−1p < (j +1)1q−1p < 4−1p , we get ‖x(1)‖`pq = 1. similarly, one mayobserve that ‖x(2)‖`pq = ‖x(3)‖`pq =1.next, we observe that x (1) k +x (2) k +x (3) k =  3, k =0, 1, k = j,2j, −1, k =3j, 0, otherwise; x (1) k +x (2) k −x(3) k =  3, k = j, 1, k =0,3j, −1, k =2j, 0, otherwise; x (1) k −x(2) k +x (3) k =  3, k =2j, 1, k =0,3j, −1, k = j, 0, otherwise; x (1) k −x(2) k −x(3) k =  3, k =3j, 1, k = j,2j, −1, k =0, 0, otherwise.we first compute that ‖x(1)+x(2)+x(3)‖`pq =max{3,(j+1) 1 q −1 p(3p+1) 1 p ,(2j+1) 1 q −1 p(3p+2) 1 p ,(3j+1) 1 q −1 p(3p+3) 1 p}. notice that • (j +1) 1 q −1 p(3p +1) 1 p < ( 3p+1p 4 )1 p < (3p) 1 p =3. • (2j +1) 1 q −1 p(3p +2) 1 p < (j +1) 1 q −1 p(3p +2) 1 p < ( 3p+2 4 )1 p < 3. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 5 • (3j +1) 1 q −1 p(3p +3) 1 p < (j +1) 1 q −1 p(3p +3) 1 p < ( 3p+3 4 )1 p < 3. hence, we obtain ‖x(1) +x(2) +x(3)‖`pq =3.similarly, we have ‖x(1) ±x(2) ±x(3)‖`pq = sup m∈z∩[0,3j],n∈z∩[0,3j/2] |sm,n| 1 q −1 p ( ∑ k∈sm,n |x(1) k ±x(2) k ±x(3) k |p )1 p =3 for every combination of ± signs. consequently, ∑±‖x(1)±x(2)±x(3)‖2`pq 22 ∑3 i=1‖x(i)‖`pq = 3 and min‖x(1) ± x(2) ± x(3)‖`pq = 3, so we come to theconclusion that c (3) nj (`pq(z))= c (3) j (`pq(z))=3. case 2: d > 1. let j ∈ z be a nonnegative, even integer such that j > 4 qd(q−p) −1, which isequivalent to (j +1) d(1 q −1 p ) < 4 −1 p . we then construct x(1),x(2),x(3) ∈ `pq(zd) as follows: • x(1) =(x(1) k )k∈zd is defined by x (1) k = 1, k =(0,0, . . . ,0),(j,0, . . . ,0),(2j,0, . . . ,0),(3j,0, . . . ,0), 0, otherwise; • x(2) =(x(2) k )k∈zd is defined by x (2) k =  1, k =(0,0, . . . ,0),(j,0, . . . ,0), −1, k =(2j,0, . . . ,0),(3j,0, . . . ,0), 0, otherwise; • x(3) =(x(3) k )k∈zd is defined by x (3) k =  1, k =(0,0, . . . ,0),(2j,0, . . . ,0), −1, k =(j,0, . . . ,0),(3j,0, . . . ,0), 0, otherwise. as in the case where d =1, one may observe that ‖x(1)‖`pq = sup m∈zd,n∈ω |sm,n| 1 q −1 p ( ∑ k∈sm,n |x(1) k |p )1 p =max{1,(j +1)d( 1 q −1 p ) 2 1 p ,(2j +1) d(1 q −1 p ) 3 1 p ,(3j +1) d(1 q −1 p ) 4 1 p} =1. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 6 we also get ‖x(2)‖`pq = ‖x(3)‖`pq =1. moreover, through similar observation as in the 1-dimensionalcase, we have ‖x(1) ±x(2) ±x(3)‖`pq =3for every possible combinations of ± signs. it thus follows that c (3) j (`pq(z d))= sup{min‖x1 ±x2 ±x3‖`pq : x1,x2,x3 ∈ s`pq}=3 and c (3) nj (`pq(z d))= sup {∑ ±‖x1 ±x2 ±x3‖ 2 ` p q 22 ∑3 i=1‖xi‖`pq : xi 6=0, i =1,2,3 } =3. � we now state the general result for n ≥ 3. (the proof is also valid for n =2, which amounts tothe work of [3].) theorem 2.2. for 1≤ p < q < ∞, we have c(n) nj (` p q(zd))= c (n) j (` p q(zd))= n. proof. as for n =3, we shall consider the case where d =1 first, and then the case where d > 1later. case 1: d =1. let j ∈z be a nonnegative, even integer such that j > 2(n−1)( qq−p)−1, which isequivalent to (j +1) 1 q −1 p < 2 −(n−1) p . we construct x(i) ∈ `pq ∈z for i =1,2, . . . ,n as follows: • x(1) =(x(1) k )k∈z is defined by x (1) k = 1, k ∈ s (1) 1 , 0, otherwise, where s (1) 1 = {0, j,2j,3j, . . . ,(2 n−1 −1)j}; • x(i) =(x(i) k )k∈z for 2≤ i ≤ n is defined by x (i) k =  1, k ∈ s(i)1 , −1, k ∈ s(i)−1, 0, otherwise, with the following rules: write p = {0, j,2j, . . . ,(2n−1 −1)j} as p = p (i) 1 ∪p (i) 2 ∪·· ·∪p (i) 2i−1 https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 7 where p(i)1 consists of the first 2n−12i−1 terms of p , p(i)2 consists of the next 2n−12i−1 terms of p ,and so on. then s(i)1 and s(i)−1 are given by s (i) 1 = p (i) 1 ∪p (i) 3 ∪·· ·∪p (i) 2i−1−1, s (i) −1 = p (i) 2 ∪p (i) 4 ∪·· ·∪p (i) 2i−1 . for example, for i =2, x(2) =(x(2) k )k∈z is defined by x (2) k =  1, k ∈ s(2)1 , −1, k ∈ s(2)−1, 0, otherwise, where s (2) 1 = { 0, j,2j,3j, . . . , (2n−1 2 −1 ) j } s (2) −1 = {(2n−1 2 ) j, (2n−1 2 +1 ) j, . . . ,(2n−1 −1)j } ; note that the largest absolute value of the terms of x(i) in the above construction will beequal to 1 for each i = 1, . . . ,n. next, since the number of possible combinations of ± signs in x(1) ± x(2) ± ···± x(n) is 2n−1, the above construction will give us 1+1+ · · ·+1 = n as thelargest absolute value of x(1)±x(2)±···±x(n) for every combination of ± signs. this means that,if x(1) ±x(2) ±···±x(n) =(xk)k∈z, then max k∈z |xk|= n.let us now compute the norms. for x(1), we have ‖x(1)‖`pq = sup m∈z,n∈ω |sm,n| 1 q −1 p ( ∑ k∈sm,n |x(1) k |p )1 p = sup m∈z∩[0,(2n−1−1)j],n∈z∩[0,(2n−1−1)j/2] |sm,n| 1 q −1 p ( ∑ k∈sm,n |x(1) k |p )1 p =max{1,(j +1) 1 q −1 p2 1 p ,(2j +1) 1 q −1 p3 1 p , . . . ,((2n−1 −1)j +1) 1 q −1 p2 n−1 p }. for each r =1,2, . . . ,2n−1 −1, we have (rj +1)1q−1p ≤ (j +1)1q−1p and (r +1)1p ≤ 2n−1p , so that (rj +1) 1 q −1 p(r +1) 1 p ≤ (j +1) 1 q −1 p2 n−1 p < 2 −n−1 p 2 n−1 p =1. hence we obtain ‖x(1)‖`pq =1. similarly, one may verify that ‖x(2)‖`pq = ‖x (3)‖`pq = · · ·= ‖x (n)‖`pq =1. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 8 next, we shall compute the norms of x(1)±x(2)±···±x(n). write x(1)+x(2)+· · ·+x(n) =(xk)k∈zwhere xk :=  a1, k =0, a2, k = j, a3, k =2j,... a2n−1, k =(2 n−1 −1)j, 0, otherwise, with a1 = n and |ai| < n for i =2,3, . . . ,(2n−1)j. accordingly, we have ‖x(1) +x(2) + · · ·+x(n)‖`pq = sup m∈z,n∈ω |sm,n| 1 q −1 p ( ∑ k∈sm,n |xk|p )1 p = sup m∈z∩[0,(2n−1−1)j],n∈z∩[0,(2n−1−1)j/2] |sm,n| 1 q −1 p ( ∑ k∈sm,n |xk|p )1 p =max { n,(j +1) 1 q −1 p(np +a p 2) 1 p ,(2j +1) 1 q −1 p(np +a p 2 +a p 3) 1 p , . . . ,((2n−1 −1)j +1) 1 q −1 p ( np + 2n−1∑ i=2 a p i )1 p } . since (rj +1)1q−1p ≤ (j +1)1q−1p for each r =1,2, . . . ,2n−1 −1, we obtain (rj +1) 1 q −1 p ( np + r+1∑ i=2 a p i )1 p ≤ (j +1) 1 q −1 p ( np + r+1∑ i=2 a p i )1 p < 2 −(n−1) p ( np + r+1∑ i=2 a p i )1 p < 2 −(n−1) p (np +np + · · ·+np︸ ︷︷ ︸ r +1 times ) 1 p =2 −(n−1) p (r +1) 1 p(np) 1 p ≤ 2− (n−1) p 2 (n−1) p n = n. it thus follows that ‖x(1) +x(2) + · · ·+x(n)‖`pq = n. as we have remarked earlier, the largest absolute value of x(1) ± x(2) ±···± x(n) is equal to n for every combination of ± signs. moreover, it is clear that for k /∈ {0,2j, . . . ,(2n−1 −1)j}, the https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 9 k-th term of x(1) ±x(2) ±···±x(n) is equal to 0. hence, we obtain ‖x(1) ±x(2) ±···±x(n)‖`pq = sup m∈z,n∈ω |sm,n| 1 q −1 p ( ∑ k∈sm,n |x(1) k ±x(2) k ±···±x(n) k |p )1 p = sup m∈z∩[0,(2n−1−1)j],n∈z∩[0,(2n−1−1)j/2] |sm,n| 1 q −1 p ( ∑ k∈sm,n |x(1) k ±x(2) k ±···±x(n) k |p )1 p = n. consequently, we get ∑ ±‖x (1) ±x(2) ±···±x(n)‖2 ` p q 2n−1 ∑n i=1‖xi‖`pq = 2n−1n2 2n−1n = n and min‖x(1) ±x(2) ±···±x(n)‖`pq = n,whence c (n) nj (`pq(z))= c (n) j (`pq(z))= n. case 2: d > 1. here we choose j ∈ z to be a nonnegative, even integer such that j > 2 (n−1 d )( q q−p) −1 or, equivalently, (j +1) d(1 q −1 p ) < 2 −(n−1) p . then, using the sequences x(i) =(x (i) k1 )k1∈z ∈ ` p q(z), i =1, . . . ,n, in the case where d =1, we now define x(i) := (x(i) k )k∈zd ∈ ` p q(zd) for i =1, . . . ,n, where x (i) k = x (i) k1 , k =(k1,0,0, . . . ,0), 0, otherwise. we shall then obtain c (n) nj (`pq(z d))= c (n) j (`pq(z d))= n, as desired. � corollary 2.2.1. for 1≤ p < q < ∞, the space `pq is not uniformly non-`1n. corollary 2.2.2. for 1≤ p < q < ∞, the space `pq is not uniformly n-convex. acknowledgement. the work is part of the first author’s thesis. both authors are supported byp2mi 2021 program of bandung institute of technology. https://doi.org/10.28924/ada/ma.2.2 eur. j. math. anal. 10.28924/ada/ma.2.2 10 references [1] b. beauzamy, introduction to banach spaces and their geometry, 2nd ed., north holland, amsterdamnewyork-oxford, 1985. https://pascal-francis.inist.fr/vibad/index.php?action=getrecorddetail&idt= pascal82x0319279.[2] h. gunawan, d.i. hakim, a.s. putri, on geometric properties of morrey spaces, ufimsk. mat. zh. 13 (2021) 131–136. https://doi.org/10.13108/2021-13-1-131.[3] h. gunawan, e. kikianty, c. schwanke, discrete morrey spaces and their inclusion properties, math. nachr. 291(2018) 1283–1296. https://doi.org/10.1002/mana.201700054.[4] h. gunawan, e. kikianty, y. sawano, and c. schwanke, three geometric constants for morrey spaces, bull. korean.math. soc. 56 (2019) 1569-1575. https://doi.org/10.4134/bkms.b190010.[5] r.c. james, uniformly non-square banach spaces, ann. math. 80 (1964) 542-550. https://doi.org/10.2307/ 1970663.[6] m. kato, y. takahashi, and k. hashimoto, on n-th von neumann-jordan constants for banach spaces, bull. kyushuinst. tech. 45 (1998), 25-33. https://ci.nii.ac.jp/naid/110000079659.[7] l. maligranda, l. nikolova, l.-e. persson, t. zachariades, on n-th james and khintchine constants of banachspaces, math. inequal. appl. 1 (2007) 1–22. https://doi.org/10.7153/mia-11-01.[8] w.a. wojczynski, geometry and martingales in banach spaces, part ii, in: probability in banach spaces iv, j.kuelbs, ed., marcel-dekker, 1978, 267–517. https://doi.org/10.1201/9780429462153. https://doi.org/10.28924/ada/ma.2.2 https://pascal-francis.inist.fr/vibad/index.php?action=getrecorddetail&idt=pascal82x0319279 https://pascal-francis.inist.fr/vibad/index.php?action=getrecorddetail&idt=pascal82x0319279 https://doi.org/10.13108/2021-13-1-131 https://doi.org/10.1002/mana.201700054 https://doi.org/10.4134/bkms.b190010 https://doi.org/10.2307/1970663 https://doi.org/10.2307/1970663 https://ci.nii.ac.jp/naid/110000079659 https://doi.org/10.7153/mia-11-01 https://doi.org/10.1201/9780429462153 1. introduction 2. main results references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 5doi: 10.28924/ada/ma.3.5 updated and weaker convergence criteria of newton iterates for equations samundra regmi1, ioannis k. argyros2,∗, santhosh george3, michael i. argyros4 1department of mathematics, university of houston, houston, tx 77204, usa sregmi5@uh.edu 2department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 3department of mathematical and computational sciences, national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 4university of oklahoma, department of computer science, norman, ok 73019, usa michael.i.argyros-1@ou.edu ∗correspondence: iargyros@cameron.edu abstract. newton iteration is often used as a solver for nonlinear equations in abstract spaces.some of the main concerns are general: criteria for convergence, error estimations on consecutiveiterates, and the location of a solution. a plethora of authors has addressed these concerns by pre-senting results based on the celebrated kantorovich theory. this article contributes in this directionby extending earlier results but without additional conditions. these extensions become possibleusing a more precise majorization than the one given in earlier articles. numerical experimentationcomplements the theoretical results involving a partial differential and an integral equation. 1. introduction nonlinear equation f (x) = 0, (1.1) plays a important role due to the fact that many applications can be brought to look like it. thecelebrated newton iteration (ni) in the following form xn+1 = xn −f ′(xn)−1f (xn), ∀ n = 0, 1, 2, . . . (1.2) is often applied to solve equation (1.1) iteratively. here, f : ω ⊂ m1 −→ m2 is differentiable perfréchet and operates between banach spaces m1 and m2, whereas set ω 6= ∅.kantorovich inaugurated the semi-local convergence of ni (slcni) analysis of ni in abstractspaces by applying the contraction mapping principle due to banach. he presented two different received: 29 apr 2022. key words and phrases. iterative processes; newton iteration; banach space; semi-local convergence.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 2 proofs based on majorization and recurrent relations [12]. the newton-kantorovich theorem givesthe slcni. numerous authors applied this result, in applications and also as a theoretical tool.even a simple equation given in [1–4, 7, 10, 11] shows that convergence criteria may not besatisfied. however, ni may be convergent (see the numerical section, example 4.1). that iswhy these criteria are weakened in [2–4]. but no new conditions are added. in this study twoadditional features are presented. one involves an explicit upper bound on the smallness of initialapproximation. moreover by choosing a bit larger bound the convergence order of ni is recovered.consequently, new results can always replace corresponding ones by kantorovich [7] and others[5, 8–11], since preceding results imply the one in this study but not necessarily vice versa. methodin this study uses smaller lipschitz or hölder parameters to achieve these extensions which arespecializations of earlier ones. that is no additional effort is needed. the generality of this ideaallows its application on other processes [3, 4, 11].contributions by others can be found in section 4, where comparisons take place. the majoriza-tion of ni is discussed in section 2. slcni appears in section 3. the numerical experimentationis given in section4. conclusions complete this study in section 5. 2. majorization of ni let k0,k,l0,l denote positive numbers, q ∈ (0, 1] and t stand for a positive variable. theseparametrs are connected in section 3 to initial data d = (ω,y,f,f ′,x0). define sequence {sn}by s0 = 0,s1(t) = s1 = t s2(t) = s2 = s1 + k(s1 − s0)1+q (1 + q)(1 −k0s q 1 ) , sn+2(t) = sn+2 = sn+1 + l(sn+1 − sn)1+q (1 + q)(1 −l0s q n+1) , ∀n = 1, 2, . . . . (2.1) sequence {xn} is majorized by {sn} (see section 3). that is why convergence is studied first forsequence {sn}. lemma 2.1. suppose k0t q < 1 and l0sqn+1 < 1 ∀n = 0, 1, 2, . . . . (2.2) then, sequence {sn} is strictly increasing and converges to some limit point s∗ ∈ (0, ( 1l0 ) 1 q ]. the point s∗ is the unique least upper bound of sequence {sn}. proof. the result follows from definition of sequence {sn} and hypothesis (2.1). �let � be a positive constant. moreover, introduce parameters by α = 1+�, β = l (1+q) (1+�), γ = � (1+�)l0 , δ = β(s2 − s1), λ = γ 1 q , h = δ1+q and u = ( 1 k0 ) 1 q . furthermore, consider functions withcommon domain in t = [0,u) given as https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 3 f1(t) = ( ktq (1 + q)(1 −k0tq) + t )q −γ, f2(t) klαt1+q (1 + q)2(1 −k0tq) − 1 and f3(t) = (s2 + β −1 q h 1 −h )q −γ. it follows by these definitions f1(0) = −γ < 0, f2(0) = −1 < 0, f3(0) = −γ < 0 and f1(t) −→ ∞, f2(t) −→ ∞ and f3(t) −→ ∞ as t −→ u−. so, function fi, i = 1, 2, 3 have zeros in interval t by ivt (intermediate value theorem). let ηi denote the smallest such zero of functions fi ininterval t0 = (0,u), respectively.it also follows by these choices of zeros ηi k0s q 1 < 1, s q 2 < γ,f1(t) < 0 at t = η1 (2.3) δ < 1, f2(t) < 0 at t = η2 (2.4) and f3(t) < 0 at t = η3. (2.5) define parameter η0 = min{ηi}. (2.6) suppose η ≤ η0. (2.7) if η0 = η1 or η0 = η2, suppose hypothesis (2.7) holds as a strict inequality.a second stronger convergence result follows. but hypotheses are easier to verify. lemma 2.2. suppose hypothesis (2.7) holds. then, sequence {sn} is strictly increasing and convergent to some s∗ ∈ (0,γ0), where γ0 = s2 + β −1q h 1−h . moreover, for σn+2 = sn+2 − sn+1 ∀n = 0, 1, 2, . . . σn+2 ≤ βσ 1+q n+1 ≤ β −1 q δ(1+) n (2.8) and s∗ − sn+1 ≤ β −1 q δ(1+q) n 1 −δ1+q . (2.9) proof. the assertions (ij) : 0 < 1 1 −l0s q j+1 ≤ α (2.10) https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 4 is shown using induction. assertion (i1) is true by the choice of η1 and estimates (2.3). it followsby (i1) and sequence {sn} that 0 < s3 − s2 ≤ β(s2 − s1)1+q, or s3 < s2 + β −1 q δ(1+q) 1 ≤ γ. so, assertion (2.8) holds for n = 1. suppose assertion (2.10) holds ∀j = 1, 2, . . .n. then, 0 < σn+1 ≤ βσ1+qn and sn+1 ≤ sn + βσ1+qn ≤ . . . ≤ s2 + β (1+q)−1 1+q−1 σ 1+q 2 + β (1+q)2−1 1+q−1 σ (1+q)2 2 + . . . + β (1+q)n−1−1 1+q−1 σ (1+q)n−1 2 = s2 + β −1 q (δ1+q + δ2(1+q) + . . . + δ(n−1)(1+q)) (δ < 1) = s2 + β −1 q δ1+q 1 − (δ1+q)n−1 1 −δ1+q < s2 + β −1 q δ1+q 1 −δ1+q = s2 + β −1 q h 1 −h = γ0. hence, estimate s q n+1 ≤ γholds if f3(t) ≤ 0 at t = η3, which is estimate (2.5). the induction for assertion (2.10) is completed.it follows that estimate (2.8) holds. notice δ = βσ2 = lα 1 + q kσ 1+q 1 (1 + q)(1 −k0s q 1 ) < 1 (2.11) also holds since it is equivalent to the second estimate in (2.4). let n = 2, 3, . . . . then, it followsin turn by assertion (2.8) sj+n − sj+1 ≤ σj+n + σj+n−1 + . . . + σj+2 ≤ β− 1 q (δ(1+q) j+n−2 + δ(1+q) j+n−1 + . . . + δ(1+q) j ) ≤ β− 1 q δ(1+q) n δ1+q 1 −δ2n−1 1 −δ1+q . (2.12) then, assertion (2.9) follows from estimate (2.12) if n −→∞. � remark 2.3. an at least as large parameter as η3 can replace it in condition (2.7) as follows. define sequences of functions ϕn on the interval t by ϕn(t) = (s2(t) + β −1 q (δ(t)1+q + δ(t)(1+q) 2 + . . . + δ(t)(1+q) n−1 )q −γ. (2.13) https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 5 it follows by these definition that ϕn+1(t) −ϕn(t) ≥ 0, so ϕn(t) ≤ ϕn+1(t) ∀t ∈ t. (2.14) moreover these functions have zeros in t0. these zeros are assured to exist by (ivt), since by the definitions of fucntions ϕn give ϕn(0) = −γ < 0 and ϕn(t) −→ ∞ as t −→ u−. denote the smallest such zeros of functions ϕn in t by rn, respectively. according to the proof of lemma 2.2, lim n−→∞ ϕn(t) ≤ f3(t) ∀t ∈ t. (2.15) so, this limit exists as a well defined function denoted by ψ. then, this function has zeros in t0, since ψ(0) = −γ and ψ(t) −→ ∞ as t −→ u−. denote by η4 the smallest such zero in (0,u). clearly, the proof of lemma 2.2 goes through if instead of f3(t) ≤ 0, it is shown that ψ3(t) ≤ 0 ∀t ∈ t. (2.16) define parameter η̄0 = min{η1,η2,η4}. notice that by (2.15) ψ(η3) ≤ f3(η3) = 0, so η3 ≤ η4 and consequenlty η0 ≤ η̄0. if η̄0 replaces η0 in condition (2.6), then assertion (2.16) follows. condition (2.7) becomes η ≤ η̄0. (2.17) hence, the range of initial approximations η is further extended. 3. convergence of nm the notation u(w,ρ),u[w,ρ] means the open and closed balls with radius ρ > 0 and center w ∈ x, respectively. the parameters k0,l0,k,l and t are connected with operator f as follows.consider conditions (a):suppose (a1) there exist x0 ∈ ω, t ≥ 0 such that f ′(x0)−1 ∈ l(m2,m1), ‖f ′(x0)−1f (x0)‖≤ t. ‖f ′(x0)−1(f ′(x1) −f ′(x0))‖≤ k0‖x1 −x0‖qand ‖f ′(x0)−1(f ′(x0 + τ(x1 −x0)) −f ′(x0))‖≤ k‖τ(x1 −x0)‖q.(a2) ‖f ′(x0)−1(f ′(x) −f ′(x0))‖≤ l0‖x −x0‖q, ∀x ∈ ω. set b1 = u(x0, ( 1l0 ) 1q ) ∩ ω.(a3) ‖f ′(x0)−1(f ′(x + τ(y −x)) −f ′(x))‖≤ l‖τ(y −x)‖q ∀x,y ∈ b1 and ∀τ ∈ [0, 1).(a4) conditions of lemma 2.1 or lemma 2.2 hold(a5) u[x0,t∗] ⊂ ω. https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 6 notice that k0 ≤ k ≤ l0.next, conditions a are applied to show the main convergence result for ni. theorem 3.1. under conditions a sequence ni is convergent to a solution x∗ ∈ u[x0,s∗] of equation f (x∗) = 0. moreover, upper bounds ‖x∗ −xn‖≤ s∗ − sn (3.1) hold ∀n = 0, 1, 2, . . . . proof. the items ‖xi+1 −xi‖≤ si+1 − si, (3.2)and u[xi+1,s ∗ − si+1] ⊆ u[xi,s∗ − si ], (3.3)are shown by induction ∀i = 0, 1, 2, . . . . let u ∈ u[x1,s∗ − s1]. it follows by condition (a1) ‖x1 −x0‖ = ‖f ′(x0)−1f (x0)‖≤ t = s1 − s0, ‖u −x0‖≤‖u −x1‖ + ‖x1 −x0‖≤ s∗ − s1 + s1 − s0 = s∗. hence, point u ∈ u[x0,s∗−s0]. that is items (3.2) and (3.3) hold for i = 0. assume these assertionshold if i = 0, 1, . . . ,n. it follows for each ξ ∈ [0, 1] ‖xi + ξ(xi+1 −xi ) −x0‖ ≤ si + ξ(si+1 − si ) ≤ s∗, and ‖xi+1 −xi‖≤ i+1∑ j=1 ‖xj −xj−1‖≤ i+1∑ j=1 (sj − sj−1) = si+1. it follows by induction hypotheses, lemmas and conditions (a1) and (a2) ‖f ′(x0)−1(f ′(xi+1) −f ′(x0))‖ ≤ k̄‖xi+1 −x0‖q, ≤ k̄(si+1 − s0)q ≤ k̄s q i+1 < 1. hence, the inverse of linear operator f ′(xi+1) exists. therefore, hence, f ′(v)−1 ∈ l(m2,m1) and ‖f ′(xi+1)−1f ′(x0)‖≤ 1 1 − k̄sq i+1 ) , (3.4) follows as a consequence of a lemma on invertible linear operators due to banach [2, 7], where k̄ = { k0, i = 0 l0, i = 1, 2, . . . .ni gives https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 7 f (xi+1) = f (xi+1) −f (xi ) −f ′(xi )(xi+1 −xi ), = ∫ 1 0 (f ′(xi + ξ(xi+1 −xi ))dξ−f ′(xi ))(xi+1 −xi ). (3.5) then, using induction hypotheses, identity (a3) and condition (??) ‖f ′(x0)−1f (xi+1)‖ ≤ l̄ ∫ 1 0 (‖xi+1 −xi‖)q (3.6) ≤ l̄ 1 + q (si+1 − si )1+q, where l̄ = { k, i = 0 l, i = 1, 2, . . . .it follows by ni, estimates (3.4), (3.6) and the definition (2.1) of sequence {sn} ‖xi+2 −xi+1‖ ≤ ‖f ′(xi+1)−1f ′(x0)‖‖f ′(x0)−1f (xi+1)‖, ≤ k̃(si+1 − si )2 2(1 − l̃si+1) = si+2 − si+1, where k̃ = { k, i = 0 l, i = 1, 2, . . . . and l̃ = { k0, i = 0 l0, i = 1, 2, . . . . moreover, if v ∈ u[xi+2,s∗ − si+2] it follows ‖v −xi+1‖ ≤ ‖v −xi+2‖ + ‖xi+2 −xi+1‖ ≤ s∗ − si+2 + si+2 − si+1 = s∗ − si+1. hence, point w ∈ u[xi+1,s∗ − si+1] completing the induction for items (3.2) and (3.3). noticethat scalar majorizing sequence {si} is fundamental as convergent. hence, the sequence {xi} isalso convergent to some x∗ ∈ u[x0,s∗]. furthermore, let i −→ ∞ in estimate (3.6), to conclude f (x∗) = 0. �next, the uniqueness ball for a solution is presented. notice that not all condition a are used. proposition 3.2. under center-lipschitz condition (a2) further suppose the existence of a solution p ∈ u(x0, r) ⊂ ω of equation (1.1) such that operator f ′(p) is invertible for some r > 0; a parameter r1 ≥ r given by r1 = ( 1 + q l0 − rq )1 q . (3.7) then, the poiny p solves uniquely equation f (x) = 0 in the domain b2 = u(x0, r1) ∩ ω. https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 8 proof. define linear operator q = ∫1 0 f ′(p̄ + ξ(p − p̄))dξ for some point p̄ ∈ b2 satisfying f (p̄) = 0. by using the definition of r1, set b2 and condition (a2) ‖f ′(x0)−1(f ′(x0) −q)‖ ≤ ∫ 1 0 l0((1 −ξ)‖x0 −p‖q + ξ‖x0 − p̄‖q)dξ, < l0 1 + q (r q 1 + r q) = 1, concluding that p = p̄, where the invertability of linear operator is also used together with theapproximation 0 = f (p) −f (p̄) = q(p− p̄). � remark 3.3. (1) if conditions a hold, set p = x∗ and r = s∗ in proposition 3.2. (2) lipschitz condition (a3) can be replaced by ‖f ′(x0)−1(f ′(z1 + τ(z2 −z1)) −f ′(z1))‖≤ d‖τ(z1 −z2)‖q (3.8) for all z1 ∈ b1 and z2 = z1 −f ′(z1)−1f (z1) ∈ b1. this even smaller parameter d can replace l in the previous results. the existence of iterate z2 is assured by (a2). 4. numerical experimentation three experimenta are considered in this section. example 4.1. the parameters using example of the introduction are k0 = µ+53 ,k = l0 = µ+11 6 . moreover,ω0 = u(1, 1 −µ) ∩u(1, 1l0 ) = u(1, 1 l0 ). set l = 2(1 + 1 3−µ) l0 < l1 and l < l1 for all µ ∈ (0, 0.5). the kantorovich criterion η ≤ 1 l1 is violated, since η > 1 l1 ∀µ ∈ (0, 0.5), where l1 is the lipschitz constant on ω. interval can be enlarged if condition of lemma 2.1 is verified. then, for µ = 0.4, we have the following; 1 l0 = 0.3846, table 1. sequence (2.1) n 1 2 3 4 5 6 7 sn+1 0.2000 0.2594 0.2744 0.2755 0.2755 0.2755 0.2755 hence conditions of lemma 2.1 hold. hence condition (2.2) holds, and the interval is extended form ∅ to [0.4,o.5]. example 4.2. let us consider the two point pbvp(tpbvp) u′′ + u 3 2 = 0 u(0) = u(1) = 0. https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 9 the interval [0, 1] is divided into j subintervals. set m = 1 j . denote by w0 = 0 < w1 < ... < wj = 1 the points of subdivision with corresponding values of the function u0 = u(w0), . . . ,uj = u(wj). then, the discretization of u′′ is given by u′′k ≈ uk−1 − 2uk + uk+1 m2 , ∀k = 2, 3, . . . j − 1. notice that u0 = uj = 0. it follows that the following system of equations is obtained m2u 3 2 1 − 2u1 + u2 = 0, uk−1 + m 2u 3 2 k − 2uk + uk+1 = 0, ∀k = 2, 3, . . . , j − 1 uj−2 + m 2u 3 2 j−1 − 2uj−1 = 0. this system can be converted into an operator equation as follows: define operator g : rj−1 −→ rj−1 whose derivative is given as g′(u) =  3 2 m2u 1 2 1 − 2 1 0 . . . 0 1 3 2 m2u 1 2 2 − 2 1 0 . . . 0 . . . . . . . . . . . . ... ... ... ... ... 0 · · · 1 0 3 2 m2u 1 2 j−1 − 2  . let z ∈ rj−1 be arbitrary. the norm is ‖z‖ = max1≤k≤j−1‖zk‖, where as the norm for g ∈ rj−1 ×rj−1 is given as ‖g‖ = max 1≤k≤j−1 j−1∑ i=1 ‖gk,i‖. then, if u,z ∈rj−1 for |uk| > 0, |zk| > 0, ∀k = 1, 2, . . . , j − 1 to obtain in turn ‖g′(u) −g′(z)‖ = ‖diag{ 3 2 (u 1 2 k −z 1 2 k )}‖ = 3 2 m2 [ max 1≤k≤j−1 |uk −zk| ]1 2 = 3 2 m2‖u −z‖ 1 2 . choose as an initial guess vector 130 sin πx to obtain after four iterations u0 = [3.35740e + 01, 6.5202e + 01, 9.15664e + 01, 1.09168e + 02, 1.15363e + 02, 1.09168e + 02, 9.15664e + 01, 6.52027e + 01, 3.35740e + 01]tr ]. then, the parameters are ‖q′(u0)−1‖≤ 2.5582e + 01,η = 9.15311e − 05, q = 0.5,k0 = l0 = k = l = 3200 = 0.015. then, k0η p = 1.4351e − 04 and the following table shows that the conditions of lemma 2.1 are satisfied. example 4.3. let m1 = m2 = c[0, 1] be the set of continuous real functions on [0, 1]. the norm-max is used. set ω = u[x0, 3]. consider hammerstein nonlinear integral operator h [3, 6] on https://doi.org/10.28924/ada/ma.3.5 eur. j. math. anal. 10.28924/ada/ma.3.5 10table 2. sequence (2.1) n 1 2 3 4 5 6 vn+1 0.1435e-03 0.1435e-03 0.1435e-03 0.1435e-03 0.1435e-03 0.1435e-03 ω as h(v)(z1) = v(z1) −y(z1) − ∫ 1 0 v(z1,z2)v3(z2)dz1 = 2, v ∈ c[0, 1],z1 ∈ [0, 1]. (4.1) where function y ∈ c[0, 1], and v is a kernel related by green’s function v(z1,z2) = { (1 −z1)z2, z2 ≤ z1 z2(1 −z1), z1 ≤ z2. (4.2) it follows by this definition that h′ is [h′(v)(z)](z1) = z(z1) − 3 ∫ 1 0 v(z1,z2)v2(z2)z(z2)dz2 (4.3) z ∈ c[0, 1],z1 ∈ [0, 1]. pick x0(z1) = y(z1) = 1. it then follows from (4.1)-(4.3) that h′(x0)−1 ∈ l(m2,m1), ‖i −h′(x0)‖ < 0.375, ‖h′(x0)−1‖≤ 1.6, η = 0.2, l0 = 2.4, l1 = 3.6, and ω0 = u(x0, 3) ∩ u(x0, 0.4167) = u(x0, 0.4167), so l = 1.5. notice that l0 < l1 and l < l1. set k0 = k = l0. the kantorovich convergence criterion (a3) is not satisfied, since 2l1η = 1.44 > 1. therefore convergence of ni is not guaranteed. however, the new condition (2.7) is satisfied, since 2lη = 0.6 < 1. 5. conclusions an updated and weaker unified framework is presented for ni. the new analysis is finer thanbefore. convergence order 1 + q is also recovered by choosing a larger upper bound on t. newlipschitz or hölder parameters are smaller and specilizations of previous parameters. the newtheory can always replace previous ones due to weaker criterion. the strategy can be applied onother iterations [2, 3, 7, 11]. references [1] j. appell, e.d. pascale, j.v. lysenko, p.p. zabrejko, new results on newton-kantorovich approximations with appli-cations to nonlinear integral equations, numer. funct. anal. optim. 18 (1997) 1–17. https://doi.org/10.1080/ 01630569708816744.[2] i.k. argyros, unified convergence criteria for iterative banach space valued methods with applications, mathematics.9 (2021) 1942. https://doi.org/10.3390/math9161942.[3] i.k. argyros, the theory and applications of iteration methods, 2nd edition, crc press, boca raton, 2022. https: //doi.org/10.1201/9781003128915. https://doi.org/10.28924/ada/ma.3.5 https://doi.org/10.1080/01630569708816744 https://doi.org/10.1080/01630569708816744 https://doi.org/10.3390/math9161942 https://doi.org/10.1201/9781003128915 https://doi.org/10.1201/9781003128915 eur. j. math. anal. 10.28924/ada/ma.3.5 11 [4] s. singh, e. martínez, p. maroju, r. behl, a study of the local convergence of a fifth order iterative method, indianj. pure appl. math. 51 (2020) 439–455. https://doi.org/10.1007/s13226-020-0409-5.[5] f. cianciaruso, e. de pascale, newton–kantorovich approximations when the derivative is hölderian: old and newresults, numer. funct. anal. optim. 24 (2003) 713–723. https://doi.org/10.1081/nfa-120026367.[6] j.a. ezquerro, m.a. hernandez, newton’s scheme: an updated approach of kantorovich’s theory, cham switzerland,2018.[7] l.v. kantorovich, g.p. akilov, functional analysis, pergamon press, oxford, 1982.[8] f.a. potra, v. pták, nondiscrete induction and iterative processes, research notes in mathematics, 103. pitman(advanced publishing program), boston, 1984.[9] p.d. proinov, new general convergence theory for iterative processes and its applications to newton–kantorovichtype theorems, j. complex. 26 (2010) 3–42. https://doi.org/10.1016/j.jco.2009.05.001.[10] r. verma, new trends in fractional programming, nova science publisher, new york, usa, 2019.[11] t. yamamoto, historical developments in convergence analysis for newton’s and newton-like methods, j. comput.appl. math. 124 (2000) 1–23. https://doi.org/10.1016/s0377-0427(00)00417-9.[12] p.p. zabrejko, d.f. nguen, the majorant method in the theory of newton-kantorovich approximations and the ptákerror estimates, numer. funct. anal. optim. 9 (1987) 671–684. https://doi.org/10.1080/01630568708816254. https://doi.org/10.28924/ada/ma.3.5 https://doi.org/10.1007/s13226-020-0409-5 https://doi.org/10.1081/nfa-120026367 https://doi.org/10.1016/j.jco.2009.05.001 https://doi.org/10.1016/s0377-0427(00)00417-9 https://doi.org/10.1080/01630568708816254 1. introduction 2. majorization of ni 3. convergence of nm 4. numerical experimentation 5. conclusions references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 7doi: 10.28924/ada/ma.2.7 on the stratonovich estimator for the itô diffusion jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte, 376 fretwell bldg, 9201 university city blvd., charlotte, nc 28223-0001, usa correspondence: j.bishwal@uncc.edu abstract. for the parameter appearing non-linearly in the drift coefficient of homogeneous itô sto-chastic differential equation having a stationary ergodic solution, the paper obtains the strong con-sistency of an approximate maximum likelihood estimator based on stratonovich type approximationof the continuous girsanov likelihood, under some regularity conditions, when the corresponding dif-fusion is observed at equally spaced dense time points over a long time interval in the high frequencyregime. pathwise convergence of stochastic integral approximations and their connection to discretedrift estimators is studied. often it is shown that discrete drift estimators converge in probability. weobtain convergence of the estimator with probability one. ornstein-uhlenbeck process is consideredas an example. 1. introduction and preliminaries parameter estimation in diffusion processes based on discrete observations is being paid a lot ofattention now a days in view of its application in many fields such as biology, physics, oceanograpgyand especially in finance, see kutoyants (2004) and bishwal (2008, 2021).consider the itô stochastic differential equation dxt = f (θ,xt)dt + dwt, t ≥ 0 x0 = x 0 (1.1) where {wt,t ≥ 0} is a one dimensional standard wiener process, θ ∈ θ, θ is a compact subsetof r, f is a known real valued function defined on θ × r, the unknown parameter θ is to beestimated on the basis of observation of the proces {xt,t ≥ 0}. let θ0 be the true value of theparameter which is in the interior of θ. we assume that the process {xt,t ≥ 0} is observed at 0 = t0 < t1 < ... < tn = t with ∆ti := ti − ti−1 = tn = h, i = 1, 2, . . . ,n and t = dn1/2for some fixed real number d > 0. we estimate θ from the observations {xt0,xt1, . . . ,xtn}. this received: 5 dec 2021. key words and phrases. itô stochastic differential equation; stratonovich integral; diffusion process; discrete ob-servations; high frequency; approximate maximum likelihood estimators; conditional least squares estimator; strongconsistency; monte carlo methods. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 2 model was first studied by dorogovcev (1976) who obtained weak consistency of the conditionalleast squares estimator (clse) under some regularity conditions as t →∞ and t n → 0. kasonga(1988) obtained the strong consistency of the clse under some regularity conditions as n → ∞assuming that t = dn1/2 for some fixed real number d > 0.note that the conditional least squares estimator (clse) of θ is defined as θn,t := arg min θ∈θ qn,t (θ) where qn,t (θ) = n∑ i=1 [ xti −xti−1 − f (θ,xti−1 )h ]2 ∆ti . note that the clse, the euler-maruyama estimator and the iamle are the same estimator(see shoji (1997)). for the ornstein-uhlenbeck process, bishwal and bose (2001) studied therates of weak convergence of approximate maximum likelihood estimators, which are of conditionalleast squares type. for the ornstein-uhlenbeck process bishwal (2010a) studied uniform rate ofweak convergence for the minimum contrast estimator, which has close connection to stratonovich-milstein scheme. bishwal (2009a) studied berry-esseen inequalities for conditional least squaresestimator discretely observed nonlinear diffusions. bishwal (2009b) studied stratonovich basedapproximate m-estimator of discretely sampled nonlinear diffusions. bishwal(2011a) studied mil-stein approximation of posterior density of diffusions. bishwal (2010b) studied conditional leastsquares estimation in nonlinear diffusion processes based on poisson sampling. bishwal (2011b)obtained some new estimators of integrated volatility using the stochastic taylor type schemeswhich could be useful for option pricing in stochastic volatility models. in mathematical finance,almost sure optimal hedging has received recent attention. gobet and landon (2014) studied theoptimal discretization error in the context of hedging error in a multidimensional itô model wherethe convergence is studied in an almost sure sense and the discrete trading dates are stoppingtimes which includes the sampling scheme of karandikar (1995) who studied pathwise convergenceof stochastic integrals. bishwal (2011c) studied higher order approximation of hedging error inthe mean square sense. almost sure hedging and optimality of discretization error motivates ouralmost sure consistency in estimation problem.florens-zmirou (1989) studied minimum contrast estimator, based on an euler-maruyama typefirst order approximate discrete time scheme of the sde (1.1) which is given by zti −zti−1 = f (θ,zti−1 )(ti − ti−1) + wti −wti−1, i ≥ 1, z0 = x 0. the log-likelihood function of {zti, 0 ≤ i ≤ n} is given by c n∑ i=1 [ zti −zti−1 − f (θ,zti−1 )h ]2 ∆ti . https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 3 where c is a constant independent of θ. a contrast for the estimation of θ is derived from the abovelog-likelihood by substituting {zti, 0 ≤ i ≤ n} with {xti, 0 ≤ i ≤ n}. the resulting contrast is hn,t = c n∑ i=1 [ xti −xti−1 − f (θ,xti−1 )h ]2 ∆ti . and the resulting minimum contrast estimator, called the euler estimator, is θ̌n,t := arg min θ∈θ hn,t (θ) florens-zmirou (1989) showed l2 consistency of the estimator as t →∞ and tn → 0.if continuous observation of {xt} on the interval [0,t ] were available, then the likelihood functionof θ would be lt (θ) = exp {∫ t 0 f (θ,xt)dxt − 1 2 ∫ t 0 f 2(θ,xt)dt } , (1.2) (see liptser and shiryayev (1977)). in our case we have discrete data and we have to approximatethe likelihood to get the mle. taking itô type approximation of the stochastic integral and rectanglerule approximation of the ordinary integral in (1.2) and obtain the approximate likelihood function ln,t (θ) = exp { n∑ i=1 f (θ,xti−1 )(xti −xti−1 ) − h 2 n∑ i=1 f 2(θ,xti−1 ) } . (1.3) an approximate maximum likelihood estimate (amle) based on ln,t is defined as θ̂n,t := arg max θ∈θ ln,t (θ). weak consistency and other properties of this estimator were studied by yoshida (1992) as t →∞and t n → 0.note that the clse, the euler estimator and the amle1 are the same estimator (see shoji(1997)).in order to obtain a better estimator, which may have faster rate of convergence, we propose anew algorithm. note that the itô and the stratonovich integrals are connected by∫ t 0 f (θ,xt)dxt = ∫ t 0 f (θ,xt) o dxt − 1 2 ∫ t 0 ḟ (θ,xt)dt. (see ikeda and watanabe (1989)). we transform the itô integral in (1.2) to stratonovich integraland apply stratonovich type approximation of the stochastic integral and rectangular rule typeapproximation of the ordinary integrals and obtain the approximate likelihood l̃n,t (θ) = exp { 1 2 n∑ i=1 (f (θ,xti−1 ) + f (θ,xti ))(xti −xti−1 ) − h 2 n∑ i=1 (ḟ (θ,xti−1 ) + f 2(θ,xti−1 )) } . (1.4) https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 4 the stratonovich approximate maximum likelihood estimator (samle) based on ∼ln,t is defined as θ̃n,t := arg max θ∈θ ∼ ln,t (θ). this estimator is known to have faster rate of convergence (in the mean square sense) than theconditional least squares estimator, see bishwal (2009b).for monte carlo simulations in finance, one would be interested for pathwise convergence ofthe estimator. in this paper prove the strong consistency of the samle under some regularityconditions given below as n → ∞. we shall use the following notations : ∆xi = xti − xti−1 , ∆wi = wti − wti−1 , c is a generic constant independent of h,n and other variables (perhaps itmay depend on θ). prime denotes derivative w.r.t. θ and dot denotes derivative w.r.t. x. supposethat θ0 denote the true value of the parameter and θ0 ∈ θ. we assume the following conditions:(a1) the parameter space θ is compact.(a2) |f (θ,x)| ≤ k(θ)(1 + |x|), |f (θ,x) − f (θ,y)| ≤ k(θ)|x −y|. |f (θ,x) − f (φ,y)| ≤ c(x)|θ−φ| for all θ,φ ∈ θ,x,y ∈r where sup θ∈θ |k(θ)| = k < ∞,e|c(x0)|m = cm < ∞ for some m > 16. (a3) the diffusion process x is stationary and ergodic with invariant measure ν, i.e., for any gwith e[g(·)] < ∞ 1 n n∑ i=1 g(xti ) → eν[g(x0)] a.s. as t →∞ and h → 0. further e|x0|m < ∞ for some m > 16.(a4) e|f (θ,x0) − f (θ,x0)|2 = 0 iff θ = θ0.(a5) f is twice continuously differentiable function in x with e sup t |ḟ (xt)|2 < ∞, e sup t |f̈ (xt)|2 < ∞. 2. main results we shall use the following theorem to prove the strong consistency of the samle. theorem 2.1 (frydman (1980). suppose the random function dn satisfy the following conditions: (c1) with probability one, dn(θ) → d(θ) uniformly in θ ∈ θ as n →∞. (c2) the limiting nonrandom function d is such that d(θ0) ≥ d(θ) for all θ ∈ θ. https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 5 (c3) d(θ) = d(θ0) iff θ = θ0. then θn → θ0 a.s. as n →∞, where θn = supθ∈θ dn(θ). we need the following lemmas in order to prove our main result. lemma 2.1 under (a1)(a5), sup θ∈θ 1 2t { n∑ i=1 [ v(θ,xti−1 ) + v(θ,xti ) ] ∆wi − h 2 n∑ i=1 [ v̇(θ,xti−1 ) + v̇(θ,xti ) ]} → 0 a.s. as t →∞, t n → 0. proof. let v(θ,x) := f (θ,x) − f (θ0,x). the fourier expansion of v(θ,x) in l(θ) be given by v(θ,x) = ∞∑ m=1 am(x)e πjmθ, j = √ −1, x ∈r where ak(x) are the fourier coefficients. thus 1 2t { n∑ i=1 [ v(θ,xti−1 ) + v(θ,xti ) ] ∆wi − h 2 n∑ i=1 [ v̇(θ,xti−1 ) + v̇(θ,xti ) ]} = 1 2t { ∞∑ m=1 n∑ i=1 [ am(xti−1 ) + am(xti ) ] eπjmθ∆wi − h 2 ∞∑ m=1 n∑ i=1 [ ȧm(xti−1 ) + ȧm(xti ) ] eπjmθ } where |am(x)| ≤ cm|x|, ∞∑ m=1 m1+γc4m < ∞. let am,n(s) := 1 2 n∑ i=1 [ am(xti−1 ) + am(xti ) ] i(ti−1−ti ](s) where i(ti−1−ti ], i = 1, 2, ...,n are indicator functions. then 1 2 n∑ i=1 [ am(xti−1 ) + am(xti ) ] ∆wi = ∫ t 0 am,n(s) o dws and h 2 n∑ i=1 [ ȧm(xti−1 ) + ȧm(xti ) ] = ∫ t o ȧm,nds. but ∫ t 0 am,n(s) o dws − 1 2 ∫ t o ȧm,nds = ∫ t 0 am,n(s)dws. https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 6 by exponential inequality for martingales, we have p {∫ t 0 am,n(s)dws − α 2 ∫ t o a2m,nds > β } ≤ e−αβ for any α,β > 0. thus p { 1 t ∫ t 0 am,n(s)dws > β t + α 2t ∫ t o a2m,nds } ≤ e−αβ and p {∣∣∣∣ 1t ∫ t 0 am,n(s)dws ∣∣∣∣ > βt + αh8t n∑ i=1 [ am(xti−1 ) + am(xti ) ]2} ≤ 2e−αβ. since h 2t n∑ i=1 [ am(xti−1 ) + am(xti ) ]2 ≤ c2m ht n∑ i=1 [ (xti−1 ) 2 + (xti ) 2 ] and by (a3) h 2t n∑ i=1 [ (xti−1 ) 2 + (xti ) 2 ] → e(x20 ) > 0 a.s., there exists a random variable v such that h 2t n∑ i=1 [ (xti−1 ) 2 + (xti ) 2 ] < v a.s. for all t > 0,n = 1, 2, . . . . where p (v < ∞) = 1.denote zm,n := 1 tn ∫ tn 0 am,n(s)dws. recall that t = tn. choose α := ma tδn , β := t γ n mb , where δ < γ < 1 and 1 2 < b < 1+γ 2 .then p ( |zm,n| > 1 t 1−γ n m b + mac2mv 2tδn ) < 2e−m a−bt γ−δ n . https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 7 this p ( ∞∑ m=1 z2m,n > ∞∑ m=1 ( 1 t 1−γ n m b + mac2mv 2tδn )2) ≤ ∞∑ m=1 p ( z2m,n > ( 1 t 1−γ n m b + mac2mv 2tδn )2) = ∞∑ m=1 p ( |zm,n| > 1 t 1−γ n m b + mac2mv 2tδn ) ≤ 2 ∞∑ m=1 e−m a−bt γ−δ n ≤ 2e−t γ−δ n ∞∑ m=1 e−m a−b . hence ∞∑ n=1 p ( ∞∑ m=1 z2m,n > ∞∑ m=1 ( 1 t 1−γ n m b + mac2mv 2tδn )2) ≤ 2 ∞∑ n=1 e−t 1−γ n ∞∑ m=1 e−m a−b < ∞ since γ −δ > 0 and a−b > 0. the above implies ∞∑ n=1 p ( ∞∑ m=1 z2m,n > 2 t 2(1−γ) n ∞∑ m=1 m−2b + v 2 t2δn ∑ m m2ac4m ) < ∞. by borel-cantelli lemma, ∞∑ m=1 ( 1 2tn n∑ i=1 [ am(xti−1 ) + am(xti ) ] ∆wi − h 2tn n∑ i=1 [ v̇(θ,xti−1 ) + v̇(θ,xti ) ])2 −→ 0 a.s. as n →∞. this completes the proof of the lemma. lemma 2.2 under (a1)– (a5), with probability one, sup θ∈θ ∣∣∣∣∣ 1t n∑ i=1 ∫ ti ti−1 [f (θ0,xs) − f (θ0,xti−1 )]v(θ,xti−1 )ds ∣∣∣∣∣ → 0. proof. for m > 0, we have e supθ∈θ ∣∣∣∣∣ 1t n∑ i=1 ∫ ti ti−1 [f (θ0,xs) − f (θ0,xti−1 )]v(θ,xti−1 )ds ∣∣∣∣∣ 2m  = e { sup θ∈θ ∣∣∣∣ 1t ∫ t 0 gn(s)ds ∣∣∣∣2m } . https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 8 where gn(s) = ∑ni=1 ∫ titi−1 [f (θ0,xs) − f (θ0,xti−1 )]v(θ,xti−1 ) if ti−1 ≤ s ≤ ti .hölder’s inequality implies that e { sup θ∈θ ∣∣∣∣ 1t ∫ t 0 gn(s)ds ∣∣∣∣2m } ≤ t−2me { sup θ∈θ t 2m−1 ∫ t 0 |gn(s)|2mds } ≤ t−2me ( sup θ∈θ t 2m−1 n∑ i=1 ∫ ti ti−1 |f (θ0,xs) − f (θ0,xti−1 )| 2m|v(θ,xti−1 )| 2mds ) ≤ t−1um n∑ i=1 ∫ ti ti−1 e(|f (θ0,xs) − f (θ0,xti−1 )| 2m|c(xti−1 )| 2mds) by condition (a2) where um := supθ∈θ |θ−θ0|2m < ∞.by cauchy-schwarz’s inequality the above term is ≤ t−1um n∑ i=1 ∫ ti ti−1 (e|f (θ0,xs) − f (θ0,xti−1 )| 4m)1/2(e(c(xti−1 )| 4m)1/2ds ≤ t−1umk2m(θ0)(e|c(x0)|4m)1/2 n∑ i=1 ∫ ti ti−1 (e|xs −xti−1 )| 4m)1/2ds by condition (a2). since e|xt −xs|2m ≤ m(t − s)m, from gikhman and skorohod (1975, p.48),the above term ≤ t−1umk2m(θ0)(e|c(x0)|4m)1/2m1/2 n∑ i=1 ∫ ti ti−1 (s − ti−1)mds = umk 2m(θ0)(e|c(x0)|4mm)1/2t−1 n∑ i=1 (∆ti ) m+1 m + 1 ≤ umk 2m(θ0) m + 1 (e|c(x0)|4mm)1/2hmn−m/2, m > 4. chebyshev’s inequality and the above implies that for any � > 0, ∞∑ n=1 p { sup θ∈θ ∣∣∣∣∣ 1t n∑ i=1 ∫ ti ti−1 [f (θ0,xs) − f (θ0,xti−1 )]v(θ,xti−1 )ds ∣∣∣∣∣ > � } < ∞. hence borel-cantelli lemma yields the result. lemma 2.3 under (a1)(a6), with probability one, 1 t n∑ i=1 [f (θ,xti−1 ) − f (θ0,xti−1 )] 2∆ti → e|v(θ,x0)|2 uniformly in θ as t →∞, t n → 0. https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 9 proof. by the strong law of large numbers (ergodicity), 1 t ∫ t 0 |v(θ,xs)2ds → e|v(θ,x0)|2. a.s. as t →∞ for each θ ∈ θ. the condition (a2) implies that 1 t ∫ t 0 |v(θ,xs)2ds ≤ 1 t |θ−θ0|2 ∫ t 0 |c(xs)|2ds ≤ sup θ∈θ |θ−θ0|2 1 t ∫ t 0 |c(xs)|2ds ≤ b almost surely for some random variable b by (a1), (a2) and (a3). it also follows easily by (a1)-(a4)that ∣∣∣∣ 1t ∫ t 0 |v(θ1,xs)2ds − 1 t ∫ t 0 |v(θ2,xs)2ds ∣∣∣∣ ≤ j|θ1 −θ2| almost surely for some random variable j and θ1,θ2 ∈ θ. thus the family of functions { 1 t ∫ t 0 |v(·,xs)|2ds, t ≥ 0 } is equicontinuous. hence by arzela-ascoli theorem, the convergence is uniform. denote g2n(θ) := h 2 n∑ i=1 [ (xti−1 ) 2 + (xti ) 2 ] . now it is enough to show that 1 t ∫ t 0 |v(θ,xs)|2ds − 1 t g2n(θ) → 0 a.s. uniformly in θ. we have e { sup θ∈θ | ∫ t 0 |v(θ,xs)2ds −g2n(θ)| 2m } e { sup θ∈θ | ∫ t 0 |v(θ,xs)2ds −h n∑ i=1 |v(θ,xti−1 )| 2|2m } = e { sup θ∈θ | n∑ i=1 ∫ ti ti−1 n∑ i=1 (v(θ,xs −v(θ,xti−1 ))(v(θ,xs + v(θ,xti−1 ))ds| 2m } . https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 10 hölder inequality implies the above expectation ≤ t 2m−1e sup θ∈θ n∑ i=1 { ∫ ti ti−1 |v(θ,xs −v(θ,xti−1 )| 2m|v(θ,xs + v(θ,xti−1 ))| 2m} ≤ t 2m−1 n∑ i=1 ∫ ti ti−1 e[sup θ∈θ |v(θ,xs −v(θ,xti−1 )| 2m sup θ∈θ |v(θ,xs + v(θ,xti−1 ))| 2m]ds ≤ t 2m−1k2m22mum n∑ i=1 ∫ ti ti−1 e[|xs −xti−1| 2m(|c(xs)|2m + |c(xti−1 ))| 2m]ds ≤ t 2m−1k2m22m+1um n∑ i=1 ∫ ti ti−1 (e|xs −xti−1 )| 4m)1/2(e|c(xs)|4m + e|c(xti−1 ))| 4m)1/2ds ≤ t 2m−1k2m22m+1umm1/2(e|c(x0)|2m))1/2 n∑ i=1 ∫ ti ti−1 (s − ti−1)mds| (by stationarity) ≤ rmt 2m−1n(t/n)m+1 where um := supθ∈θ |θ − θ0|2m < ∞ and rm := k2m22m+2umm1/2(e|c(x0)|4m)1/2. hence if m > 4, e { sup θ∈θ | 1 t ∫ t 0 |v(θ,xs)|2ds − 1 t g2n(θ)| 2m } ≤ rm(t/n)m ≤ rmhm/2n−m/2. borel-cantelli argument yields the result. now we are ready to present the main result of the paper: theorem 2.2 under the conditions (a1)-(a5), the samle is strongly consistent, i.e., θ̃n,t → θ0 a.s. as t →∞, t n → 0. proof. let ∼ l n,t (θ) := log ∼ ln,t (θ) and v(θ,x) := f (θ,x) − f (θ0,x). https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 11 note that 1 t [∼ l n,t (θ) − ∼ l n,t (θ0) ] = 1 2t n∑ i=1 [f (θ,xti−1 ) + f (θ,xti )](xti −xti−1 ) − 1 2t n∑ i=1 [f (θ0,xti−1 ) + f (θ0,xti )](xti −xti−1 ) − 1 2n n∑ i=1 [ḟ (θ,xti−1 ) − ḟ (θ0,xti−1 )] − 1 2n n∑ i=1 [f 2(θ,xti−1 ) − f 2(θ0,xti−1 )] = 1 2t { n∑ i=1 [ v(θ,xti−1 ) + v(θ,xti ) ] ∆wi −h n∑ i=1 v̇(θ,xti−1 ) } − 1 2n n∑ i=1 v2(θ,xti−1 ) − 1 t n∑ i=1 ∫ ti ti−1 v(θ,xti−1 )[f (θ0,xt) + f (θ0,xti−1 )]dt − 1 t n∑ i=1 ∫ ti ti−1 [v(θ,xti )f (θ0,xt) −v(θ0,xti−1 )f (θ0,xti−1 )]dt =: i1 − i2 − i3 − i4. let dn,t (θ) := 1 t [∼ l n,t (θ) − ∼ l n,t (θ0) ] . below lemma 2.1-2.3 show that dn,t (θ) → d(θ) a.s. as t →∞, t n → 0 where d(θ) := − 1 2 e|f (θ,x0) − f (θ0,x0)|2.thus condition (c1) of theorem 2.1 is satisfied. the limiting function d(θ) satisfies the conditions(c2) and (c3) of theorem. hence as a consequence of theorem 2.1 we obtain the result. https://doi.org/10.28924/ada/ma.2.7 eur. j. math. anal. 10.28924/ada/ma.2.7 12 3. ornstein-uhlenbeck process consider the ornstein-uhlenbeck process satisfying dxt = θxtdt + dwt, t ≥ 0, x0 = 0, θ < 0. the euler estimator (conditional least squares estimator) is given by θ̌n,t = ∑n i=1 xti−1 (xti −xti−1 ) h ∑n i=1 x 2 ti−1 . strong consistency of this estimator is obtained in kasonga (1988). as a consequence of theorem2.2, we obtain the strong consistency of three estimators with θ̃n,t = (x2t −t )/2 h ∑n i=1 x 2 ti−1 , θ̄n,t,3 = x2t/2 h ∑n i=1 x 2 ti−1 , θ̂n,t,2 = −t/2 h ∑n i=1 x 2 ti−1 . which are samle, yamle (young amle) and , amce respectively as t → ∞ and t/n → 0.samle is the linear combination of amce and yamle.define the continuous mle, ymle and mce respectively θt,1 = ∫t 0 xtdxt∫t 0 x2t dt , θt,2 = x2t/2∫t 0 x2t dt , θt,3 = −t/2∫t 0 x2t dt . interpreting ∫t 0 xtdxt to be the young (1936) integral, it equals x2t/2. belfadli et al. (2011)(see also el machkouri et al. (2016)) obtained the strong consistency of θt,2 as t → ∞. theyamle θ̄n,t,2 is the euler discretization of θt,2. lanksa (1979) obtained strong consistency ofthe mce θt,3 as t → ∞ whose euler discretetization is θ̂n,t,3. liptser and shiryayev (1978)obtained strong consistency of the mle θt,1 as t →∞ whose euler discretetization is θ̌n,t,1. concluding remark it would be interesting to extend the results of the paper to diffusions drivenby persistent fractional brownian motion which are neither markov processes nor semimartingales,but preserved long memory property of the model. references [1] r. belfadli, k. es-sebaiy and y. ouknine, parameter estimation for fractional ornstein-uhlenbeck processes:non-ergodic case, front. sci. eng. 1 (2011) 41-56. https://doi.org/10.34874/imist.prsm/fsejournal-v1i1. 26873.[2] j.p.n. bishwal, parameter estimation in stochastic differential equations, springer berlin heidelberg, berlin, hei-delberg, 2008. https://doi.org/10.1007/978-3-540-74448-1.[3] j.p.n. bishwal, berry–esseen inequalities for discretely observed diffusions, monte carlo methods and applications.15 (2009) 229-239. https://doi.org/10.1515/mcma.2009.013.[4] j.p.n. bishwal, m-estimation for discretely sampled diffusions, theory stoch. processes, 15 (31) (2) (2009b) 62-83. https://doi.org/10.28924/ada/ma.2.7 https://doi.org/10.34874/imist.prsm/fsejournal-v1i1.26873 https://doi.org/10.34874/imist.prsm/fsejournal-v1i1.26873 https://doi.org/10.1007/978-3-540-74448-1 https://doi.org/10.1515/mcma.2009.013 eur. j. math. anal. 10.28924/ada/ma.2.7 13 [5] j.p.n. bishwal, uniform rate of weak convergence of the minimum contrast estimator in the ornstein–uhlenbeckprocess, methodol. comput. appl. probab. 12 (2010) 323–334. https://doi.org/10.1007/s11009-008-9099-x.[6] j.p.n. bishwal, conditional least squares estimation in diffusion processes based on poisson sampling, j. appl.probab. stat. 5 (2) (2010b) 169-180.[7] j.p.n. bishwal, milstein approximation of posterior density of diffusions, int. j. pure appl. math. 68 (4) (2011a)403-414.[8] j.p.n. bishwal, some new estimators of integrated volatility, amer. open j. stat. 1 (2) (2011b) 74-80.[9] j.p.n. bishwal, stochastic moment problem and hedging of generalized black–scholes options, appl. numer. math.61 (2011) 1271–1280. https://doi.org/10.1016/j.apnum.2011.08.005.[10] j.p.n. bishwal, parameter estimation in stochastic volatility models, springer nature switzerland ag (forthcoming).(2021).[11] j.p.n. bishwal, a. bose, rates of convergence of approximate maximum likelihood estimators in the ornstein-uhlenbeck process, computers math. appl. 42 (2001) 23–38. https://doi.org/10.1016/s0898-1221(01) 00127-4.[12] m. el machkouri, k. es-sebaiy, y. ouknine, least squares estimator for non-ergodic ornstein-uhlebeck processesdriven by gaussian processes, j. korean stat. soc. 45 (2016) 329-341.[13] d. florens-zmirou, approximate discrete-time schemes for statistics of diffusion processes, statistics. 20 (1989)547–557. https://doi.org/10.1080/02331888908802205.[14] r. frydman, a proof of the consistency of maximum likelihood estimators of nonlinear regression models withautocorrelated errors, econometrica. 48 (1980) 853-860. https://doi.org/10.2307/1912936.[15] e. gobet, n. landon, almost sure optimal hedging strategy, ann. appl. probab. 24 (2014). https://doi.org/10. 1214/13-aap959.[16] n. ikeda, s. watanabe, stochastic differential equations and diffusion processes, second edition, north-holland,amsterdam (kodansha ltd., tokyo). 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prep; dual protection; hiv/aids; stability analysis.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.21 https://orcid.org/0000-0002-6442-4211 eur. j. math. anal. 10.28924/ada/ma.3.21 2 use for men who have sex with men is 70% [3]. proper use (correctly and consistently) as well asquality concerns have been directly attributed to the success of this approach.in 2012, the u.s food and drug administration (fda) approved the use of truvada for prep asan oral pill taken once a day [14]. numerous efficacy trials( by iprex, partners prep, tdf2,e.t.c) have since been conducted to ascertain the potential of prep to prevent hiv infection. theiprex trial demonstrated that prep has the potential of reducing the risk of hiv infection amongtransgender women, bisexual men, as well as men who have sex with men [8]. two major studies;partners prep, and tdf2 demonstrated the effectiveness of prep among heterosexual men andwomen. out of all these studies, none displayed a 100% effectiveness [11]. adherence has beenfound to be directly correlated with the effectiveness of prep [11]. in the absence of adherence,which guarantees efficacy, prep failures have been characterized by; system failures, people fail-ures, doctor failures, drug failures, as well as assay failures [9]. these failures expose prep usersto the risk of hiv infection hence the need for additional protection whenever prep has beenutilized.the nature of storage, date of manufacture, religious as well as socio-cultural beliefs also influencehow each hiv prevention venture is utilized. the challenges experienced when various approachesare employed in an attempt to control the spread of hiv infection in a high risk population form thebasis for the need to use dual protection in order to achieve maximum protection. a combinationprevention approach as proposed by [6], based on proven efficacy interventions, provides one withthe best opportunity to curb the spread of hiv among the high risk population.in this study, we propose a mathematical model of dual protection against hiv infection by the useof condom and prep, and adherence to art treatment, while focusing on the high risk populationcollectively. earlier studies have either narrowed down to a particular category of persons at highrisk of infection [3], or have used a combination of prevention techniques where one techniqueacts as a supplement to the other [4], [5]. the study will focus on the impact of dual protectionon reducing the number of new infections, and that of art adherence in ensuring those who areinfected remain less infectious. 2. model formulation and description the population is subdivided into the classes; susceptible, infected, and aids individuals. thesusceptible class has been further subdivided into two compartments on the basis of degree ofrisk of infection. these include susceptible individuals at high risk of infection, denoted by (sh),and those at low risk, denoted by (sl). the high risk population incorporates mainly commercialsex workers, men who have sex with men (msm), and hiv-discordant couples [8]. the infectedclass is subdivided into two compartments; those who are unaware of their hiv status (i), andthose who have been diagnosed and consequently enrolled for treatment (td). the individualswho are unaware of their hiv status may progress to the td compartment after successful hiv https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 3 awareness campaigns that will persuade them to get tested, or when they develop hiv symptomsand consequently enroll for art treatment. if art treatment fails, the individual progresses to theaids compartment. this happens when there is lack of adherence to art, which allows the virusto multiply, thus increasing the plasma viral load. this results in weakening of the immune systemand hence the aids symptoms begin to manifest. the aids compartment comprises of those whoposses full blown symptoms, and are mostly bedridden, they thus do not significantly contribute tothe spread of the disease. exit from the aids class is through natural death.thus, considering apopulation of size n(t), at a time t, n(t) = sh(t) + sl(t) + i(t) + td(t) + a(t). (1) the following interventions have been incorporated in the model;(a) 0 ≤ φ1 ≤ 1 measures prep effectiveness, including its awareness and proper use as a meansto prevent susceptible individuals from being infected. thus, (1 −φ1) measures prep failure.(b) 0 ≤ φ2 ≤ 1measures condom effectiveness as a result of proper use, following adequateawareness campaigns and availability. thus, (1 −φ2) measures condom failure.(c) 0 ≤ φ3 ≤ 1 measures the efficacy of art treatment, including uptake with proper adherence,with the aim of reducing the plasma viral load and reconstructing the individual’s immune systemhence making them less infectious.movement of individuals from the susceptible to infected and then to the aids classes is illustratedby the compartmental model shown in figure 1. sh sl a td i �� (� − �)� (� − ��)(� − ��)��� (� − ��)��� �sh �sl ��� (� + �)� �i �� ��� (� − ��)�� figure 1. compartmental model. https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 4 the following symbols will be used to represent various phenomena as described in table 1. symbol description λ constant rate of recruitment of susceptible upon becoming sexually active. δ proportion of susceptible individuals at high risk of infection. (1 −δ) proportion of susceptible population at low risk of hiv infection. λ rate of acquisition of an infection by susceptibles.it is given by; λ = ( β1i+β2td n ), where β1, and β2 are the mean contact rates for thesusceptible individuals with i and td respectively. µ natural removal rate by death. σ aids induced mortality. α represents the proportion of infected individuals who upon beingtested and found to be hiv positive,they enroll for art treatment. γ2 represents the proportion of infected individuals who do not get testedhence remain undiagnosed until they begin to exhibit aids symptoms.table 1. table showing symbols and their description from the dynamics described above, the following system of ordinary differential equations isformulated. dsh dt = δλ − (1 −φ1)(1 −φ2)λsh −µsh dsl dt = (1 −δ)λ − (1 −φ2)λsl −µsl di dt = (1 −φ1)(1 −φ2)λsh + (1 −φ2)λsl −αi −γ2i −µi dtd dt = αi − (γ3 + µ)td da dt = γ2i + γ3td − (µ + σ)a. (2) 3. model analysis it can be shown that the solutions for the system of ordinary differential equations (2) areall positive and bounded for all t > 0, with positive initial conditions in the feasible region γ ={ (sh(t),sl(t), i(t), (td(t),a(t)) ∈r5+ : n(t) ≤ λ µ } . it therefore suffices to study the dynamicsof the system (2) in this region.the mathematical model developed in (2) has two unique equilibrium points, that is, the diseasefree equilibrium (d.f.e), and the endemic equilibrium (e.e). the d.f.e is obtained by setting https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 5 i = td = a = 0 in (2) to yield e0 = ( δλ µ , (1 −δ)λ µ , 0, 0, 0 ) . (3) the basic reproduction number (r0) of the system (2), computed using the next generation matrixapproach [10] is given by r0 = β3 q1 + αβ4 q1q2 . (4) by [10, theorem 2], the following result is thus established. theorem 3.1. the disease free equilibrium of the model (2), e0 = (δλµ , (1−δ)λµ , 0, 0, 0), is locally asymptotically stable whenever r0 < 1 and unstable otherwise. proof. the proof follows immediately from the computation of r0 above and theorem 2 of van dendriessche and watmough [10]. � mathematically, theorem (3.1) implies that whenever there is a small perturbation on the system,the system returns to the disease free equilibrium. epidemiologically, this implies that when a fewhiv infectious individuals are introduced in a population that is fully susceptible to hiv infection,the disease dies out whenever r0 < 1, otherwise, the disease will spread. it is therefore necessaryto show that eliminating hiv in a population is independent of the size of the initial sub-populationby proving the global asymptotic stability of the disease free equilibrium. theorem 3.2. the disease free equilibrium e0 = ( δλ µ , (1−δ)λ µ , 0, 0, 0 ) of the system (2) is globally asymptotically stable whenever r0 < 1. proof. castillo chavez’s theorem [1] is used to analyze the global asymptotic stability of the math-ematical model (2) such that e0 = (x∗, 0), x = (sh,sl) and z = (i,td,a).now; f (x,0) = ( δλ −µsh (1 −δ)λ −µsl ) and g(x,z) = pz − g̃(x,z). matrix p is given by h1β1sh n + (1 −φ2)β1sl n − (α + γ2 + µ) h1β2sh n + (1 −φ2)β2sl n 0 α −(γ3 + µ) 0 γ2 γ3 −(σ + µ)  , where h1 = (1 −φ1)(1 −φ2), and pz is given by h1β1ish n + (1 −φ2)β1isl n − (α + γ2 + µ) + h1β2tdsh n + (1 −φ2)β2tdsl n αi − (γ3 + µ)td γ2i + γ3td − (σ + µ)a  . https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 6 moreover, g(x, z) is given by (1 −φ1)(1 −φ2) ( β1i + β2td n ) sh + (1 −φ2) ( β1i + β2td n ) sl − (α + γ2 + µ)i αi − (γ3 + µ)td γ2i + γ3td − (σ + µ)a  , and therefore g̃(x,z) = pz − g(x,z)=  g̃1(x,z) g̃2(x,z) g̃3(x,z) =  0 0 0  . hence conditions h1 and h2 are satisfied. also from theorem (3.1), e0 is locally asymptotically stable whenever r0 < 1. thereforefollowing castillo chavez’s theorem, e0 is globally asymptotically stable whenever r0 < 1, asdesired. � this implies that with a large perturbation of the disease free equilibrium, solutions of the modelrepresented by the system (3.2) converge to d.f.e whenever r0 < 1. epidemiologically, this impliesthat if a sufficiently large number of hiv infected individuals are introduced in a population thatis fully susceptible to hiv infection, the disease will die out whenever r0 < 1. 3.1. existence of the endemic steady state. theorem 3.3. an endemic equilibrium point e1 = (s∗∗h ,s ∗∗ l , i ∗∗,t∗∗d ,a ∗∗), of the system (2) exists whenever r0 > 1. proof. equating the right hand side of each equation in the system (2) to zero and simplifyingyields; δλ − (1 −φ1)(1 −φ2) ( β1i ∗∗ + β2t ∗∗ d n ) s∗∗h −µs ∗∗ h = 0, (5) (1 −δ)λ − (1 −φ2) ( β1i ∗∗ + β2t ∗∗ d n ) s∗∗l −µs ∗∗ l = 0, (6) (1 −φ1)(1 −φ2) ( β1i ∗∗ + β2t ∗∗ d n ) s∗∗h + (1 −φ2) ( β1i ∗∗ + β2t ∗∗ d n ) s∗∗l − (α + γ2 + µ)i ∗∗ = 0, (7) αi∗∗ − (γ3 + µ)t∗∗d = 0, (8) γ2i ∗∗ + γ3t ∗∗ d − (µ + σ)a ∗∗ = 0. (9) from equation (8), t∗∗d = αq2 i∗∗.substituting for t∗∗td in equation (9) and simplifying gives a∗∗ = ( γ2q3 + αγ3q2q3 ) i∗∗. https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 7 using equation (5) and substituting t∗∗d gives δλn − (1 −φ1)(1 −φ2) ( β1 + β2α q2 ) i∗∗s∗∗h −µns ∗∗ h = 0 ⇒ s∗∗h = δλn a1i ∗∗ + µn , where a1 = (1 − φ1)(1 − φ2) (β1 + β2α q2 ). in a similar manner, s∗∗l is expressed as s∗∗l = (1 −δ)λn a2i ∗ + µn , where a2 = (1 −φ2) (β1 + β2α q2 ).using equation (7) and substituting for s∗∗h and s∗∗l ,we obtain a1i ∗∗δλ a1i ∗∗ + µn∗∗ + a2i ∗∗(1 −δ)λ a2i ∗∗ + µn∗∗ −q1i∗∗ = 0 (10) thus from equation (10),( a1δλ a1i ∗∗ + µn∗∗ + a2(1 −δ)λ a2i ∗∗ + µn∗∗ −q1 ) i∗∗ = 0. (11) from equation (11), i∗∗ = 0 corresponds to the disease free equilibrium point of the system (2),denoted by (e0). the other solution of (11) when i∗∗ 6= 0 corresponds to the endemic equilibriumpoint of the system such that, a1δλ a1i ∗∗ + µn∗∗ + a2(1 −δ)λ a2i ∗∗ + µn∗∗ −q1 = 0. (12) multiplying through by (a1i∗∗ + µn∗∗)(a2i∗∗ + µn∗∗) yields ci∗∗2 + di∗∗ + e = 0. (13) where: c = −q1a1a2,d = (a1a2δλ + a1a2(1 −δ)λ) − (q1a1µn + q1a2µn), and e = a1δλµn + a2(1 −δ)λµn −q1µnµn.the endemic equilibrium of the system exists if the roots of equation (13) are real and positive.descarte’s rule of signs is used to check the possible number of real roots of the polynomial. thenumber of positive real roots is equal to the number of sign changes in the coefficients of theterms of a polynomial [15]. considering that all the parameters used are positive, the sign of c isnegative. the sign of e is then checked as follows; e = a1δλµn + a2(1 −δ)λµn −q1µnµn = (1 −φ1)(1 −φ2) ( β1 + β2α q2 ) δλµn + (1 −φ2) ( β1 + β2α q2 ) (1 −δ)λµn −q1µnµn = (1 −φ1)(1 −φ2)(β1 + β2α)δλµn + (1 −φ2)(β1 + β2α)(1 −δ)λµn −q1q2µnµn https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 8 using r0 = β3 q1 + αβ4 q1q2 and the limiting value of n = λ µ , we obtain e = (r0 − 1)λ2. thus e >0iff r0 > 1. since c is negative, and e is positive, we see that there is at least one sign changeregardless of the sign of d. this implies that equation (13) has at least one positive real root.hence an endemic equilibrium point of the system (2) exists whenever r0 > 1. � 3.2. local stability of the endemic equilibrium. at the endemic equilibrium, there is persistenceof hiv infection in the population. theorem 3.4. the endemic equilibrium point e1 = (s∗∗h ,s ∗∗ l , i ∗∗,t∗∗d ,a ∗∗) of system (2) is locally asymptotically stable if r0 > 1. proof. the jacobian matrix of the system (2) evaluated at endemic equilibrium is j(e1) =  −b1 0 −b2 −b3 0 0 −b4 −b5 −b6 0 b7 b8 b9 −q1 b10 0 0 0 α −q2 0 0 0 γ2 γ3 −q3  where b1 = (1−φ1)(1−φ2)(β1q2+β2α)µi∗∗+q2µλ q2λ ,b2 = (1−φ1)(1−φ2)β1a1δλi∗∗ a1i∗∗+λ ,b3 = (1−φ1)(1−φ2)β2a1δλi∗∗ a1i∗∗+λ b4 = (1−φ2)(β1q2+β2α)µi∗∗+q2µλ q2λ ,b5 = (1−φ2)(1−δ)λβ1a2i∗∗ a2i∗∗+λ ,b6 = (1−φ2)(1−δ)λβ2a2i∗∗ a2i∗∗+λ b7 = (1−φ1)(1−φ2)(β1q2+β2α)µi∗∗+q2µλ q2λ ,b8 = (1−φ2)(β1q2+β2α)µi∗∗+q2µλ q2λ b9 = (1−φ1)(1−φ2)β1a1δλi∗∗ a1i∗∗+λ + (1−φ2)(1−δ)λβ1a2i∗∗ a2i∗∗+λ ,b10 = (1−φ1)(1−φ2)β2a1δλi∗∗ a1i∗∗+λ + (1−φ2)(1−δ)λβ2a2i∗∗ a2i∗∗+λclearly, −q3 is an eigenvalue of the jacobian matrix j(e1). the other eigenvalues can be computedby finding the solution to the equation p (λ) = ∣∣∣∣∣∣∣∣∣∣∣ λ + b1 0 −b2 −b3 0 λ + b4 −b5 −b6 b7 b8 λ− (b9 + q1) b10 0 0 α λ + q2 ∣∣∣∣∣∣∣∣∣∣∣ =0 the characteristic equation of j(e1)is then given by; p (λ) = λ4 + c0λ 3 + c1λ 2 + c2λ + c3 = 0 (14) where; c0 = b1 + b4 −b9 −q1 + q2 c1 = b1b4 + b2b7 + b5b8 −b1b9 −b4b9 −αb10 −b1q1 −b4q1 + b1q2 + b4q2 −b9q2 −q1q2 c2 = −αb3b7 +b2b4b7 +b1b5b8−αb6b8−b1b4b9−alphab1b10−alphab4b10−b1b4q1 +b1b4q2 + b2b7q2 + b5b8q2 −b1b9q2 −b4b9q2 −b1q1q2 −b4q1q2 c3 = −αb3b4b7 −αb1b6b8 −αb1b4b10 + b2b4b7q2 + b1b5b8q2 −b1b4b9q2 −b1b4q1q2the number of negative zeros of equation (14) depends on the signs of c0,c1,c2 and c3. descarte’s https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 9 rule of signs is applied to study the number of negative real roots of the polynomialp (λ1) com-prising of the coefficients c0,c1,c2 and c3 given by; p (λ1) = c0λ 3 + c1λ 2 + c2λ + c3 = 0 (15) descarte’s rule of signs states that the number of negative real zeros of p (λ) is either equal tothe variations in sign of p (−λ) or less than this by an even number [15]. the possibilities ofnegative real zeros of p (λ), is as summarized in table 2. the maximum number of variationsof signs in p (−λ) is 3, hence the characteristic polynomial (15) has three negative roots. thus p (−λ) = λ4 − c0λ3 + c1λ2 − c2λ + c3 = 0 has negative roots.therefore, given that cases 1-8 intable 1 are satisfied, model (2) is locally asymptotically stable if r0 > 1. � table 2. the zeros of the characteristic equation (14) cases c0 c1 c2 c3 r0 > 1 sign change no. of roots1 + − − + r0 > 1 2 2,02 + − + + r0 > 1 2 2,03 − − + − r0 > 1 2 2,04 + + − − r0 > 1 1 05 − − + + r0 > 1 1 06 + + + − r0 > 1 1 07 − + − + r0 > 1 3 3,18 − − − − r0 > 1 0 0 this implies that for a small pertubation of the e1, solutions of the mathematical model representedby the system (2) always converge to e1, whenever r0 > 1. epidemiologically, it implies that ifa few hiv infected individuals are introduces in a fully susceptible population, the disease willpersist provided r0 > 1. 4. sensitivity analysis in mathematical modeling, sensitivity refers to the degree to which a given input parameterin a mathematical model influences its output. sensitive parameters are thus those that cause asignificant impact on the disease transmission dynamics. sensitivity analysis will aid in identify-ing the parameters which greatly impact on the value of the basic reproductive number r0, andhence ought to be targeted when coming up with intervention strategies. the sensitivity of modelparameters is calculated using the normalized forward sensitivity index. the normalized forwardsensitivity index of the basic reproductive number is given by sr0w = ∂r0 ∂w × w r0 , where w is the https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 10 parameter whose sensitivity is to be determined [7]. r0 is given by r0 = (1 −φ1)(1 −φ2)β1δ + (1 −φ2)(1 −δ)β1 α + γ2 + µ + α(1 −φ1)(1 −φ2)β2δ + (1 −φ2)(1 −δ)β2 (α + γ2 + µ)(γ3 + µ) . (16) forβ1,sr0β1 = β1(γ3 + µ)β1(γ3 + µ) + αβ2 . forβ2,sr0β2 = αβ2β1(γ3 + µ) + αβ2 . for α,sr0α = [β2(α + γ2 + µ) − (β1(γ3 + µ) + αβ2)]α(α + γ2 + µ)(β1(γ3 + µ) + αβ2) .forγ2,sr0γ2 = (αγ2 + γ22 + µγ2) ln |α + γ2 + µ|. forγ3,sr0γ3 = −αβ2γ3(β1(γ3 + µ)2 + αβ2(γ3 + µ). for δ,sr0 δ = −φ1δ 1 −δφ1 for µ,sr0µ = [(α + γ2 + µ)(γ3 + µ)β1 + ((γ3 + µ)β1 + αβ2)(α + γ2 + γ3 + 2µ)]µ(α + γ2 + µ)(γ3 + µ)((β1(γ3 + µ) + αβ2)) . based on the sensitivity indices in table 3, the most sensitive parameter to the value of r0 is β1, table 3. sensitivity indices for the model parameters parameter description sensitivity index δ proportion of high risk sussceptibles −0.36986 φ1 prep effectiveness −0.041095 φ2 condom effectiveness −0.11111 γ3 art failure −0.27182 β1 mean contact rate with i 0.72345 β2 mean contact rate with td 0.27654 α progression from i to td −0.40225 γ2 progression from i to a −0.05123 µ natural mortality rate −0.02969 the mean contact rate with undiagnosed infectives. this implies that in order to control the spreadof hiv in a high risk population, efforts should be geared towards reducing the number of thosewho are undiagnosed by testing them and enrolling them on art treatment. this in turn lowerstheir infectivity as well as chances of progressing to the aids class. https://doi.org/10.28924/ada/ma.3.21 eur. j. math. anal. 10.28924/ada/ma.3.21 11 5. conclusion in this study, a mathematical model is formulated, based on a system of ordinary differentialequations, incorporating the impact of dual protection and art adherence in preventing the spreadof hiv among persons at high risk of infection.stability analysis of the model was done and depicted that when r0 < 1, the disease freeequilibrium is both locally and globally asymptotically stable. the endemic equilibrium of themathematical model exists and was shown to be locally asymptotically stable whenever r0 > 1,implying that there is persistence of hiv infection in the population provided that r0 is greaterthan unity. sensitivity analysis was conducted, depicting that the most sensitive parameter is β1,the mean contact rate with the un-diagnosed infectives. therefore, in order to control the spreadof hiv among the high risk population, efforts ought to be channeled towards the undiagnosedpopulation by frequently testing and enrolling them on art treatmentwhich guarantees low viral load within the infected individual, making them less infective. thusdual protection and art adherence are essential in the fight against the spread of hiv among thehigh risk population. references [1] c. castillo-chavez, z. feng, w. huang, on the computation of r0 and its role on global stability, in: c. castillo-chavez, s. blower, p. van den driessche, d. kirschner, a.-a. yakubu (eds.), mathematical approaches for emergingand reemerging infectious diseases: an introduction, springer new york, new york, ny, 2002: pp. 229-250. https://doi.org/10.1007/978-1-4757-3667-0_13.[2] s.c. weller, k. davis-beaty, condom 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https://aidsinfo.nih.gov/news/1254/fda-approves-{\@@par }rst-drug-for-reducing-the-risk-of-sexually-acquired-hiv-infection https://doi.org/10.2307/4145072 https://doi.org/10.2307/4145072 1. introduction 2. model formulation and description 3. model analysis 3.1. existence of the endemic steady state 3.2. local stability of the endemic equilibrium 4. sensitivity analysis 5. conclusion references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 19-33doi: 10.28924/ada/ma.1.19 the generalized viscosity implicit rules of asymptotically nonexpansive mappings in hilbert spaces sang b mendy, john t mendy∗ , alieu jobe university of the gambia, brikama campus, gambia sangbm1@gmail.com, jt.mendy@utg.edu.gm, alieueejobe@gmail.com ∗correspondence: jt.mendy@utg.edu.gm abstract. the generalized viscosity implicit rules of nonexpansive asymptotically mappings in hilbertspaces are considered. the strong convergence theorems of the rules are proved under certain as-sumptions imposed on the sequences of parameters. an application of it in the convex minimizationproblem is considered. the results presented in this paper improve and extend some recent corre-sponding results in the literature. 1. background let h be a real hilbert space and m be a nonempty closed convex subset of h,t : m→mbe a nonexpansive mapping with a nonempty fixed point set f(t )the following iteration method is known as the viscosity approximation method: for arbitrarilychosen u0 ∈m un+1 = αnψ(un) + (1 −αn)tun,n ≥ 0, (1.1)where ψ : m → m is a contraction and {αn} is a sequence in (0, 1). under some certainconditions, the sequence {un} converges strongly to a point z ∈ f (t ) which solves the variationalinequality (v i) 〈(i −ψ)z,u −z〉≥ 0,u ∈ f (t ), (1.2) where i is the identity of h. many authors studied iterative sequence for the implicit midpointrule because of it’s significant for solving ordinary differential equations; see [?][12], john t [9], [7]and the references therein. recently, xu et al [3] proposed the following viscosity implicit midpointrule (vimr) for nonexpansive mappings: un+1 = αnψ(un) + (1 −αn)t (un + un+1 2 ) ,n ≥ 0, (1.3) received: 23 aug 2021. key words and phrases. viscosity; hilbert space; convex minimization; asymptotically nonexpansive mapping; varia-tional inequality; fixed point. 19 https://adac.ee https://doi.org/10.28924/ada/ma.1.19 https://orcid.org/0000-0002-3774-0761 eur. j. math. anal. 1 (2021) 20 in 2015, ke and ma [4] proposed the generalized viscosity implicit rules of nonexpansive mappingsin hilbert spaces as follows: un+1 = αnψ(un) + (1 −αn)t (snun + (1 − sn)un+1),n ≥ 0, (1.4) and un+1 = αnun + βnψ(un) + γnt (snun + (1 − sn)un+1),n ≥ 0, (1.5) they proved that the generalized viscosity implicit rules 1.4 and 1.5 converge strongly to a fixedpoint of t under certain assumptions, which also solved the v i(1.1). in 2016, motivated by the work of xu [3], zhao et al [5] proposed the following implicit midpointrule for asymptotically nonexpansive mappings: un+1 = αnψ(un) + (1 −αn)t n (un + un+1 2 ) ,n ≥ 0, (1.6) where t is an asymptotically nonexpansive mapping. they proved that the sequence {un} con-verges strongly to a fixed point of t , which, in addition, also solves the v i(1.1). in 2017, he et la [14] studied the following iterative un+1 = αnψ(un) + (1 −αn)t n(βnun + (1 −βn)un+1),n ≥ 0 (1.7) in the setting of a hilbert space and proved that the sequence {un} converges strongly to u∗ = pf(t )ψ(u ∗) which is also the unique solution of the following v i 〈(i −ψ)u,v −u〉≥ 0,∀v ∈ f (t ) (1.8) in this paper, we introduce and study the generalized viscosity implicit rules of asymptoticallynonexpansive mappings in hilbert spaces. more precisely, we consider the following implicititerative algorithm: u1 ∈mun+1 = αnun + βnψ(un) + γnt n(snun + (1 − sn)un+1) ∀n ∈n (1.9) under suitable conditions, we proved that the sequence {un} converge strongly to a fixed point ofthe asymptotically nonexpansive mapping t , which also solves the variational inequality 〈(i −ψ)u,p−u〉≥ 0 p ∈ f (t ). as applications, we apply our results to solve convexly constrained minimization problem. this wayresults in 1.5 are complemented, extended and generalized. eur. j. math. anal. 1 (2021) 21 2. preliminaries in the sequel, we always assume that h is a real hilbert space and m is a nonempty, closed,and convex subset of h. the nearest point projection from h onto m,pm, is defined by pm(u) := arg min z∈m ∥∥∥u −z∥∥∥2, u ∈h. (2.1) namely, pm(u) is the only point in m that minimizes the objective ∥∥∥u − z∥∥∥ over z ∈ m. and pm(u) is characterized as follows: pm(u) ∈m and 〈 u −pm(u),z −pm(u) 〉 ≤ 0 f or all z ∈m. (2.2) definition 2.1. . a mapping t : m→m is said to be: a): α-inverse strongly monotone if there exists α > 0 satisfying 〈u −v,tu −t v〉≥ α‖au −av‖2 ∀u,v ∈m; (2.3) b): l-lipschitz continuous if there exists l ≥ 0 satisfying ‖tu −t v‖≤ l‖u −v‖ ∀u,v ∈m; (2.4) c): nonexpansive if ‖tu −t v‖≤‖u −v‖ ∀u,v ∈m; (2.5) d): asymptotically nonexpansive if there exists a sequence {kn} ⊂ [1,∞) with lim n→∞ kn = 1 suchthat ‖t nu −t nv‖≤ kn‖u −v‖ ∀u,v ∈m and ∀n ∈n; (2.6) e): contraction if there exists the contractive constant α ∈ [0, 1) such that ‖tu −t v‖≤ α‖u −v‖ ∀u,v ∈m; (2.7) lemma 2.2. (the demiclosedness principle [10]) . let h be a hilbert space, m be a nonempty closed convex subset of h, and t : m → m be a asymptotically nonexpansive mapping with fix(t ) 6= ∅. if {un} is a sequence in m such that {un} weakly converges to u and {(i −t )un} converges strongly to 0, then u = t (u) lemma 2.3. let h be a hilbert space. then for all θ,u,v ∈h, the following inequality holds ‖u −θ‖2 ≤‖v −θ‖2 + 2〈u −v,u −θ〉 lemma 2.4. [11]). assume that {αn} is a sequence of nonnegative real numbers such that αn+1 ≤ (1 −λn)αn + δn for all n ∈ n, where {λn} ⊆ (0, 1) and {δn} ⊆ r are two sequences satisfying the following conditions: eur. j. math. anal. 1 (2021) 22 (i): ∞∑ n=1 λn = ∞ (ii): lim sup n→∞ δn λn ≤ 0 or ∞∑ n=1 |δn| < ∞ then lim n→∞ αn = 0 then the sequence {αn} converges to 0. 3. main result we now prove the following new result. theorem 3.1. let m be a nonempty closed convex subset a real hilbert space h,t : m → m be asymptotically nonexpansive mappings with the same sequence {kn} ⊆ [1,∞) such that limn→∞kn = 1,fix(t ) 6= ∅ and ψ : m → m be a contraction mapping with the contractive constant α ∈ [0, 1). define a sequence {un} in m as follows: u1 ∈mun+1 = αnun + βnψ(un) + γnt n(snun + (1 − sn)un+1) ∀n ∈n (3.1) where αn,βn,γn,sn ∈ (0, 1) satisfying the following conditions, a1: αn + βn + γn = 1 a2: ∞∑ n=0 αn = ∞ a3: 0 < � ≤ sn ≤ sn+1 < 1 for all n ≥ 0 a4: lim n→∞ γn = 1 and lim n→∞ αn = lim n→∞ βn = lim n→∞ sn = 0 lim n→∞ ‖un −t nun‖ = 0 then the sequence {un} strongly converges to a common fixed point q of t , which is also the unique solution of the following variational inequality 〈(i −ψ)u,p−u〉≥ 0 p ∈ f (t ). we now show that algorithm 3.1 is well posed. letting bn(u) = αnun + βnψ(un) + γnt n ( snun + (1 − sn)un ) ‖bn(u) −bn(v)‖ = ‖γnt n ( snun + (1 − sn)u ) −γnt n ( snun + (1 − sn)v ) ‖ = ‖γnt n(1 − sn)u −γnt n(1 − sn)v‖ ≤ γnkn(1 − sn)‖u −v‖ since lim n→∞ sn = 0, lim n→∞ kn = 1, lim n→∞ γn = 1 and 0 < � ≤ sn ≤ sn+1 < 1 for all n > 0, we mayassume that γnkn(1 − sn) ≤ 1 − � for all n > 0. this implies that bn is a contraction for each eur. j. math. anal. 1 (2021) 23 n. therefore there exists a unique fixed point for bn by banach contraction principle, which alsoimplies that (3.1) is well-defined. we now show that the sequence {un} is bounded. rewriting 3.1, we have un+1 = βnψ(un) + αnun + (1 −βn)vn (3.2) where vn = γnt n(snun + (1 − sn)un+1) 1 −βn remark 3.2. the real sequences that satisfies the above conditions are αn = 1 n , βn = 1 n and γn = 1 − 2 n proof. our prove are in six steps. first we prove that the sequence {un} defined by 3.1 is bounded. step 1: letting p ∈ fix(t ), we have the following estimates ‖un+1 −p‖ = ‖βnψ(un) + αnun + (1 −βn)vn −p‖ ≤ βn‖ψ(un) −ψ(p)‖ + βn‖ψ(p) −p‖ + αn‖un −p‖ + (1 −βn)‖vn −p‖ ≤ (αβn + αn)‖un −p‖ + βn‖ψ(p) −p‖ + (1 −βn)‖vn −p‖ (3.3) ‖vn −p‖ = ‖ γnt n(snun + (1 − sn)un+1) 1 −βn −p‖ = γnt nsn(un −p) 1 −βn + γnt n(1 − sn)(un+1 −p) 1 −βn ‖ ≤ γnknsn 1 −βn ‖un −p‖ + γnkn(1 − sn) 1 −βn ‖un+1 −p‖ (3.4) putting 3.4 in 3.3, gives the following ‖un+1 −p‖ ≤ (αβn + αn)‖un −p‖ + βn‖ψ(p) −p‖ + γnknsn‖un −p‖ + γnkn(1 − sn)‖un+1 −p‖ (1 −γnkn(1 − sn))‖un+1 −p‖ ≤ (αβn + αn + γnknsn)‖un −p‖ + βn‖ψ(p) −p‖ ‖un+1 −p‖ ≤ (αβn + αn + γnknsn) 1 −γnkn(1 − sn) ‖un −p‖ + βn 1 −γnkn(1 − sn) ‖ψ(p) −p‖ eur. j. math. anal. 1 (2021) 24 since γn,sn ∈ (0, 1), 1 −γnkn(1 − sn) > 0 and lim n→∞ kn = 1. from the condition (a1), wehave ‖un+1 −p‖ ≤ 1 − 1 −αβn −αn −γnkn 1 −γnkn(1 − sn) ‖un −p‖ + βn 1 −γnkn(1 − sn) ‖ψ(p) −p‖ ] ‖un+1 −p‖ ≤ 1 − βn(1 −α) 1 −γnkn(1 − sn) ‖un −p‖ + βn(1 −α) 1 −γnkn(1 − sn) 1 (1 −α) ‖ψ(p) −p‖ ] ‖un+1 −p‖ ≤ max { ‖un −p‖, 1 (1 −α) ‖ψ(p) −p‖ } therefore by mathematical induction, we have ‖un+1 −p‖≤ max { ‖u0 −p‖, 1 (1 −α) ‖ψ(p) −p‖ } for all n ≥ n. therefore {un} is bounded. consequently, {ψ(un)} and {vn} are also bounded. step 2: we now prove that the sequence {un+1} converges to {un} as n →∞. that is lim n→∞ ‖un+1− un‖ = 0 ‖un+1 −un‖ = ‖un+1 −t nun + t nun −un‖ = ‖βnψ(un) + αnun + (1 −βn)vn − (βn + αn + γn)t n + t nun −un‖ ≤ ‖βnψ(un) −βnt nun‖ + ‖αnun −αnt nun‖ +‖(1 −βn)vn −γnt n + t nun −un‖ ≤ βn‖ψ(un) −t nun‖ + αn‖un −t nun‖ +(1 −βn)‖vn −γnt n‖ + ‖t nun −un‖ (3.5) ‖vn −γnt nun‖ = ‖ γnsn 1 −βn t nun + γn(1 − sn) 1 −βn t nun+1 −γnt nun‖ ≤ ‖ γnsn 1 −βn ‖t nun −t nun‖ + γn(1 − sn) 1 −βn ‖t nun+1 −t nun‖ ≤ γn(1 − sn)kn 1 −βn ‖un+1 −un‖ (3.6) eur. j. math. anal. 1 (2021) 25 now putting 3.6 in 3.5, we have the following ‖un+1 −un‖ ≤ βn‖ψ(un) −t nun‖ + αn‖un −t nun‖ +(1 −βn) [γn(1 − sn)kn 1 −βn ‖un+1 −un‖ ] + ‖t nun −un‖ ≤ βn‖ψ(un) −t nun‖ + αn‖un −t nun‖ +γn(1 − sn)kn‖un+1 −un‖ ] + ‖t nun −un‖ ≤ βn‖ψ(un) −t nun‖ + (αn + 1)‖un −t nun‖ + γn(1 − sn)kn‖un+1 −un‖[ 1 −γn(1 − sn)kn ] ‖un+1 −un‖ ≤ βn‖ψ(un) −t nun‖ + (αn + 1)‖un −t nun‖ ‖un+1 −un‖ ≤ βn 1 −γn(1 − sn)kn ‖ψ(un) −t nun‖ + (αn + 1) 1 −γn(1 − sn)kn ‖un −t nun‖ let m :> max {‖ψ(un) −t nun‖}, then we have ‖un+1 −un‖≤ βnm 1 −γn(1 − sn)kn + (αn + 1) 1 −γn(1 − sn)kn ‖un −t nxn‖ ‖un+1 −un‖≤ βnm 1 −γn(1 − sn)(1 + �αn) + (αn + 1) 1 −γn(1 − sn)(1 + �αn) ‖un −t nun‖ since lim n→∞ αn = lim n→∞ βn = lim n→∞ ‖un −t nun‖ = 0, we then conclude that lim n→∞ ‖un+1 − un‖ = 0 step 3: again we then show that lim n→∞ ∥∥∥un −t (un)∥∥∥ = 0. estimating as follows we have ‖un −t nun‖ = ‖un −un+1 + un+1 −t nun‖ ≤ ‖un −un+1‖ + ‖un+1 −t nun‖ ≤ ‖un −un+1‖ + ‖βnψ(un) + αnun + (1 −βn)vn −t nun ∥∥∥ ≤ ‖un −un+1‖ + βn‖ψ(un) −t nun‖ + αn‖un −t nun‖ + (1 −βn)‖vn −γnt nun‖ (3.7) ‖vn −γnt nun‖ = ‖ γnt n(snun + (1 − sn)un+1) 1 −βn −γnt nun‖ ≤ ‖ γnsn 1 −βn ‖t nun −t nun‖ + (1 − sn)γn 1 −βn ‖t nun+1 −t nun‖ ≤ (1 − sn)γnkn 1 −βn ‖un+1 −un‖ (3.8) eur. j. math. anal. 1 (2021) 26 now substituting 3.8 into 3.7, gives the following estimation ‖un −t nun‖ ≤ ‖un −un+1‖ + βn‖ψ(un) −t nun‖ + αn‖un −t nun‖ + (1 −βn) ((1 − sn)γnkn 1 −βn ‖un+1 −un‖ ) ≤ ( 1 + (1 − sn)γnkn ) ‖un −un+1‖ + βn‖ψ(un) −t nun‖ + αn‖un −t nun‖ ≤ ( 1 + (1 − sn)γnkn ) 1 −αn ‖un −un+1‖ + βn 1 −αn ‖ψ(un) −t nun‖ ‖un −t nun‖ ≤ ( 1 + (1 − sn)γnkn ) 1 −αn ‖un+1 −xn‖ + βnm 1 −αn therefore from 3.1 condition a4, with lim n→∞ ‖un+1 −un‖ = 0, we can conclude that lim n→∞ ‖un −t nun‖ = 0 (3.9) but we know that from the following fact lim n→∞ ‖un −t (un)‖ ≤ lim n→∞ ‖un −t nun‖ + lim n→∞ ‖t nun −t xn‖ ≤ lim n→∞ ‖un −t nun‖ + lim n→∞ k1‖t n−1un −un‖ (3.10) proving that lim n→∞ ‖tn−1un −un‖ = 0, we have the following estimation ‖t n−1(un) −un‖ = ‖un −t n−1(un)‖ = ‖βn−1ψ(un−1) + αn−1un−1 + (1 −βn−1)vn−1 −(βn−1 + αn−1 + γn−1)t n−1un‖ = ‖βn−1ψ(un−1) −βn−1t n−1un + αn−1un−1 −αn−1t n−1un +(1 −βn−1)vn−1 −γn−1t n−1un‖ ≤ βn−1‖ψ(un−1) −t n−1un‖ + αn−1‖un−1 −t n−1un‖ +(1 −βn−1)‖vn−1 −γn−1tn−1xn‖ (3.11) ‖vn−1 −γn−1t n−1un‖ = ‖ γn−1t n−1(sn−1un−1 + (1 − sn−1)un) 1 −βn−1 −γn−1t n−1un‖ ≤ γn−1kn−1sn−1 1 −βn−1 ‖|un −un−1‖ (3.12) eur. j. math. anal. 1 (2021) 27 combining 3.12 and 3.11 we have the following ‖t n−1(un) −un‖ ≤ βn−1‖ψ(un−1) −t n−1un‖ + αn−1‖un−1 −t n−1un‖ +(1 −βn−1) [γn−1kn−1sn−1 1 −βn−1 ‖|un −un−1‖ ] ≤ βn−1‖ψ(un−1) −t n−1un‖ + αn−1‖un−1 −t n−1un‖ +γn−1kn−1sn−1‖un −un−1‖ ≤ βn−1‖ψ(un−1) −t n−1un‖ + αn−1‖un−1 −t n−1un‖ +γn−1kn−1sn−1‖|un −un−1‖ with the assumption of {αn},{βn} and lim n→∞ ‖un+1 −un‖ = 0, we can conclude that lim n→∞ ‖t n−1un −un‖ = 0 (3.13) therefore from 3.9 and 3.13, we can see from inequality 3.10, that lim n→∞ ‖un −t (un)‖ = 0 (3.14) step 4: in this step, we will show that wω(xn) ⊆ fix(t ), where wω(un) := {u ∈h : there exist a subsequence of {un} converges weakly to u}.suppose that u ∈ wω(un). then there exists a subsequence {uni} of {un} such that uni ⇀ xas i →∞ . from 3.14, we have lim i→∞ ∥∥∥(i −t )xni∥∥∥ = lim n→∞ ∥∥∥uni −tuni∥∥∥ = 0 . this implies that {(i −t )uni} converges strongly to 0. by using lemma 2.2, we have tu = u, and so u ∈ fix(t ). step 5: in this step, we will show that lim sup n→∞ 〈q −ψ(q),q −un〉≤ 0, (3.15) where q ∈ f (t ) is the unique fixed point of pf(t ) ◦ψ, that is, q = pf(t )(ψ(z)). since {un} is bounded, there exists a subsequence {uni} of {un} such that uni ⇀ u as i →∞ forsome u ∈h and lim sup n→∞ 〈q −ψ(q),q −un〉 = lim i→∞ 〈q −ψ(q),q −uni〉 (3.16) from step 4, we get x ∈ f (t ). by using inequality 2.2, we obtain lim sup n→∞ 〈q −ψ(q),q −un〉 = lim i→∞ 〈q −ψ(q),q −uni〉 = 〈q −ψ(q),q −u〉≤ 0 eur. j. math. anal. 1 (2021) 28 step 6: finally, setting ϕn = βnq + αnq + (1 −βn)vn we show that un → q as n → ∞. again,take q ∈ f (t ) to be the unique fixed point of the contraction pf(t ) ◦ψ. for each n ∈ n,consider ‖un+1 −q‖2 ≤ ‖ϕn −q‖2 + 2〈un+1 −ϕn,un+1 −q〉 = (1 −βn)2‖vn −q‖2 + 2〈βn(ψ(un) −q) + αn(un −q),un+1 −q〉 ≤ (1 −βn)‖vn −q‖2 + 2〈βn(ψ(un) −ψ(q)) + βn(ψ(q) −q) + αn(un −q),un+1 −q〉 ≤ (1 −βn)2‖vn −q‖2 + 2βn‖ψ(xn) −ψ(q)‖‖un+1 −q‖ + 2αn‖un −q‖‖un+1 −q‖ +2βn〈ψ(q) −q,un+1 −q〉 ≤ (1 −βn)2‖vn −q‖2 + 2βnα‖un −q‖‖un+1 −q‖ + 2αn‖un −q‖‖un+1 −q‖ +2βn〈ψ(q) −q,un+1 −q〉 ≤ (1 −βn)2‖vn −q‖2 + (2βnα + 2αn)‖un −q‖‖un+1 −q‖ +2βn〈ψ(q) −q,un+1 −q〉 (3.17) for the fact that ‖vn −q‖2 = ∥∥∥γntn(snun + (1 − sn)un+1) (1 −βn) −q ∥∥∥2 ≤ γ2ns 2 nk 2 n (1 −βn)2 ‖un −q‖2 + γ2n(1 − sn)2k2n (1 −βn)2 ‖un+1 −q‖2 + γ2nsn(1 − sn)k2n (1 −βn)2 〈un −q,un+1 −q〉 ≤ γ2ns 2 nk 2 n (1 −βn)2 ‖un −q‖2 + γ2n(1 − sn)2k2n (1 −βn)2 ‖un+1 −q‖2 + γ2nsn(1 − sn)k2n (1 −βn)2 ‖un −q‖‖un+1 −q‖ +2αnβn 〈 ψ(un) −ψ(q),tn (un + un+1 2 ) −q 〉 (3.18) now substituting 3.18 into 3.17, we have the following estimation ‖un+1 −q‖2 ≤ γ2ns 2 nk 2 n‖un −q‖ 2 + γ2n(1 − sn) 2k2n‖un+1 −q‖ 2 +γ2nsn(1 − sn)k 2 n‖un −q‖‖un+1 −q‖ +2(βnα + αn)‖un −q‖‖un+1 −q‖ + 2βn〈ψ(q) −q,un+1 −q〉 ≤ γ2ns 2 nk 2 n‖un −q‖ 2 + γ2n(1 − sn) 2k2n‖un+1 −q‖ 2 + [ γ2nsn(1 − sn)k 2 n + 2(βnα + αn) ] ‖un −q‖‖un+1 −q‖ +2βn〈ψ(q) −q,un+1 −q〉 (3.19) eur. j. math. anal. 1 (2021) 29 again using the fact that( ‖un −q‖−‖un+1 −q‖ )2 ≤ ‖un −q‖2 − 2‖un −q‖‖un+1 −q‖ +‖un+1 −q‖2 setting the left hand to zero, we have the following estimate 2‖un −q‖‖un+1 −q‖ ≤ ‖un −q‖2 + ‖un+1 −q‖2 ‖un −q‖‖un+1 −q‖ ≤ 1 2 ‖un −q‖2 + 1 2 ‖un+1 −q‖2 (3.20) putting inequality 3.20 in inequality 3.19, gives the following ‖un+1 −q‖2 ≤ γ2ns 2 nk 2 n‖un −q‖ 2 + γ2n(1 − sn) 2k2n‖un+1 −q‖ 2 + γ2nsn(1 − sn)k2n 2 ‖un −q‖2 + (βnα + αn)‖un −q‖2 + γ2nsn(1 − sn)k2n 2 ‖un+1 −q‖2 + (βnα + αn)‖un+1 −q‖2 +2βn〈ψ(q) −q,un+1 −q〉 ‖un+1 −q‖2 ≤ [γ2nsnk2n(sn + 1) + 2(βnα + αn) 2 ] ‖un −q‖2 + [γ2n(1 − sn)2k2n(2 − sn) + 2(βnα + αn) 2 ] ‖un+1 −q‖2 +2βn〈ψ(q) −q,un+1 −q〉 thus we have( 1 − [γ2n(1 − sn)2k2n(2 − sn) + 2(βnα + αn) 2 ]) ‖un+1 −q‖2 ≤ [γ2nsnk2n(sn + 1) + 2(βnα + αn) 2 ] ‖un −q‖2 + 2βn〈ψ(q) −q,un+1 −q〉 ‖un+1 −q‖2 ≤ γ2nsnk 2 n(sn + 1) + 2(βnα + αn) 2 − [ γ2n(1 − sn)2k2n(2 − sn) + 2(βnα + αn) ]‖un −q‖2 + 4βn 2 − [ γ2n(1 − sn)2k2n(2 − sn) + 2(βnα + αn) ]〈ψ(q) −q,un+1 −q〉 ‖un+1 −q‖2 ≤ ( 1 − 2 −γ2n(1 − sn)2k2n(2 − sn) −γ2nsnk2n(sn + 1) 2 − [ γ2n(1 − sn)2k2n(2 − sn) + 2(βnα + αn) ])‖un −q‖2 + 4βn 2 − [ γ2n(1 − sn)2k2n(2 − sn) + 2(βnα + αn) ]〈ψ(q) −q,un+1 −q〉 eur. j. math. anal. 1 (2021) 30 therefore from condition lim n→∞ αn = lim n→∞ βn = lim n→∞ sn = 0 in 3.1, we concludes that ‖un+1 −q‖2 ≤ ( 1 − 2 − 2γ2nk2n 2 − 2γ2nk2n ) ‖un −q‖2 lim n→∞ ‖un+1 −q‖2 = 0 this complete the proof. � theorem 3.3. let m be a nonempty closed convex subset a real hilbert space h,t : m → m be asymptotically nonexpansive mappings with the same sequence {kn} ⊆ [1,∞) such that limn→∞kn = 1,fix(t ) 6= ∅ and ω be a constant. define a sequence {un} in m as follows: u1 ∈mun+1 = αnun + βnω + γnt n(snun + (1 − sn)un+1) ∀n ∈n (3.21) where αn,βn,γn,sn ∈ (0, 1) satisfying conditions a1 −a4 and ψ(un) = ω lim n→∞ ‖t nun −un‖ = 0 then the sequence {un} strongly converges to a common fixed point q of t , which is also the unique solution of the following variational inequality 〈(i −ψ)u,p−u〉≥ 0 p ∈ f (t ). taking sn = 0the following corollaries holds: corollary 3.4. let m be a nonempty closed convex subset a real hilbert space h,t : m → m be asymptotically nonexpansive mappings with the same sequence {kn} ⊆ [1,∞) such that limn→∞kn = 1,fix(t ) 6= ∅ and ψ : m → m be a contraction mapping with the contractive constant α ∈ [0, 1). define a sequence {un} in m as follows:{ u1 ∈m un+1 = αnun + βnψ(un) + γnt n(un+1) ∀n ∈n (3.22) where αn,βn,γn ∈ (0, 1) satisfying conditions a1 −a4 without lim n→∞ sn = 0 lim n→∞ ‖t nun −un‖ = 0 then the sequence {un} strongly converges to a common fixed point q of t , which is also the unique solution of the following variational inequality 〈(i −ψ)u,p−u〉≥ 0 p ∈ f (t ). eur. j. math. anal. 1 (2021) 31 corollary 3.5. let m be a nonempty closed convex subset a real hilbert space h,t : m → m be asymptotically nonexpansive mappings with the same sequence {kn} ⊆ [1,∞) such that limn→∞kn = 1,fix(t ) 6= ∅ and u ∈m be a constant. define a sequence {un} in m as follows:{ u1 ∈m un+1 = αnun + βnω + γnt n(un+1) ∀n ∈n (3.23) where αn,βn,γn ∈ (0, 1) satisfying conditions a1 −a4 without lim n→∞ sn = 0 lim n→∞ ‖t nun −un‖ = 0 then the sequence {un} strongly converges to a common fixed point q of t , which is also the unique solution of the following variational inequality 〈(i −ψ)u,p−u〉≥ 0 p ∈ f (t ). 4. application to convex minimization problems in this section, we study the problem of finding a minimizer of a convex function φ defined from areal hilbert space m to r.consider the optimization problem min x∈c φ(x) (4.1) where φ : m → r is a convex and differentiable function. assume 4.1 is consistent, and let ω 6= ∅ be its set of solutions. the gradient projection algorithm generates a sequence {un} via theiterative procedure: un+1 = pm(un −δ∇φ(u)) (4.2)if ∇φ is θ−inverse strongly monotone mapping and δ(0, 2θ). the following basic results are wellknown. remark 4.1. it is well known that if φ : m → r be a real-valued differentiable convex functionand u∗ ∈m, then the point u∗ is a minimizer of φ on m if and only if dφ(u∗) = 0. definition 4.2. a function φ : m→r is said to be strongly convex if there exists α > 0 such thatfor every u,v ∈m and λ ∈ (0, 1), the following inequality holds: φ(λu + (1 −λ)v) ≤ λφ(u) + (1 −λ)φ(v) −α‖u −v‖2. (4.3) lemma 4.3. let e be normed linear space and φ : m → r a real-valued differentiable convex function. assume that φ is strongly convex. then the differential map dψ : m → m is strongly monotone, i.e., there exists a positive constant k such that 〈dφ(u) −dφ(v),u −v〉≥ k‖u −v‖2 ∀u,v ∈m. (4.4) the prove of the following theorem follows from 3.1 eur. j. math. anal. 1 (2021) 32 theorem 4.4. let m be a nonempty closed convex subset a real hilbert space h. for the minimization problem 4.1, assume that φ is (gateaux) differentiable and the gradient ∇φ is a θ−inverse-strongly monotone mapping for some positive real number θ. let ψ : m → m be a contraction with coefficient α ∈ [0, 1). for a given u1 ∈m, let {un} be a sequence generated by:{ u1 ∈m un+1 = αnun + βnψ(un) + γnpm(1 −δ∇φ)(snun + (1 − sn)(un+1)) ∀n ∈n (4.5) where αn,βn,γn,sn ∈ (0, 1) satisfying the following conditions a1: αn + βn + γn = 1 a2: lim n→∞ k2n − 1 αn = 0 a3: ∞∑ n=0 αn = ∞ a4: lim n→∞ γn = 1 and lim n→∞ αn = lim n→∞ βn = lim n→∞ sn = 0 then {un} converges strongly to a solution (u∗) of the minimization problem 4.1, which is also the unique solution of the variational inequality 〈(i −ψ)u,p−u〉≥ 0 p ∈ f (t ). conflict of interest: the authors declare that they have no competing interests. availability of data and materials: no data were used to support this study. funding: no funding was given towards this manuscript. authors contributions: all authors have contributed equally and significantly in writing this paper and also readand approved the final manuscript. acknowledgement: the authors are very grateful to the editor and anonymous referees for their helpfulcomments. references [1] h. attouch, viscosity approximation methods for minimization problems, siam j. optim. 6 (3) (1996) 769-806. https://doi.org/10.1137/s1052623493259616.[2] a. moudafi, viscosity approximation methods for fixed-points problems, j. math. anal. appl. 241 (1) (2000) 46-55. https://doi.org/10.1006/jmaa.1999.6615.[3] h.k. xu, m.a. alghamdi, n. shahzad, the viscosity technique for the implicit midpoint rule of nonexpansive mappingsin hilbert spaces, fixed point theory appl. 2015 (2015) 41. https://doi.org/10.1186/s13663-015-0282-9. https://doi.org/10.1137/s1052623493259616 https://doi.org/10.1006/jmaa.1999.6615 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asymptotically nonexpansivemappings in hilbert spaces, appl. math. sci. 11 (2017), 549-560. https://doi.org/10.12988/ams.2017.718. https://doi.org/10.1186/s13663-015-0439-6 http://doi.org/10.22436/jnsa.009.06.86 http://doi.org/10.22436/jnsa.009.06.86 https://doi.org/10.12988/ams.2017.718 http://doi.org/10.30538/oms2017.0011 https://doi.org/10.1112/s0024610702003332 https://doi.org/10.1112/s0024610702003332 https://dx.doi.org/10.1073/pnas.53.5.1100 https://doi.org/10.12988/ams.2017.718 1. background 2. preliminaries 3. main result 4. application to convex minimization problems references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 151-163doi: 10.28924/ada/ma.1.151 some aspects of geometric constants in modular spaces zhijian yang, qi liu, muhammad sarfraz, yongjin li∗ department of mathematics, sun yat-sen university, guangzhou, 510275, p. r. china yangzhj55@mail2.sysu.edu.cn, liuq325@mail2.sysu.edu.cn, sarfraz@mail2.sysu.edu.cn, stslyj@mail.sysu.edu.cn ∗correspondence: stslyj@mail.sysu.edu.cn abstract. in this paper, we generalize the typical geometric constants of banach spaces to modularspaces. we study the equivalence between the convexity of modular and normed spaces, and obtainthe relationship between ρ-neumann-jordan constant and ρ-james constant. in particular, we extendthe convexity and smoothness modular, and obtain the criterion theorems of the uniform convexity andstrict convexity. 1. introduction in the recent years, the geometric theory of banach spaces has been fully developed, especiallythe geometric constant, which is a powerful tool to characterize the geometric properties of thespace sphere. as early as 1936, clarkson introduced the convexity modular of space [1]. in 1963,lindenstrauss introduced the smoothness modular, and obtained the close relationship betweenthe two constants [2]. in 1937, in order to better characterize jordan and von-nuemann’s famouswork in inner product spaces, clarkson defines the von-nuemann constant [3] which is the minimumconstant c for all x,y ∈ x and (x,y) 6=(0,0) of the following equations: 1 c ≤ ‖x +y‖2 +‖x −y‖2 2(‖x‖2 +‖y‖2) ≤ c. in 1964, james introduced james constant [4] in order to study the normal structure of space.after the appearance of these constants, many scholars paid attention to them and obtained manywonderful properties [5].modular space problems have been considered by h. nakano, musielak and orlicz [6] underthe additional hypothesis of convexity or subadditivity of the modular ρ : x → [0,+∞). moreoverthe case of semi-ordered linear spaces and that of b-norms have been chiefly investigated. under received: 13 sep 2021. key words and phrases. banach spaces; geometric constants; modular spaces.151 https://adac.ee https://doi.org/10.28924/ada/ma.1.151 eur. j. math. anal. 1 (2021) 152 weaker assumptions, they investigated the structure of the spaces under consideration. neitherconvexity nor subadditivity of the modular be assumed. in introducing the norm, a certain naturalconnection between the modular and the norm convergence will be required: norm convergenceshould imply modular convergence.through their researches, they found that although modular spaces are not generally normedspaces, they still have many wonderful properties, such as convergence, completeness, convexity andadditivity. in view of these properties, poom kumam extended jordan von-neumann constant andjames constant in banach spaces to modular spaces, and obtained uniform convexity and uniformnon-squareness of modular spaces [10].in this paper, based on the idea of generalizing geometric constants in banach spaces to modularspaces, we generalize the properties of von-neumann constant and james constant in [10]. bydefining convexity modules and smoothness modular, we derive the relationships between jamesconstant, convexity modular and the strict convexity of modular spaces. 2. preliminaries we first give some basic facts about modular spaces formulated by musielak and orlicz [6]. definition 1.[8] let x be a vector space over f(r or c). then a function ρ : x → [0,∞] is calleda modular on x if for arbitrary x,y in x,(i) ρ(x)=0 if and only if x =0,(ii) ρ(αx)= ρ(x) for every scalar α with |α|=1,(iii) ρ(αx +βy)≤ ρ(x)+ρ(y) if α+β =1 and α,β ≥ 0.if (iii) is replaced by (iv): ρ(αx +βy)≤ αρ(x)+βρ(y) if α,β ≥ 0 and α+β =1. we now callthat ρ is a convex modular.a modular ρ can be used to define a corresponding modular space, i.e, the vector space xρ asgiven by xρ = {x ∈ x : ρ(λx)→ 0 as λ → 0}, where xρ is a linear subspace of x.in general, the modular ρ is not necessarily subadditive and therefore it does not behave as anorm or a distance. but we can associate it to a modular f-norm.the modular space xρ can be equipped with a f-norm defined by ‖x‖ρ = inf { α > 0;ρ( x λ )≤ α } , when ρ is convex. then norm ‖ · ‖ρ is frequently called the luxemburg norm. if ρ is convex, thenthe functional ‖x‖ρ = inf{α > 0;ρ(xλ) ≤ 1} is a norm in xρ which is equivalent to the f-norm ‖ ·‖ρ. eur. j. math. anal. 1 (2021) 153 proposition 1. let xρ be a modular space. then ρ is convex if and only if xρ is a normed spacewith ρ as norm. proof. the proof of sufficiency is obvious.conversely, assume ρ is convex, then we can obtain ρ(x)=0 if and only if x = 0.(i) according to the definition 1, if α > 0, then ρ ( 1 α x ) = ρ ( 1 α x + 1−α α ·0 ) ≤ 1 α ρ(x)+ 1−α α ρ(0)= 1 α ρ(x) and αρ ( 1 α x ) = αρ ( 1 α x ) +(1−α)ρ(0)≥ ρ(α · 1 α +(1−α) ·0)= ρ(x). this show that ρ(1 α x)≥ 1 α ρ(x) and hence ρ(1 α x ) = 1 α ρ(x) for α > 0.suppose α 6=0, then |α| > 0. according to the definition 1, we have ρ ( |α| · 1 |α| αx ) = |α|ρ ( 1 |α| αx ) = |α|ρ(x) which shows that ρ(αx)= |α|ρ( 1|α|αx) = |α|ρ(x).(ii) since ρ(x +y)= ρ ( 2 (x 2 + y 2 )) =2ρ (x 2 + y 2 ) ≤ ρ(x)+ρ(y), then xρ is a normed space with ρ as norm. 3. the ρ-neumann–jordan constant and the ρ-james constant in 2006, poom kumam [10] generalized two typical constants cnj(x)= sup{‖x +y‖2 +‖x −y‖2 2‖x‖2 +2‖y‖2 : x,y ∈ x,(x,y) 6=(0,0) } and j(x)= sup{min{‖x +y‖,‖x −y‖} : x,y ∈ x,‖x‖= ‖y‖=1} and introduced two new geometric constants cnj(xρ) and j(xρ) defined on modular spaces. definition 2.[10] the ρ-neumann-jordan constant cnj(xρ) of a modular space xρ is defined by cnj(xρ)=2sup { ρ2( x+y 2 )+ρ2( x−y 2 ) ρ2(x)+ρ2(y) : x,y ∈ xρ,ρ(x)=1,ρ(y)≤ 1 } . definition 3.[10] the ρ-james constant j(xρ) of a modular space xρ is defined by j(xρ)=2sup { min{ρ( x +y 2 ),ρ( x −y 2 )} : x,y ∈ xρ,ρ(x)=1,ρ(y)≤ 1 } . in the following section, we extend the proposition 3.5 in [10] and obtain inequalities of cnj(xρ)and j(xρ). theorem 1. let xρ be a modular space, then eur. j. math. anal. 1 (2021) 154 (i) 0 < j (xρ)≤ 4 and 1≤ cnj (xρ)≤ 8, in particular, if ρ is convex, then 1≤ j (xρ)≤ 2 and 1≤ cnj (xρ)≤ 2;(ii)1 2 j2(xρ) ≤ cnj(xρ) ≤ 64j2(xρ) +4, in particular, if ρ is convex, then 12j2(xρ) ≤ cnj(xρ) ≤ 4 j2(xρ) +1. proof. (i) let y =0, then j(xρ)≥ 2sup{ρ( x 2 ) : x ∈ xρ,ρ(x)=1}. since ρ(x) = 1, then ρ(x 2 ) > 0 implies j(xρ) > 0. since ρ(x±y2 ) ≤ ρ(x)+ ρ(y) ≤ 2, then 0 < j(xρ)≤ 4.let x = y , then cnj(xρ)≥ 2sup { ρ2(x+x 2 )+ρ2(x−x 2 ) ρ2(x)+ρ2(x) : x ∈ xρ,ρ(x)=1 } ≥ 1. since ρ2(x +y 2 )+ρ2( x −y 2 )≤ 2(1+ρ(y))2, we have ρ2 ( x+y 2 ) +ρ2 ( x−y 2 ) ρ2(x)+ρ2(y) ≤ 2 ( 1+ 2ρ(y) 1+ρ2(y) ) ≤ 4, thus 1≤ cnj(xρ)≤ 8.in particular, if ρ is convex and let x = y , then j(xρ)≥ 2sup{ρ(x 2 ) : x ∈ xρ,ρ(x)=1}=1. since ρ(x±y 2 )≤ 1 2 ρ(x)+ 1 2 ρ(y)≤ 1, then 1≤ j(xρ)≤ 2. we also can prove 1≤ cnj(xρ)≤ 2by the same way.(ii) since ρ2 ( x +y 2 ) +ρ2 ( x −y 2 ) ≤ 2[1+ρ(y)]2 ≤ 4 ( 1+ρ2(y) ) , then ρ2 ( x+y 2 ) +ρ2 ( x−y 2 ) ρ2(x)+ρ2(y) −2≤ 2(1+ρ(y))2 1+ρ2(y) −2= 4ρ(y) 1+ρ2(y) . since 1 4 (ρ2( x+y 2 )+ρ2( x−y 2 ))≤ 1+ρ2(y), then 4ρ(y) 1+ρ2(y) ≤ 16ρ(y) ρ2( x+y 2 )+ρ2( x−y 2 ) , that is ρ2 ( x+y 2 ) +ρ2 ( x−y 2 ) ρ2(x)+ρ2(y) −2 ≤ 16ρ(y) ρ2 ( x+y 2 ) +ρ2 ( x−y 2 ) ≤ 16 ρ2 ( x+y 2 ) +ρ2 ( x−y 2 ). finally 1 2 cnj(xρ)−2≤ 161 2 j2(xρ) implies that cnj(xρ)≤ 64j2(xρ) +4. eur. j. math. anal. 1 (2021) 155 according to the proof of proposition 3.5 in [12], we can prove 1 2 j2(xρ)≤ cnj(xρ), thus 1 2 j2(xρ)≤ cnj(xρ)≤ 64 j2(xρ) +4. in particular, if ρ is convex, then ρ2( x +y 2 )+ρ2( x −y 2 )≤ 1 2 (1+ρ(y))2 ≤ 1+ρ2(y), thus ρ2 ( x+y 2 ) +ρ2 ( x−y 2 ) ρ2(x)+ρ2(y) − 1 2 ≤ ρ(y) 1+ρ2(y) ≤ 1 ρ2 ( x+y 2 ) +ρ2 ( x−y 2 ). therefore cnj (xρ)≤ 4 j2 (xρ) +1. example 1. (i) consider x = r2, ρ(x) = { 0,x =0 1 ‖x‖1 ,x 6=0 , where ‖x‖1 = ‖(x1,x2)‖1 = |x1|+ |x2|. obviously, xρ is a modular space.we choose x0 = (1 2 , 1 2 ) ,y0 = ( 1 2 ,− 1 2 ), then ρ(x0)= ρ(y0)=1,ρ ( x0 +y0 2 ) = ρ ( x0 −y0 2 ) =2, thus j (xρ)≥ 4. since j (xρ)≤ 4, then j (xρ)=4. according to (ii) of theorem 1, we know that cnj(xρ)=8 in this example.(ii) consider x = r2,ρ(x)= ‖x‖1. obviously, xρ is a modular space and ρ is convex. we have j (xρ)= sup{min{‖x +y‖1,‖x −y‖1} : x,y ∈ xρ,‖x‖=1,‖y‖≤ 1}.we choose x0 = (1,0),y0 = (0,1), then ‖x0‖1 = ‖y0‖1 = 1 and ‖x0 + y0‖1 = ‖x0 − y0‖1 = 2,thus j (xρ)=2. according to (ii) of theorem 1, we can get that cnj(xρ)=2 in this example. 4. the ρ-convex modular and the ρ-smooth modular in order to study the uniform convexity of banach spaces, clarkson introduced the modular ofconvexity δx(ε)= inf{1− 1 2 ‖x +y‖ : ‖x‖= ‖y‖=1,‖x −y‖≥ ε } . goebel called ε0 = sup{ε ∈ [0,2] : δx(ε)= 0} as the characteristic of convexity. based on thegeometric intuitionistic meaning of convexity of banach spaces and its application in fixed pointtheory, this paper gives the ρ-convex modular of modular spaces with reference to the definition of δx(ε). definition 4. the ρ-convex modular δxρ(ε) of a modular space xρ is defined by δxρ(ε)= inf { 1−ρ ( x+y 2 ) : x,y ∈ xρ,ρ(x),ρ(y)≤ 1,ρ(x −y)≥ ε } ,0≤ ε ≤ 2. in particular, if ρ is convex, the ρ-uniform convexity of xρ is defined as ε0 (xρ)= sup { ε ∈ [0,2] : δxρ(ε)=0 } . eur. j. math. anal. 1 (2021) 156 remark 1. we can easily prove that −1≤ δxρ(ε)≤ 1 and δxρ(0)≤ 0.in banach spaces, the convexity modular δx(ε) and the smoothness modular ρx(t)= sup{‖x +y‖+‖x −y‖ 2 −1 : ‖x‖=1,‖y‖=1,t ≥ 0 } are conjugate concepts. therefore, this paper gives the definition of ρ-smooth modular of modularspaces by referring to the definition of smoothness modular ρx(t). definition 5. the ρ-smooth modular ρxρ(t) of a modular space xρ is defined by ρxρ(t)= sup { ρ( x +y 2 )+ρ( x −y 2 )−1 : x,y ∈ xρ,ρ(x)≤ 1,ρ(y)≤ t } ,t ≥ 0. remark 2. it is true that min{0,t −1}≤ ρxρ(t)≤ 1+2t and ρxρ(t) is increasing of t. theorem 2. let x be a modular space, then(i) j(xρ) < 2� if and only if δxρ(�) > 1− �, in particular, if ρ is convex, then j(xρ) < � if andonly if δxρ(�) > 1− �2;(ii) j(xρ)=2sup{� ∈ (0,2) : δxρ(�)≤ 1−�}, in particular, if ρ is convex, then j(xρ)= sup{� ∈ (0,2) : δxρ(�) < 1− � 2 }. proof. (i) note α = j(xρ) < 2�, thus min { ρ( x +y 2 ),ρ( x −y 2 ) } ≤ α 2 , shows that 1−ρ(x+y 2 )≥ 1− α 2 > 1− �. therefore δxρ(�) > 1− �.note β = δxρ(�) > 1− �, then 1−ρ(x+y2 )≥ β implies ρ(x+y2 )≤ 1−β < �. thus min { ρ( x +y 2 ),ρ( x −y 2 ) } = ρ( x +y 2 ). then j (xρ)=2sup { ρ ( x +y 2 ) : x,y ∈ xρ,ρ(x)=1,ρ(y)≤ 1 } ≤ 2−2β < 2ε.in particular, if ρ is convex and let λ = j(xρ) < �, then j(xρ) < � if and only if ∀x,y ∈ xρ,ρ(x),ρ(y)≤ 1, we have ρ(x +y)≤ λ or ρ(x −y)≤ λ. according to the definition of δxρ(�), we obtain ρ(x+y)≥ � > λ, thus ρ(x−y)≤ λ shows that δxρ(�)≥ 1− α 2 > 1− � 2 . (ii) note �0 =sup{� ∈ (0,2) : δxρ(�)≤ 1− �}.suppose �0 < 2 . ∀� ∈ (�0,2), for any x,y ∈ xρ and ρ(x),ρ(y)≤ 1, we have ρ(x −y) > � or ρ(x −y)≤ �. if ρ(x −y) > �, then δxρ(�)≥ 1− � implies ρ(x+y2 )≤ �. thus j(xρ)≤ 2�. eur. j. math. anal. 1 (2021) 157 since δxρ(�)≤ 1− �, then j(xρ)≤ 2�0 shows that j(xρ)=2sup{� ∈ (0,2) : δxρ(�)≤ 1− �}. in particular, if ρ is convex and let α = j(xρ)∈ [1,2], then ∀x,y ∈ xρ,ρ(x),ρ(y)≤ 1, we have ρ(x +y)≤ α or ρ(x −y)≤ α. what’s more, ∀η > 0, there exist x′,y ′ ∈ xρ and ρ(x′),ρ(y ′)≤ 1 such that ρ ( x′ +y ′ ) > α−η and ρ(x′ −y ′) > α−η. fix η > 0, then 1−ρ(x′ +y ′ 2 ) < 1− α−η 2 implies δxρ(�) < 1− α−η2 , therefore sup { � ∈ (0,2) : δxρ(�) < 1− � 2 } ≥ α−η. ∀� ∈ (0,2), if � ≤ α, thus sup { � ∈ (0,2) : δxρ(�) < 1− � 2 } ≤ α. if � > α, then ρ(x +y)≤ α shows that δxρ(�)≥ 1− α2 . in (0,2), we know sup { � ∈ (0,2) : δxρ(�) < 1− � 2 } ≤ α, thus α−η ≤ sup{� ∈ (0,2) : δxρ(�) < 1− �2}≤ α.let η → 0, then sup { � ∈ (0,2) : δxρ(�) < 1− � 2 } = α. theorem 3. let xρ be a modular space, then(i) j(xρ)≤ ρxρ(1)+1;(ii) cnj(xρ)≤ 2(√12+(1+ρxρ(1))2 −2)2. proof. (i) we can deduce that j (xρ)≤ sup { ρ ( x +y 2 ) +ρ ( x −y 2 ) : x,y ∈ xρ,ρ(x)=1,ρ(y)≤ 1 } = ρxρ(1)+1. (ii) we know that a2 +b2 ≤ (a+b)2 −4(a+b)+8 for 0 < a,b ≤ 2. thus ρ2 ( x +y 2 ) +ρ2 ( x −y 2 ) ≤ ( ρ ( x +y 2 ) +ρ ( x −y 2 ))2 −4 ( ρ ( x +y 2 ) +ρ ( x −y 2 )) +8. since ρ ( x +y 2 ) +ρ ( x −y 2 ) ≥ √ ρ2 ( x +y 2 ) +ρ2 ( x −y 2 ) , eur. j. math. anal. 1 (2021) 158 then ρ2 ( x +y 2 ) +ρ2 ( x −y 2 ) +4 √ ρ2 ( x +y 2 ) +ρ2 ( x −y 2 ) −8 ≤ ( ρ ( x +y 2 ) +ρ ( x −y 2 ))2 ≤ ( 1+ρxρ(1) )2 .thus ρ2 ( x +y 2 ) +ρ2 ( x −y 2 ) ≤ (√ 12+ ( 1+ρxρ(1) )2 −2)2 which shows that 1 2 cnj(xρ)≤ (√ 12+ ( 1+ρxρ(1) )2 −2)2 . 5. convexity and non-squareness clarkson introduced uniform convexity in 1936, proved that lp(1≤ p < ∞) spaces are uniformlyconvex banach spaces and uniformly convex banach spaces have radon-nikodym properties. dueto the geometrical intuitiveness of convexity, poom kumam [10] gave the definitions of ρr -uniformlyconvex, ρ-uniformly non-square and ρ-strictly convex of modular spaces in 2006.on the basis of literature [10], this paper studies the relationships between convexity, non-squareness and geometric constants of modular spaces. definition 6.[10] for r > 0, a modular space xρ is said to be ρr -uniformly convex if for each � > 0,there exists δ > 0 such that for any x,y ∈ xρ, the conditions ρ(x)≤ r , ρ(y)≤ r and ρ(x−y)≥ r�imply that ρ(x+y 2 )≤ (1−δ)r . definition 7.[10] the modular space xρ is said to be ρ-uniformly non-square if there exists δ ∈ (0,1)such that for any x,y ∈ xρ with ρ(x)=1 and ρ(y)≤ 1, ρ(x+y2 )≤ 1−δ or ρ(x−y2 )≤ 1−δ. definition 8.[10] the modular space xρ is said to be ρ-strictly convex if for any x,y ∈ xρ, theconditions ρ(x)≤ 1, ρ(y)≤ 1 and x 6= y imply that ρ(x+y 2 ) < 1. theorem 4. let xρ be a modular space, then the following conditions are equivalent.(i) j(xρ) < 2;(ii) �0(xρ) < 2 for all 0 < � ≤ 2;(ii) xρ is ρ-uniformly non-square. proof. suppose j(xρ) < 2. there exists � > 0, for any x,y ∈ xρ with ρ(x)= 1 and ρ(y)≤ 1,such that ρ( x +y 2 )≤ j(xρ) 2 − � < 1− � or ρ(x −y 2 )≤ j(xρ) 2 − � < 1− �, implies xρ is ρ-uniformly non-square.suppose xρ is ρ-uniformly non-square, then we can prove j(xρ) < 2 by the same way. thus(i) and (iii) are equivalent.next, we know that �0(xρ) < 2 if and only if δxρ(2) > 0. let α = δxρ(2), then ∀x,y ∈ xρ and ρ(x)= 1,ρ(y)≤ 1, we can get ρ(x±y2 )≤ 1−α, thus xρ is ρ-uniformly non-square. thus eur. j. math. anal. 1 (2021) 159 (ii) and (iii) are equivalent. remark 3. in fact, this theorem is a generalization of theorem 3.8 in [11]. theorem 5. let x be a modular space, then(i) x is ρ1-uniformly convex if and only if δxρ(�) > 0 for 0 < � ≤ 2;(ii) if δxρ(2)=1, then xρ is ρ-strictly convex. proof. (i) denote δε = δxρ(�). then δxρ(�) > 0 if and only if ∀x,y ∈ xρ,ρ(x),ρ(y) ≤ 1 and ρ(x −y)≥ �, we have ρ(x+y 2 )≤ 1−δ�. thus xρ is ρ1-uniformly convex.(ii) since δxρ(2)=1, then ∀x,y ∈ xρ,ρ(x),ρ(y)≤ 1 and ρ(x −y)≥ 2, we have ρ( x +y 2 )=0 < 1, implies xρ is ρ-strictly convex. 6. midpoint convexity in the following section, we discuss a special type of modular and study its properties in termsof geometric constants. definition 9.[6] let (x,‖ ·‖) be a normed space and xρ be a modular space. then ρ is said to bestrongly midpoint convex with non-negtive constant c if ρ( x+y 2 )≤ ρ(x)+ρ(y) 2 − c 4 ‖x −y‖2. theorem 6. let (x,‖ · ‖) be a normed space and xρ be a modular space. if there exists c ≥ 0such that c‖x‖2 ≤ 1 2 ρ(x) for all x ∈ bxρand ρ is strongly midpoint convex with constant c, then cnj (xρ)≤ 3. proof. since ρ(x+y 2 ) ≤ ρ(x)+ρ(y) 2 − c 4 ‖x −y‖2 and ρ(x−y 2 ) ≤ ρ(x)+ρ(−y) 2 − c 4 ‖x +y‖2, then ρ2 ( x +y 2 ) ≤ 1 4 (ρ(x)+ρ(y))2 − c 4 ‖x −y‖2(ρ(x)+ρ(y))+ c2 16 ‖x −y‖4 and ρ2 ( x −y 2 ) ≤ 1 4 (ρ(x)+ρ(y))2 − c 4 ‖x +y‖2(ρ(x)+ρ(y))+ c2 16 ‖x +y‖4 . therefore, for x ∈ sxρ and y ∈ bxρ, we have ρ2 ( x +y 2 ) +ρ2 ( x −y 2 ) ≤ 1 2 (1+ρ(y))2 − c 4 (1+ρ(y)) ( ‖x +y‖2 +‖x −y‖2 ) + c2 16 ( ‖x +y‖4 +‖x −y‖4 ) . next, we only need to prove ρ(y)+ c2 16 ( ‖x +y‖4 +‖x −y‖4 ) − c 4 ( ‖x +y‖2 +‖x −y‖2 ) (1+ρ(y))−1−ρ2(y)≤ 0. eur. j. math. anal. 1 (2021) 160 let t = ‖x + y‖2 + ‖x − y‖2,s = ‖x + y‖‖x − y‖ and i1 = ρ(y)+ c216 (t2 −2s2) − c4t(1+ ρ(y))−1−ρ2(y), then i1 ≤ t2 −2s2 16 c2 − t 4 c = c ( t2 −2s2 ) 16 ( c − 4t t2 −2s2 ) . since c‖x‖2 ≤ 1 2 ρ(x), then c‖x+y 2 ‖2 ≤ 1 2 ρ ( x+y 2 ) ≤ 1 and c‖x−y 2 ‖2 ≤ 1 2 ρ ( x−y 2 ) ≤ 1.therefore 4t t2 −2s2 = ∥∥x+y 2 ∥∥2 +‖x−y 2 ‖2∥∥x+y 2 ∥∥4 +∥∥x−y 2 ∥∥4 ≥ 1‖x+y 2 ‖2+‖ x−y 2 ‖2 ≥ c, then i1 ≤ 0. example 3. if ρ is convex, then c =0 which satisfies the condition of theorem 6, and cnj (xρ)≤ 2 < 3. theorem 7. let (x,‖·‖) be a normed space and xρ be a modular space. if there exist c,λ,µ,γ > 0such that 2µγ ≤ 1≤ 1 2λ + √ 6 8µ and µρ(x)≤ c‖x‖2 ≤ λρ(x) for all x ∈ xρ. what’more, ρ is strongly midpoint convex with constant c, then cnj (xρ)≤ 2. proof. by following the ideas in theorem 6, we can get ρ2 ( x+y 2 ) +ρ2 ( x−y 2 ) 1+ρ2(y) ≤ 1 2 + 1 1+ρ2(y) { ρ(y)− c 4 (1+ρ(y)) ( | x +y ∥∥2+∥∥x −y‖2)} + 1 1+ρ2(y) { c2 16 ( ‖x +y ∥∥4+∥∥x −y‖4)}. let t = ‖x +y‖2 +‖x −y‖2,s = ‖x +y‖‖x −y‖ and i2 = t2 −2s2 16 c2 − (1+ρ(y))t 4 c − 1 2 (1−ρ(y))2, thus we only need to prove i2 ≤ 0. since ρ(y)≤ 1, then (1+ρ(y))t − √ (1+ρ(y))2t2 +2(t2 −2s2)(1−ρ(y))2 ≤ t and (1+ρ(y))t +√(1+ρ(y))2t2 +2(t2 −2s2)(1−ρ(y))2 ≥ t +√3t2 −4s2.thus t t2 −2s2 = 1 4 · ‖x+y 2 ∥∥2 +‖x−y 2 ‖2 ‖x+y 2 ‖4 + ∥∥x−y 2 ∥∥4 and t + √ 3t2 −4s2 t2 −2s2 = 1 4 · ‖x +y‖2 +‖x−y 2 ‖2 + √ 3‖x+y 2 ‖4 +3‖x−y 2 ‖4 +2‖x+y 2 ‖2‖x−y 2 ‖2 ‖x+y 2 ‖4 +‖x−y 2 ‖4 , eur. j. math. anal. 1 (2021) 161 then t+ √ 3t2−4s2 t2−2s2 ≥ 1 4‖x+y 2 ‖2+4‖x−y 2 ‖2 + √ 3 4 √ ‖x+y 2 ‖4+‖x−y 2 ‖4 ≥ c λ(ρ(x+y2 )+ρ( x−y 2 )) + √ 3c 4µ √ ρ2(x+y2 )+ρ 2(x−y2 ) ≥ c 4λ + √ 3c 8 √ 2µ ≥ c 2 ,and t t2 −2s2 ≤ 1 2 · 1∥∥x+y 2 ∥∥2 +∥∥x−y 2 ∥∥2 ≤ c4µ(ρ(x+y 2 ) +ρ ( x−y 2 )) ≤ c 4µγ ≤ c 2 . therefore (1+ρ(y))t − √ (1+ρ(y))2t2 +2(t2 −2s2)(1−ρ(y))2 t2 −2s2 ≤ c 2 ≤ (1+ρ(y))t + √ (1+ρ(y))2t2 +2(t2 −2s2)(1−ρ(y))2 t2 −2s2thus i2 = t2 −2s2 4 ( c 2 − (1+ρ(y))t + √ (1+ρ(y))2t2 +2(t2 −2s2)(1−ρ(y))2 t2 −2s2 ) ( c 2 − (1+ρ(y))t − √ (1+ρ(y))2t2 +2(t2 −2s2)(1−ρ(y))2 t2 −2s2 ) ≤ 0. example 4. consider ρ(x) = 4c‖x‖2 and let λ = µ = 1 4 , then µ2ρ(x) ≤ c‖x‖2 ≤ λρ(x) and µ2 = λ 4 . what’more, cnj (xρ)=2c sup { ‖x +y‖4 +‖x −y‖4 ‖x +y‖2 +‖x −y‖2 : x,y ∈ xρ,ρ(x)=1,ρ(y)≤ 1 } ≤ 2c sup { ‖x +y ∥∥2+∥∥x −y‖2 : x,y ∈ xρ,ρ(x)=1,ρ(y)≤ 1} ≤ 4c sup{‖x‖2 +‖y‖2 : x,y ∈ xρ,ρ(x)=1,ρ(y)≤ 1} =sup{ρ(x)+ρ(y) : x,y ∈ xρ,ρ(x)=1,ρ(y)≤ 1}=2. theorem 8. let (x,‖ · ‖) be a normed space, xρ be a modular space and α0 ∈ (0,2√2]. if thereexists c > 0 such that c ≥ 4α0√ ‖x0 +y0‖4 +‖x0 −y0‖4 for some x0 ∈ sxρ,y0 ∈ bxρ and ρ is strongly midpoint convex with positive constant c, then cnj (xρ) ≥ α20. in particular, if α0 =2 √ 2, then cnj (xρ)=8. proof. since 2ρ(x)≥ c‖x‖2, then ρ2(x±y 2 ) ≥ c 2 ∥∥x±y 2 ∥∥2. therefore ρ2 ( x+y 2 ) +ρ2 ( x−y 2 ) 1+ρ2(y) ≥ c2 16 ( ‖x +y‖4 +‖x −y‖4 ) 1+ρ2(y) , eur. j. math. anal. 1 (2021) 162 shows that ρ2(x+y2 )+ρ2(x−y2 ) 1+ρ2(y) ≥ α 2 0 1+ρ2(y) , then cnj (xρ) ≥ α20. if α0 = 2√2, then cn (xρ) ≥ 8implies cnj (xρ)=8. 7. data availability no data were used to support this study. 8. conflicts of interest the author(s) declare(s) that there is no conflict of interest regarding the publication of this paper. 9. funding statement this work was supported by the national natural science foundation of p. r. china (nos.11971493 and 12071491). references [1] j. lindenstrauss, on the modulus of smoothness and divergent series in banach spaces, michigan math. j. 10 (1963). https://doi.org/10.1307/mmj/1028998906.[2] j.a. clarkson, uniformly convex spaces, trans. amer. math. soc. 40 (1936) 396–396. https://doi.org/10.1090/ s0002-9947-1936-1501880-4.[3] j.a. clarkson, the von neumann-jordan constant for the lebesgue spaces, the annals of 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(2003).[13] c. yang, a note on jordan-von neumann constant and james constant, j. math. anal. appl. 357 (2009) 98–102. https://doi.org/10.1016/j.jmaa.2009.04.002.[14] c. yang, an inequality between the james type constant and the modulus of smoothness, journal of mathematicalanalysis and applications. 398 (2013) 622–629. https://doi.org/10.1016/j.jmaa.2012.07.063. https://doi.org/10.1307/mmj/1028998906 https://doi.org/10.1090/s0002-9947-1936-1501880-4 https://doi.org/10.1090/s0002-9947-1936-1501880-4 https://doi.org/10.2307/1968512 https://doi.org/10.2307/1970663 https://doi.org/10.2307/1970663 https://www.uv.es/llorens/documento.pdf https://doi.org/10.4064/sm-18-1-49-65 https://doi.org/10.4064/sm-18-1-49-65 https://doi.org/10.1007/978-3-319-14051-3 https://doi.org/10.1016/j.jmaa.2009.04.002 https://doi.org/10.1016/j.jmaa.2012.07.063 eur. j. math. anal. 1 (2021) 163 [15] g. nordlander, the modulus of convexity in normed linear spaces, ark. mat. 4 (1960) 15–17. https://doi.org/ 10.1007/bf02591317.[16] k. nikodem, z. pales, characterizations of inner product spaces by strongly convex functions, banach j. math. anal.5 (2011) 83–87. https://doi.org/10.15352/bjma/1313362982. https://doi.org/10.1007/bf02591317 https://doi.org/10.1007/bf02591317 https://doi.org/10.15352/bjma/1313362982 1. introduction 2. preliminaries 3. the -neumann–jordan constant and the -james constant 4. the -convex modular and the -smooth modular 5. convexity and non-squareness 6. midpoint convexity 7. data availability 8. conflicts of interest 9. funding statement references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 10doi: 10.28924/ada/ma.2.10 weak and strong convergence theorems of modified projection-type ishikawa iteration scheme for lipschitz α-hemicontractive mappings imo kalu agwu∗, donatus ikechi igbokwe department of mathematics, micheal okpara university of agriculture, umudike, umuahia abia state, nigeria agwuimo@gmail.com, igbokwedi@yahoo.com ∗correspondence: agwuimo@gmail.com abstract. in this paper, we establish weak and strong convergence theorems of a two-step modifiedprojection-type ishikawa iterative scheme to the fixed point of α-hemicontractive mappings withoutany compactness assumption on the operator or the space. our results extend, improve and generalizeseveral previously known results of the existing literature. 1. introduction let h be a real hilbert space with inner product 〈, .,〉 and induced norm ‖, .,‖, k a nonemptyconvex and closed subset of h and t : k −→ k a selfmap on k. we use f (t ) to denote the setof fixed point of t , n to denote the set of natural numbers and xn → x (respectively xn ⇀ x) todenote the strong (weak) convergence of the sequence {xn}∞n=0 to the point x. definition 1.1. let t : k −→ k be a maaping. then i. t is said to be l-lipschitizian if there exists l > 0 such that ‖ts −tz‖≤‖s −z‖,∀s,z ∈ k. (1.1) from the definition, it easy to observe that every nonexpansive mapping is lipschitizian with l = 1. ii. t is called k-strictly pseudocontraction (see, for example, [9]) if there exists k ∈ (0, 1] such that for all s,z ∈ k, the inequality ‖ts −tz‖2 ≤‖s −z‖2 + k‖(i −t )s − (i −t )z‖2 (1.2) hods. note that if k = 1 in (1.2), then t is a pseudocontraction. it well-known that in real hilbert spaces, the class of nonexpansive mapping is a proper subclass of the class of received: 1 nov 2021. key words and phrases. strong convergence; modified ishikawa iterative scheme; weak convergence; α-hemicontractive operator; fixed point; real hilbert space. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 2 k-strictly pseudocontive mapping. also, the class of k-strictly pseudocontive mapping is a proper subclass of the class of pseudocontive mapping. iii. t is called demicontractive mapping (see, for example, [?]) if f (t ) = {x ∈ k : x = tx} 6= ∅ and ∀(s ×q) ∈ (k ×f (t )), there exists k ∈ [0, 1) such that the inequality ‖ts −tq‖2 ≤‖s −q‖2 + k‖s −ts‖2 (1.3) hods. iv. t is said to satisfy condition a (see, for example [?]) f (t ) = {x ∈ k : x = tx} 6= ∅ and there exists λ > 0 such that 〈s −ts,s −q〉≥ λ‖s −ts‖2,∀(s ×q) ∈ (k ×f (t )). (1.4) it is worthy to mention that the class of k-strictly pseudocontions with a nonempty fixed point set is a proper subclass of the class demicontractions. t is called hemicontraction (see, for example, [17]) if k = 1 in (1.3). the class of pseudocontractive maps is a proper subclass of the class of hemicontractive maps. again, the class of demicontractive maps is a proper subclass of the class of hemicontractive maps (see, for example, [?]). these two classes of mappings have been studied extensively by many researchers (see, for example, [?], [13], [17] and the references therein). v. t is called α-demicontraction (see, for examole, [13] ) if f (t ) = {x ∈ k : x = tx} 6= ∅ and ∀(s ×q) ∈ (k ×f (t )), there exist λ > 0 and α ≥ 1 such that the inequality 〈s −ts,s −αq〉≥ λ‖s −ts‖2,∀(s ×q) ∈ (k ×f (t )). (1.5) holds. clearly, (1.5) is equivalent to ‖ts −αq‖2 ≤‖s −αq‖2 + k‖s −ts‖2, (1.6) where k = 1 − 2λ ∈ [0, 1). v. t is called α-hemicontraction (see, for examole, [17] ) if f (t ) = {x ∈ k : x = tx} 6= ∅ and ∀(s ×q) ∈ (k ×f (t )), there exists α ≥ 1 such that the inequality ‖ts −αq‖2 ≤‖s −αq‖2 + ‖s −ts‖2 (1.7) holds. observe that (1.7) is equivalent to 〈s −ts,s −αq〉≥ 0,∀(s ×q) ∈ (k ×f (t )). (1.8) in [ [17], example 2.2], osilike and onah gave an example of α-hemicontractive mapping with α > 1 which is not hemicontractive mapping, and also showed that there are hemicontractive (1-hemicontractive) mappings which are not α-hemicontraction for α > 1(see [ [17], example 2.1] for details). again, osilike and onah [17] presented an example of a mapping which is hemicontractive (1-hemicontractive) and alpha-hemicontractive mapping for α > 1 but https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 3 neither demicontractive (1-demicontractive) nor α-demicontractive mapping for α > 1(see [17], example 2.3 for details). for further cheracterisation of α-hemicontractive mapping, interested reader should consult [17]. a mapping t : h −→ h is called ν-strongly monotone if there exists ν > 0 such that 〈s −ts,s −z〉≥ ν‖s −z‖2,∀s,z ∈ h.. (1.9) iterative method for approximating fixed point of l-lipschitz pseudocontractive mapping has beenan active area of investigation in recent times (see, for example, [?], [?], [20], [14], [26], [27] and thereferences contained in them). in [24], voluhan introduced the modified projection-type ishikawaiterative method in the following way: let h be a hilbert space, k nonempty, closed and convexsubset of h and t : k −→ k be an l-lipshitz pseudocontractive mapping. for an arbitrary x0 ∈ k, define the sequence {xn}∞n=0 iteratively as follows. xn+1 = pk[(1 −αn −γn)xn + γntyn] yn = (1 −βn)xn + βntxn,n ≥ 1, (1.10) where {αn}∞n=0,{βn}∞n=0,{γn}∞n=0 ∈ (0, 1) and pk is a projection map from h onto k. using(1.10), she proved the following theorem. theorem 1.1. let h be a hilbert space, d a nonempty closed convex subset of h and t : d −→ d an l-lipschitz pseudocontractive mapping such that f (t ) 6= ∅. for any given x0 ∈ h, let {xn}∞n=0 be the sequence defined by (1.10). assume the sequences {αn}∞n=0,{βn}∞n=0,{γn}∞n=0 ∈ (0, 1) satisfy (1) βn(1 −αn) > γn,∀n ≥ 1;(2) limn→∞αn = 0 and ∑∞n=0αn = ∞;(3) 0 < α ≤ γn ≤ βn ≤ β < 1√ 1 + l2 + 1 ,∀n ≥ 1. then, the sequence {xn}∞n=0 strongly converges to the fixed point of t . remark 1.1. if αn = 0,∀n ≥ 1, and pk is an identity, (1.10) reduces to the well-known ishikawa iteration method  xn+1 = (1 −γn)xn + γntyn yn = (1 −βn)xn + βntxn,n ≥ 1, (1.11) which has been used by several researchers to approximate the fixed points of different operators or operator equations in different spaces. motivated and inspired by the works in [17], [24] and some ongoing research in this direction, itis our purpose in this paper to extend the results in [24] and other related results from lipschitzpseudocontractive mapping to the more general α-hemicontractive mapping. our results is more https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 4 general and also more applicable because fewer and simpler conditions are required to attainconvergence. 2. preliminary the following definitions and lemmas will be needed to prove our main results. definition 2.1. (see [27]) let h and k be as defined above. for each x ∈ h, there exists a unique nearest point of k, denoted by pkx, such that ‖x −pkx‖≤‖x −y‖,∀y ∈ k. such a pk is called metric projection from h onto k. it is well-known that pk is firmly nonexpansive mapping from h onto k; that is, ‖pkx −pky‖2 ≤〈pkx −pky,x −y〉,∀x,y ∈ h. also, for any x ∈ h and z ∈ k,z = pkx if and only if 〈x −z,z −y〉≥ 0,∀y ∈ k. definition 2.2. the banach space z is said to have opial property, if for each weakly convergent sequence {zn}∞n=0with weak limit z ∈ z, the following inequality holds: lim sup n→∞ ‖zn −z‖ < ‖zn −y‖,∀y ∈ zwithz 6= y. note that all finite dimensional banach spaces, all hilbert spaces and `p(0 ≤ p < ∞) satisfy the opial property. but lp(1 < p < ∞.p 6= 2) do not satisfies the opial property. definition 2.3. (see [27]) let e be a real banach space. a mapping t, with domain d(t ) ∈ e, is said to be demiclosed at 0 if for any sequence zn ⊂ e,zn � q ∈ d(t ) and ‖zn−tzn‖→ 0, then tq = q. lemma 2.1. (see [27]) let h be a real hilbert space. then, the following inequality holds: ‖λx + (1 −λ)y‖2 ≤ λ‖x‖2 + (1 −λ)‖y‖2 −λ(1 −λ)‖x −y‖,∀λ ∈ [0, 1],∀x,y ∈ h. lemma 2.2. (see [27]) let {sn}n∈n be a sequence of nonnegative real numbers satisfying the inequality: sn+1 ≤ (1 −γn)sn + δn,∀n ≥ 1, where {γn}n∈n and {δn}n∈n satisfy the following conditions:(i) {γn}n∈n ⊂ (0, 1);(ii) ∑∞n=1γn = ∞. suppose ∑∞ n=1δn < ∞, then,limn→∞ sn = 0. https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 5 lemma 2.3. (see [4]) let e be a real hilbert space. then, for all x,y ∈ h, the following inequalities hold: i. ‖x −y‖2 ≤‖x‖2 − 2〈y, (x + y)〉 + ‖y‖2; ii. ‖x −y‖2 ≤‖x‖2 − 2〈y, (x + y)〉. lemma 2.4. (see [?]) let d be a sunset of a real hilbert space, t : d −→ h be a nonexpansive mapping and z a weak cluster point of the sequence {yn}∞n=0. if ‖tyn −yn‖→ 0, then z ∈ f (t ) proposition 2.5. (see [27]) let d be a nonempty subset of a real hilbert space amd γ : d −→ d an α-demicontractive mapping. assume that x ∈ d and α ≥ 1. then, γ is lipschitizian. theorem 2.6. (see [4]) a banach space e is reflexive if and only if every (normed) bounded sequence in e has a subsequence which converges weakly to an element of e. 3. convergence results now, we prove our main results. theorem 3.1. let h be a real hilbert space, k a nonempty closed convex subset of h and t : k −→ k an l-lipschitz α-hemicontractive mapping. for any arbitrary x0 ∈ h, define the sequence {xn}∞n=0 iteratively as follows: xn+1 = pk[(1 −αn −γn)xn + γntyn] yn = (1 −βn)xn + βntxn,n ≥ 1, (3.1) where the sequences {δn}∞n=0,{γn} ∞ n=0,{βn} ∞ n=0 ∈ (0, 1) satisfy the following conditions: (i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤ 1 −δ 1 + l2 ; (ii) limn→∞δn = 0 and ∑∞ n=0δn = ∞. then, the sequence {xn}∞n=0 generated by (3.1) weakly and strongly converges to the fixed point of t . proof. since f (t ) is nonempty, let αq ∈ f (t ) and x ∈ k. using (3.1), lemma 2.1 and the factthat t is l-lipschitizian, we estimate as follows: ‖xn+1 −αq‖2 = ‖pk[(1 −δn −γn)xn + γntyn] −αq‖ ≤ ‖(1 −δn −γn)xn + γntyn −αq‖ = ‖(1 −δn −γn)(xn −αq) + γn(tyn −αq) −δnαq‖ ≤ ‖(1 −δn −γn)(xn −αq) + γn(tyn −αq)‖ + δn‖αq‖. (3.2) set qn = ‖(1 −δn −γn)(xn −αq) + γn(tyn −αq)‖2 and observe that qn = ‖(1 −δn)(xn −αq) − (1 −γn)(xn −αq) + γn(tyn −αq)‖2. (3.3) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 6 since (1 −δn)(xn −αq) = (1 −δn)(1 −γn)(xn −αq) + γn(1 −δn))(xn −αq) (3.4) and γn(tyn −αq) = γn(1 −δn)(tyn −αq) + γnδn(tyn −αq), (3.5) it follows from (3.3) that qn = ‖(1 −δn)(1 −γn)(xn −αq) + γn(1 −δn))(xn −αq) − (1 −γn)(xn −αq) +γn(1 −δn)(tyn −αq) + γnδn(tyn −αq)‖2 = ‖(1 −δn)[(1 −γn)(xn −αq) + γn(tyn −αq)] + δnγn(tyn −xn)‖2. (3.6) (3.6) and lemma 2.1 imply that qn = (1 −δn)‖(1 −γn)(xn −αq) + γn(tyn −αq)‖2 + δn‖γn(tyn −xn)‖2 −δn(1 −δn)‖xn −αq‖2. (3.7) if we denote vn = ‖(1 −γn)(xn −αq) + γn(tyn −αq)‖2 and use similar technique as above, thenwe get vn = (1 −γn)‖xn −αq‖2 + γn‖tyn −αq‖2 −γn(1 −γn)‖xn −tyn‖2. (3.8) (3.7) and (3.8) imply qn = (1 −δn)[(1 −γn)‖xn −αq‖2 + γn‖tyn −αq‖2 −γn(1 −γn)‖xn −tyn‖2] +δnγ 2 n‖tyn −xn‖ 2 −δn(1 −δn)‖xn −αq‖2 = (1 −δn)(1 −γn)‖xn −αq‖2 + (1 −δn)γn‖tyn −αq‖2 −γn(1 −γn)(1 −δn)‖xn −tyn‖2 +δnγ 2 n‖tyn −xn‖ 2 −δn(1 −δn)‖xn −αq‖2 ≤ (1 −δn)(1 −γn)‖xn −αq‖2 + (1 −δn)γnl2‖yn −αq‖2 −(γn −δnγn −γ2n + γ 2 nδn)‖xn −tyn‖ 2 + δnγ 2 n‖tyn −xn‖ 2 −δn(1 −δn)‖xn −αq‖2 = (1 −δn)(1 −γn)‖xn −αq‖2 + γnl2‖yn −αq‖2 −δnγnl2‖yn −αq‖2 −(γn −δnγn −γ2n)‖xn −tyn‖ 2 −δn(1 −δn)‖xn −αq‖2. (3.9) observr that |xn −tyn‖ ≤ (‖xn −αq‖ + l‖yn −αq‖)2 = ‖xn −αq‖2 + l(2‖xn −αq‖‖yn −αq‖) + l2‖yn −αq‖2 ≤ ‖xn −αq‖2 + l‖xn −αq‖2 + l‖yn −αq‖2 + l2‖yn −αq‖2 = (1 + l)‖xn −αq‖2 + l(1 + l)‖yn −αq‖2. (3.10) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 7 (3.9) and (3.10) imply qn ≤ (1 −δn)(1 −γn)‖xn −αq‖2 + γnl2‖yn −αq‖2 −δnγnl2‖yn −αq‖2 −(γn −δnγn −γ2n)[(1 + l)‖xn −αq‖ 2 + l(1 + l)‖yn −αq‖2] −δn(1 −δn)‖xn −αq‖2 = (1 −δn)(1 −γn)‖xn −αq‖2 − (1 + l)(γn −δnγn −γ2n)‖xn −αq‖ −[(γn −δnγn −γ2n)l−l 2γ2n]‖yn −αq‖ 2 −δn(1 −δn)‖xn −αq‖2 (3.11) again, from (3.1), we get ‖yn −αq‖2 = ‖(1 −βn)(xn −αq) + βn(txn −αq)‖2 (3.12) since t is α-hemicontractive mapping, it follows from (3.12) and lemma 2.1 that ‖yn −αq‖2 ≤ (1 −βn)‖xn −αq‖2 + βn[‖xn −αq‖2‖2 + ‖xn −txn‖2] −βn(1 −βn)‖xn −txn‖2 = (1 −βn)‖xn −αq‖2 + β2n‖xn −txn‖ 2. (3.13) putting (3.13) into (3.11), we have qn ≤ (1 −δn)(1 −γn)‖xn −αq‖2 − (1 + l)(γn −δnγn −γ2n)‖xn −αq‖ −[(γn −δnγn −γ2n)l−l 2γ2n]{(1 −βn)‖xn −αq‖ 2 + β2n‖xn −txn‖ 2} −δn(1 −δn)‖xn −αq‖2 ≤ (1 −δn)(1 −γn)‖xn −αq‖2 − [(γn −δnγn −γ2n)(1 + l) + δn(1 −δn) −l 2γ2n]‖xn −αq‖ 2 −β2n[(γn −δnγn −γ 2 n)l−l 2γ2n]‖xn −txn‖ 2. (3.14) since from condition (i), (γn −δnγn −γ2n) −l2γ2n ≥ 0, it follows from (3.14) that qn ≤ (1 −δn)2‖xn −αq‖2 (3.15) (3.2) and (3.15) imply |xn+1 −αq‖ ≤ (1 −δn)‖xn −αq‖2 + δn‖αq‖ ≤ max{‖xn −αq‖2,‖αq‖},∀n ∈n. it is easy to see, using mathematical induction, that |xn+1 −αq‖ ≤ max{‖xn −αq‖2,‖αq‖} = ‖x0 −αq‖2. (3.16) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 8 hence, {xn}∞n=0 is bounded.furthermore, since from (3.1), ‖xn+1 −αq‖2 = ‖pk[(1 −δn −γn)xn + γntyn] −αq‖2 ≤ ‖(1 −δn −γn)xn + γntyn −αq‖2 = ‖xn −αq −γn(xn −tyn) −δnxn‖2, it follows from lemma 2.3(i) that ‖xn+1 −αq‖2 ≤ ‖xn −αq −γn(xn −tyn)‖2 − 2δn〈xn,xn+1 −αq〉. (3.17) since ‖xn −αq −γn(xn −tyn)‖2 = ‖(1 −γn)(xn −αq) + γn(αq −tyn)‖2 = (1 −γn)‖xn −αq‖2 + γn‖αq −tyn‖2 −γn(1 −γn)‖tyn −xn‖2 ≤ (1 −γn)‖xn −αq‖2 + γnl2‖yn −αq‖2 −γn(1 −γn)‖tyn −xn‖2, (3.18) it follows from (3.10) that ‖xn −αq −γn(xn −tyn)‖2 ≤ (1 −γn)‖xn −αq‖2 + γnl2‖yn −αq‖2 −γn(1 −γn){(1 + l)‖xn −αq‖2 + l(1 + l)‖yn −αq‖2} = (1 −γn)‖xn −αq‖2 + γnl2‖yn −αq‖2 −γn(1 −γn)(1 + l)‖xn −αq‖2 −γn(1 −γn)l‖yn −αq‖2 −γnl2‖yn −αq‖2 + γ2nl 2‖yn −αq‖2 = (1 −γn)‖xn −αq‖2 −γn(1 −γn)(1 + l)‖xn −αq‖2 −[γn(1 −γn)l−l2γ2n]‖yn −αq‖ 2. (3.19) (3.13) and (3.19) imply ‖xn −αq −γn(xn −tyn)‖2 ≤ (1 −γn)‖xn −αq‖2 −γn(1 −γn)(1 + l)‖xn −αq‖2 −[γn(1 −γn)l−l2γ2n]{(1 −βn)‖xn −αq‖ 2 + β2n‖xn −txn‖ 2} ≤ (1 −γn)‖xn −αq‖2 −γnl[1 −γn −γnl]{(1 −βn)‖xn −αq‖2 +β2n‖xn −txn‖ 2}. (3.20) by condition (i), 1 −γn −γnl > 0,∀n ≥ 0. consequently, ‖xn −αq −γn(xn −tyn)‖2 ≤ ‖xn −αq‖2 −(1 −γn −γnl)β2nγnl‖xn −txn‖ 2. (3.21) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 9 (3.17)and (3.21) imply ‖xn+1 −αq‖2 ≤ ‖xn −αq‖2 − (1 −γn −γnl)β2nγnl‖xn −txn‖ 2 −2δn〈xn,xn+1 −αq〉. since {xn} is bounded, there exists a constant b > 0 such that −2〈xn,xn+1 −αq〉≤ b. thus, ‖xn+1 −αq‖2 ≤ ‖xn −αq‖2 − (1 −γn −γnl)β2nγnl‖xn −txn‖ 2 δnb. the last inequality implies that ‖xn+1 −αq‖2 −‖xn −αq‖2 + (1 −γn −γnl)β2nγnl‖xn −txn‖ 2 ≤ δnb. (3.22) now, we consider the following two cases:case a: suppose there exists n0 ∈n such that {‖xn −αq‖} is non-increasing. then, {‖xn −αq‖}is convergent. clearly, ‖xn+1−αq‖−‖xn−αq‖→ 0. in view, of condition (ii) and (3.22), we have ‖xn−txn‖→ 0. by lemma 2.4, it is obvious that ωω(xn) ⊂ f (t ), where ωω(xn){x : ∃xnk ⇀ αx?}is the weak limit set of {xn}. this implies that the sequence {xn} converges weakly to a fixed point αx? of t .suppose there exists some subsequences {xnk}∞k=0 ⊂{xn}∞n=0 such that xnk ⇀ αy? weakly and αy? 6= αx?. since limn→∞‖xn −αv‖ exists for αv ∈ f (t ), by virtue of opial condition on h, wehave lim n→∞ ‖xn −αx?‖ = lim n→∞ ‖xnj −αx ?‖ < lim n→∞ ‖xnj −αy ?‖ = lim n→∞ ‖xnk −αy ?‖ < lim n→∞ ‖xnk −αx ?‖ = lim n→∞ ‖xnj −αy ?‖, which is a contradiction. consequently, αy? = αx?. this implies that {xnj}∞j=0 converges wealy toa common fixed point of t.next, we prove that {xn}∞n=0 converges strongly to x?/ let ξn = γntyn + (1−γnxn). then, from(3.1), we obtain xn+1 = pk[ξn −δnxn],n ≥ 0. this implies that xn+1 = pk[ξn + δnξn + δnξn −δnxn = pk[(1 −δn)ξn + δn(ξn −xn)]. (3.23) observe that ‖ξn −αx?‖2 = ‖xn −αx? −γn(xn −tyn)‖2. (3.24)by using the same argument as in (3.20), with αx? = αq, we get, from (3.24), that ‖ξn −αx?‖ = ‖xn −αx?‖. (3.25) again, from (3.1), we obtain ‖yn −xn‖ = βn‖xn −txn‖→ 0 as n →∞,βn ∈ (0, 1). (3.26) https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 10 in addition, since t is lipschitz, it follows that ‖ξn −xn‖ = ‖γn[(tyn −txn) − (xn −txn)]‖ ≤ γn‖tyn −txn‖−γn‖xn −txn‖ ≤ γnl‖yn −xn‖ + γn‖xn −txn‖→ 0 as n →∞. (3.27) now, using (3.23), we get ‖xn+1 −αx?‖2 ≤ ‖(1 −δn)ξn + δn(ξn −xn) −αx?‖2 = ‖(1 −δn)(ξn −αx?) + δn(ξn −xn) −δnαx?‖2, which by lemma 2.3 yields ‖xn+1 −αx?‖2 ≤ ‖(1 −δn)(ξn −αx?) + δn(ξn −xn)‖2 − 2δn〈αx?,xn+1 −αx?〉 = (1 −δn)‖ξn −αx?‖2 + δn‖ξn −xn‖2 −δn(1 −δn)‖xn −αx?‖2 −2δn〈αx?,xn+1 −αx?〉 ≤ (1 −δn)‖ξn −αx?‖2 + ‖ξn −xn‖2 − 2δn〈αx?,xn+1 −αx?〉 = (1 −δn)‖ξn −αx?‖2 − 2δn〈αx?,xn+1 −αx?〉 ( by (3.27)) (3.28) ≤ (1 −δn)‖ξn −αx?‖2 (3.29) (3.29) and lemma 2.2 imply that xn → αx? as n →∞.case b: assume that {‖xn − αq‖}∞n=0 is not a monotonically increasing sequence. set vn = ‖xn −αq‖2 and let τ : n−→n be a mapping defined by τn = max{k ∈n : k ≤ n,vn ≤ vn+1},∀n ≥ n0, for some n0 large enough. obviously, {τn}∞n=0 is a nondecreasing sequence given that τn → ∞ as n →∞ and vτn ≤ vτn+1 for all n ≥ n0. from (3.22), ‖xτ(n) −txτ(n)‖ 2 ≤ δτ(n)b (1 −γτ(n) −γτ(n)l)β2τ(n)γτ(n)l → 0 as n →∞. (3.30) therefore, limn→∞‖xτ(n) − txτ(n)‖ = 0. using similar argument as case a above, we concludethat {xτ(n)}→ αx? →∞.from (3.28), we have 0 ≤‖xτ(n)+1 −αx ?‖2 −‖xτ(n) −αx ?‖2 ≤ δτ(n)[2〈αx ? −xτ(n)+1 −‖xτ(n) −αx ?‖2], (3.31) for δτ(n) ∈ (0, 1). hence, limn→∞‖xτ(n) − αx?‖2 = 0. this implies that limn→∞vτ(n) = limn→∞vτ(n)+1 = 0. in addition, for n ≥ n0, it is easy to see that vτ(n) = vτ(n)+1 if n 6= τ(n)(i.e., τ(n) < n) because vj > vj+1, f or τ(n) + 1 ≤ n. consequently. we obtain, for all n ≥ n0, 0 ≤ vτ(n)max{vτ(n),vτ(n)+1} = vτ(n)+1. hence, limn→∞vn = 0. that is, {xn}∞n=0 converges https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 11 strongly to αx?, and this completes the proof. � the following corollaries are immediate consequence of theorem 3.1. corollary 3.2. let h be a real hilbert space, k a nonempty closed convex subset of h and t : k −→ k an l-lipschitz hemicontractive mapping. for any arbitrary x0 ∈ h, define the sequence {xn}∞n=0 iteratively as follows: xn+1 = pk[(1 −αn −γn)xn + γntyn] yn = (1 −βn)xn + βntxn,n ≥ 1, (3.32) where the sequences {δn}∞n=0,{γn} ∞ n=0,{βn} ∞ n=0 ∈ (0, 1) satisfy the following conditions: (i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤ 1 −δ 1 + l2 ; (ii) limn→∞δn = 0 and ∑∞ n=0δn = ∞. then, the sequence {xn}∞n=0 generated by (3.32) weakly and strongly converges to the fixed point of t . corollary 3.3. let h be a real hilbert space, k a nonempty closed convex subset of h and t : k −→ k is α-demicontractive mapping. for any arbitrary x0 ∈ h, define the sequence {xn}∞n=0 iteratively as follows: xn+1 = pk[(1 −αn −γn)xn + γntyn] yn = (1 −βn)xn + βntxn,n ≥ 1, (3.33) where the sequences {δn}∞n=0,{γn} ∞ n=0,{βn} ∞ n=0 ∈ (0, 1) satisfy the following conditions: (i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤ 1 −δ 1 + l2 ; (ii) limn→∞δn = 0 and ∑∞ n=0δn = ∞. then, the sequence {xn}∞n=0 generated by (3.33) weakly and strongly converges to the fixed point of t . corollary 3.4. let h be a real hilbert space, k a nonempty closed convex subset of h and t : k −→ k is demicontractive mapping. for any arbitrary x0 ∈ h, define the sequence {xn}∞n=0 iteratively as follows:  xn+1 = pk[(1 −αn −γn)xn + γntyn] yn = (1 −βn)xn + βntxn,n ≥ 1, (3.34) where the sequences {δn}∞n=0,{γn} ∞ n=0,{βn} ∞ n=0 ∈ (0, 1) satisfy the following conditions: (i) 0 < δ ≤ δn ≤ βn ≤ γn ≤ γ ≤ 1 −δ 1 + l2 ; (ii) limn→∞δn = 0 and ∑∞ n=0δn = ∞. https://doi.org/10.28924/ada/ma.2.10 eur. j. math. anal. 10.28924/ada/ma.2.10 12 then, the sequence {xn}∞n=0 generated by (3.34) weakly and strongly converges to the fixed point of t . competing interest. the authors declare that there is no conflict of interest. references [1] f.e. browder, nonlinear mappings of nonexpansive and accretive type in banach spaces, bull. amer. math. soc.73 (1967) 875-882.[2] f.e. browder, w.v. petryshyn, construction of fixed points of nonlinear mappings in hilbert space, j. math. anal.appl. 20 (1967) 197-228. https://doi.org/10.1016/0022-247x(67)90085-6.[3] c.e. chidume, picards 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https://doi.org/10.1090/s0002-9939-00-05573-8.[27] i. k. agwu, d. i. igbokwe, hybrid-type iteration scheme for approximating fixed point of lipschitz α-hemicontractivemappings, adv. fixed point theory, 10 (2020) 3. https://doi.org/10.28919/afpt/4442. https://doi.org/10.28924/ada/ma.2.10 https://doi.org/10.1006/jmaa.1993.1309 https://doi.org/10.1016/j.na.2009.03.075 https://doi.org/10.1016/j.jmaa.2008.01.045 https://doi.org/10.1016/j.na.2008.04.017 https://doi.org/10.1090/s0002-9939-00-05573-8 https://doi.org/10.28919/afpt/4442 1. introduction 2. preliminary 3. convergence results competing interest references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 15doi: 10.28924/ada/ma.2.15 quasi-likelihood estimation in fractional levy spdes from poisson sampling jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte,376 fretwell bldg, 9201 university city blvd. charlotte, nc 28223-0001, usacorrespondence: j.bishwal@uncc.edu abstract. we study the quasi-likelihood estimator of the drift parameter in the stochastic partialdifferential equations driven by a cylindrical fractional levy process when the process is observed atthe arrival times of a poisson process. we use a two stage estimation procedure. we first estimatethe intensity of the poisson process. then we plug-in this estimate in the quasi-likelihood to estimatethe drift parameter. we obtain the strong consistency and the asymptotic normality of the estimators. 1. introduction parameter estimation in infinite dimensional stochastic differential equations was first studied byloges [20]. when the length of the observation time becomes large, he obtained consistency andasymptotic normality of the maximum likelihood estimator (mle) of a real valued drift parameterin a hilbert space valued sde. koski and loges [18] extended the work of loges [20] to minimumcontrast estimators. koski and loges [17] applied the work to a stochastic heat flow problem. seethe monograph bishwal [5] for asymptotic results on likelihood inference and bayesian inferencefor drift estimation of finite and infinite dimensional stochastic differential equations.huebner, khasminskii and rozovskii [12] started statistical investigation in spdes. they gavetwo contrast examples of parabolic spdes in one of which they obtained consistency, asymptoticnormality and asymptotic efficiency of the mle as noise intensity decreases to zero under thecondition of absolute continuity of measures generated by the process for different parameters (thesituation is similar to the classical finite dimensional case) and in the other they obtained theseproperties as the finite dimensional projection becomes large under the condition of singularity ofthe measures generated by the process for different parameters. the second example was extendedby huebner and rozovskii [13] and the first example was extended by huebner [11] to mle forgeneral parabolic spdes where the partial differential operators commute and satisfy differentorder conditions in the two cases. received: 19 feb 2022. key words and phrases. cylindrical fractional levy process, stochastic partial differential equations, space-time colornoise, convoluted levy field, infinite divisibility, poisson sampling, quasi maximum likelihood estimator, consistency,asymptotic normality. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 2 huebner [10] extended the problem to the ml estimation of multidimensional parameter. lototskyand rozovskii [21] studied the same problem without the commutativity condition. small noiseasymptotics of the nonparmetric estimation of the drift coefficient was studies by ibragimov andkhasminskii [14].bishwal [3] proved the bernstein-von mises theorem (bvt) and obtained asymptotic properties ofregular bayes estimator of the drift parameter in a hilbert space valued sde when the correspond-ing ergodic diffusion process is observed continuously over a time interval [0,t ]. the asymptoticsare studied as t → ∞ under the condition of absolute continuity of measures generated by theprocess. results are illustrated for the example of an spde.bishwal [4] obtained bvt and spectral asymptotics of bayes estimators for parabolic spdeswhen the number of fourier coefficients becomes large. in that case, the measures generatedby the process for different parameters are singular. here we treat the case when the measuresgenerated by the process for different parameters are absolutely continuous under some conditionson the order of the partial differential operators. bishwal [9] studied the asymptotic properties ofthe posterior distributions and bayes estimators when one has either fully observed process orfinite-dimensional projections. the asymptotic parameter is only the intensity of noise. in thispaper we treat the more general model with non-gaussian noise with long memory.on the other hand, recently long memory processes, i.e. processes with slowly decaying auto-correlation and processes with jumps have received attention in finance, engineering and physics.the simplest continuous time long memory process is the fractional brownian motion discovered bykolmogorov [15] and later on studied by levy [19] and mandelbrot and van ness [27]. continuoustime long memory jump process is fractional levy process. hence fractional levy process can alsobe called the kolmogorov-levy process.we generalize fractional spde process to include non-normal innovations. we consider hurstparameter greater than half. this model is interesting as it preserves both jumps and long memory.a normalized fractional brownian motion {wht ,t ≥ 0} with hurst parameter h ∈ (0, 1) is acentered gaussian process with continuous sample paths whose covariance kernel is given by e(wht w h s ) = 1 2 (s2h + t2h −|t − s|2h), s,t ≥ 0. the process is self similar (scale invariant) and it can be represented as a stochastic integralwith respect to standard brownian motion. for h = 1 2 , the process is a standard brownian motion.for h 6= 1 2 , the fbm is not a semimartingale and not a markov process, but a dirichlet process.the increments of the fbm are negatively correlated for h < 1 2 and positively correlated for for h < 1 2 and in this case they display long-range dependence. the parameter h which is alsocalled the self similarity parameter, measures the intensity of the long range dependence. thearima(p,d,q) with autoregressive part of order p, moving average part of order q and fractionaldifference parameter d ∈ (0, 0.5) process converge in donsker sense to fbm. see mishura [22].the fractional levy ornstein-uhlenbeck (fou) process, is an extension of fractional ornstein-uhlenbeck process with fractional levy motion (flm) driving term. in finance, it could be useful https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 3 as a generalization of fractional vasicek model, as one-factor short-term interest rate model whichcould take into account the long memory effect and jump of the interest rate. the model parameteris usually unknown and must be estimated from data.fractional levy process (flp) is defined as mh,t = 1 γ(h + 1 2 ) ∫ r [(t − s)h−1/2+ − (−s) h−1/2 + ]dms, t ∈r where {mt,t ∈ r} is a levy process on r with e(m1) = 0, e(m21 ) < ∞ and without browniancomponent.here are some properties of the fractional levy process:1) the covariance of the process is given by cov(mh,t,mh,s) = e(m21 ) 2γ(2h + 1) sin(πh) [|t|2h + |s|2h −|t − s|2h]. 2) mh is not a martingale. for a large class of levy processes, mh is neither a semimartingale.3)mh is hölder continuous of any order β less than h − 12 .4) mh has stationary increments.5) mh is symmetric.6) m is self-similar, but mh is not self-similar.7) mh has infinite total variation on compacts.thus flp is a generalization and a natural counterpart of fbm. fractional stable motion is aspecial case of flp. first we discuss estimation in partially observed models and then we discussestimation in directly observed model in finite dimensional set up. in finance, the log-volatilityprocess can be modeled as a fractionally integrated moving average (fima) process which isdefined as yh(t) = ∫ t −∞ gh(t −u)dmu, t ∈r where gh(t) = 1 γ(h − 1 2 ) ∫ t 0 g(t − s)sh− 3 2 ds, t ∈r which is the riemann-liouville fractional integral of order h and the kernel g is the kernel of ashort memory moving average process. the log-volatility process will have slow (hyperbolic rate)decay of the auto-correlation function (acf ).the process yh(t) can be written as yh(t) = ∫ t −∞ g(t −u)dmh,u, t ∈r. we assume the following conditions on the kernel g : r → r, namely 1) g(t) = 0 for all t < 0(causality), 2) |g(t)| ≤ ce−ct for some constants c > 0 and c > 0 (short memory). https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 4 the fima process is stationary and is infinite divisible. it has long memory and jumps whichagree empirically with stochastic volatility models. the asset return can be modeled as a coga-rch process dx(t) = √ eyh(t)dlt where (lt,t ∈r is another levy process and the initial value yh(0) is independent of l.consider the kernel g(t − s) = σe−θ(t−s)i(0,∞)(t − s),θ > 0 then gh(t) = σ γ(h − 1 2 ) ∫ ∞ 0 eθ(t−s)i(0,∞)(t − s)s h−3 2 ds, t ∈r. note that uh,θ,σt = ∫ r gh(t −u)dmu, t ∈r is the fractional levy ornstein-uhlenbeck (flou) process satisfying the fractional langevin equa-tion dut = −θutdt + σdmh,t, t ∈r. the process has long memory. levy driven processes of ornstein-uhlenbeck type have beenextensively studied over the last few years and widely used in finance, see barndorff-neilsenand shephard [1]. flou process generalizes fou process to include jumps. maximum quasi-likelihood estimation in fractional levy stochastic volatility model was studied in bishwal [6].berry-esseen inequalities for the discretely observed ornstein-uhlenbeck-gamma process wasstudied in bishwal [7]. minimum contrast estimation in fractional ornstein-uhlenbeck processbased on both continuous and discrete observations was studied in bishwal [8].consider the asset return driven by fractional levy process dsh,t = σt−dlh,t, t > 0, s0 = 0, with log-volatility log σ2t = µ + xt, t ≥ 0where the levy driven ou process x satisfies dxt = −θxtdt + dmt, t > 0 with θ ∈r+ and the driving compound poisson process m is a levy process with levy symbol ψm(u) = − u2 2 + ∫ r (eiux − 1)φ0,1/λ(dx), where φ0,1/λ being a normal distribution with mean 0 and variance 1/λ. this means that m is thesum of a standard brownian motion w and a compound poisson process jt = ∑ntk=1 zk, j−t =∑−n−t k=1 z−k, t ≥ 0 where (nt,t ∈r) is an independent poisson process with intensity λ > 0 and https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 5 jump times (tk)k∈z, i.e., mt = wt + jt. the poisson process n is also independent from the i.i.d.sequence of jump sizes (zk)k∈z with z1 ∼ n(0, 1/λ). the levy process m in this case is given by mt = nt∑ k=1 (αzk + γ|zk|) −ct, t > 0 and c := γ ∫ r |x|λφ0,1/λ(dx) = √ 2λ π γ. {m−t,t ≥ 0} is defined analogously. the stationary log-volatility is given by log σ2t = µ + ∫ t −∞ e−θ(t−s)dms. we observe s at n consecutive jump times 0 = t0 < t1 < ... < tn < t < tn+1,n ∈z over the timeinterval [0,t ]. the state process x has then the following autoregressive representation xti = e −θ∆tixti−1 + nti∑ k=nti−1 +1 e−θ(ti−tk )[αzk + γ|zk|] − ∫ ti ti−1 e−θ(ti−s)cds = e−θ∆tixti−1 + αzi + ( |zi|− c θ (1 −e−θ∆ti ) ) where ∆ti := ti − ti−1, i = 1, 2, . . . ,n and nti−1 + 1 = nti = i.we do the parameter estimation in two steps. the rate λ of the poisson process n can beestimated given the jump times ti , therefore it is done at a first step. since we observe totalnumber of jumps n of the poisson process n over the t intervals of length one, the mle of λ isgiven by λ̂n := nt .to estimate the remaining parameters (α,θ,µ), we use the quasi maximum likelihood estimationprocedure in conditionally heteroscedastic time series models developed by straumann [25].assuming that s∆ti h,ti given s∆ti−1 h,ti−1 , . . . ,s ∆t1 h,t1 ,x0 is conditionally normally distributed with meanzero and variance σ2ti−/λ, the conditional log-likelihood given the initial value x0 has the repre-sentation l(ϑ|s∆h,λ) := − n 2 log(2π) − 1 2 ( n∑ i=1 log(σ2ti−/λ) − n∑ i=1 (s ∆ti h,ti )2 σ2ti−/λ ) . where s∆ti h,ti = sh,ti −sh,ti−1 is the return at time ti . since the volatility is unobservable, this log-likelihood can not be evaluated numerically. the quasi log-likelihood function for ϑ = (θ,α,γ,µ)given the data s∆h := (s∆t1h,t1,s∆t2h,t2, . . . ,s∆tnh,tn ) and the mle λ̂n is defined as l(ϑ|s∆h, λ̂n) := − 1 2 n∑ i=1 log(σ̂2h,ti (ϑ,λ̂n)) − 1 2 n∑ i=1 (s ∆ti h,ti )2 σ̂2 h,ti (ϑ,λ̂n)/λ̂n where the estimates of the volatility σ2h,ti, i = 1, 2, . . . ,n are given by σ̂2h,ti (ϑ,λn) := exp(µ + e −α∆tixh,ti−1 (ϑ,λ) − ĉ∆ti ), ß = 1, 2, . . . ,n and given the parameters ϑ and λ the estimates of the state process x are given by the recursion x̂h,ti = e −θ∆tix̂h,ti−1 + α sh,ti σ̂ti (ϑ,λ) + ( sh,ti σ̂ti (ϑ,λ) − ĉ∆ti ) , i = 1, 2, . . . ,n https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 6 note that e(|w |) = √ 2 πλ , w ∼ n(0, 1/λ).here the approximation (1−e−z ) ≈ z for small z is used and sh,ti σ̂ti (ϑ,λ) approximates the innovation zi . the recursion needs a starting value x̂h,0 which will be set equal to the mean value of thestationary distribution of x which is zero. the mean value zero of the stationary distribution of x.qmle of ϑ is defined as ϑ̂n := arg max ϑ∈θ l(ϑ|s∆h, λ̂n).let (ω,f,{ft}t≥0,p ) be the stochastic basis on which is defined the ornstein-uhlenbeck process xt satisfying the itô stochastic differential equation dxt = −θxtdt + dmht , t ≥ 0, where {mht } is a fractional levy motion with h > 1/2 with the filtration {ft}t≥0 and θ ∈ r+ isthe unknown parameter to be estimated on the basis of completely directly observed continuousobservation of the process {xt} on the time interval [0,t ]. observe that xt = ∫ t −∞ e−θ(t−s)dmhs . this process is stationary and is a process with long memory. it can be shown that xti is astationary discrete time ar(1) process with autoregression coefficient φ ∈ (0, 1) with the followingrepresentation xti = φxti−1 + �ti−1where φ = e−θ∆ and �ti−1 = ∫ ti ti−1 e−θ(ti−u)dmhu . then the problem is a ar(1) estimation with non-gaussian non-martingale error. for equidistantsampling, one can study the least squares estimator which boils down to the study of error distribu-tion for non-semimartingales. one can specialize to the case when m is a either a gamma processor an inverse gaussian process in order to have infinite number of jumps in a finite time inter-val unlike the compound poissoan case which have finite number of jumps in a finite time interval.these fractional gamma and fractional inverse gaussian ornstein-uhlenbeck (flou) processes arelou processes which include long memory. in the next section we deal with completely observedprocess.the rest of the paper is organized as follows : section 2 contains model, assumptions andpreliminaries. section 3 contains the asymptotic properties of quasi likelihood estimator. 2. flspde model and preliminaries in order to introduce fractional levy stochastic partial differential equation (flsode) we proceedas follows. let us fix θ0, the unknown true value of the parameter θ. let (ω,f,p ) be a complete https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 7 probability space and w (t,x) be a process on this space with values in the schwarz space ofdistributions d′(g) such that for φ,ψ ∈ c∞0 (g),‖φ‖−1l2(g) 〈w (t, ·),φ(·)〉 is a one dimensionalwiener process and e(〈w (s, ·),φ(·)〉〈w (t, ·),ψ(·)〉) = (s ∧ t)(φ,ψ)l2(g). this process is usually referred to as the cylindrical brownian motion (c.b.m.).we assume that there exists a complete orthonormal system {hi}∞i=1 in l2(g)) such that forevery i = 1, 2, . . . ,hi ∈ wm,20 (g) ∩c∞(g) and λθhi = βi (θ)hi, and lθhi = µi (θ)hi for all θ ∈ θ where lθ is a closed self adjoint extension of aθ, λθ := (k(θ)i −lθ)1/2m,k(θ) is a constant andand the spectrum of the operator λθ consists of eigen values {βi (θ)}∞i=1 of finite multiplicities and µi = −β2mi + k(θ).cflp mh(t) can be expanded in the series mh(t,x) = ∞∑ i=1 mh,i (t)hi (x) where {mh,i (t)}∞i=1 are independent one dimensional flps, see peszat and zabczyk [24]. thelatter series converges p-a.s. in h−ν for ν > d/2. indeed ‖mh(t)‖2−ν = ∞∑ i=1 m2h,i (t)‖hi‖ 2 −ν = ∞∑ i=1 m2h,i (t)β −2ν i and the later series converges p-a.s.consider the parabolic spde duθ(t,x) = θuθ(t,x) + ∂2 ∂x2 uθ(t,x)dt + dmh(t,x), t ≥ 0, x ∈ [0, 1] (2.1) u(0,x) = u0(x) ∈ l2([0, 1]) (2.2) uθ(t, 0) = uθ(t, 1), t ∈ [0,t ], (2.3) here θ ∈ θ ⊆ r is the unknown parameter to be estimated on the basis of the observationsof the field uθ(t,x),t ≥ 0, x ∈ [0, 1]. for x ∈ [0, 1], we observe the process {ut,t ≥ 0} attimes {t0,t1,t2, ....}. we assume that the sampling instants {ti, i = 0, 1, 2...} are generated bya poisson process on [0,∞), i.e., t0 = 0,ti = ti−1 + ξi, i = 1, 2, ... where ξi are i.i.d. positiverandom variables with a common exponential distribution f (x) = 1−exp(−λx). note that intensityparameter λ > 0 is the average sampling rate which is assumed to be known. it is also assumedthat the sampling process ti, i = 0, 1, 2, ... is independent of the observation process {xt,t ≥ 0}.we note that the probability density function of tk+i − tk is independent of k and is given by thegamma density fi (t) = λ(λt) i−1 exp(−λt)it/(i − 1)!, i = 0, 1, 2, .... (2.4)where it = 1 if t ≥ 0 and it = 0 if t < 0. https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 8 consider the fourier expansion of the process u(t,x) = ∞∑ t=1 ui (t)φi (x) (2.5) corresponding to some orthogonal basis {φi (x)}∞i=1. note that the fourier coefficients {uθi (t), i ≥ 1} are independent one dimensional ornstein-uhlenbeck processes duθi (t) = µ θ i u θ i (t)dt + β −ν i dmh,i (t) (2.6) uθi (0) = u θ 0i,recall that µi (θ) = k(θ) −β2mi . thus duθi (t) = (k(θ) −β 2m i )u θ i (t)dt + β −ν i dmh,i (t) (2.7) the random field u(t,x) is observed at discrete times t and discrete positions x. equivalently, thefourier coefficients uθi (t) are observed at discrete time points.now we focus on the fundamental semimartingale behind the o-u model. define κh := 2hγ(3/2 −h)γ(h + 1/2), kh(t,s) := κ −1 h (s(t − s)) 1 2 −h, ηh := 2hγ(3 − 2h)γ(h + 1 2 ) γ(3/2 −h) , vt ≡ vht := η −1 h t2−2h, mht := ∫ t 0 kh(t,s)dm h s . for using girsanov theorem for brownian motion, since a radon-nikodym derivative process is al-ways a martingale, a central problem is how to construct an appropriate martingale which generatesthe same filtration, up to sets of measure zero, as the non-semimartingale called the fundamental martingale.extending norros et al. [23] it can be shown that mht is a martingale, called the fundamen-tal martingale whose quadratic variation 〈mh〉t is vht . moreover, the natural filtration of themartingale mh coincides with the natural filtration of the flp mh since mht := ∫ t 0 k(t,s)dmhs holds for h ∈ (1/2, 1) where kh(t,s) := h(2h − 1) ∫ t s rh− 1 2 (r − s)h− 3 2 dr, 0 ≤ s ≤ t and for h = 1/2, the convention k1/2 ≡ 1 is used.define qi (t) := d dvt ∫ t 0 kh(t,s)ui (s)ds, i ≥ 1. https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 9 it is easy to see that qi (t) = ηh 2(2 − 2h) { t2h−1zi (t) + ∫ t 0 r2h−1dzi (s) } . define the process zi = (zi (t),t ∈ [0,t ]) by zi (t) := ∫ t 0 kh(t,s)dui (s). extending kleptsyna and le breton [16], we have:(i) zi is the fundamental semimartingale associated with the process ui .(ii) zi is a (ft) -semimartingale with the decomposition zi (t) = µi (θ) ∫ t 0 qi (s)dvs + β −ν i mht . (iii) ui admits the representation ui (t) = ∫ t 0 kh(t,s)dzi (s). (iv) the natural filtration (zi (t)) of zi and (ui (t)) of ui coincide. we focus on our obserbations now. note that for equally spaced data (homoscedastic case) vtk −vtk−1 = η −1 h ( t n )2−2h [k2−2h − (k − 1)2−2h], k = 1, 2, · · · ,n. (2.8) for h = 0.5, vtk −vtk−1 = η −1 h ( t n )2−2h [k2−2h − (k − 1)2−2h] = t n , k = 1, 2, . . . ,n. we have qi (t) = d dvt ∫ t 0 kh(t,s)ui (s)ds = κ−1 h d dvt ∫ t 0 s1/2−h(t − s)1/2−hui (s)ds = κ−1 h ηht 2h−1 d dt ∫ t 0 s1/2−h(t − s)1/2−hui (s)ds = κ−1 h ηht 2h−1 ∫ t 0 d dt s1/2−h(t − s)1/2−hui (s)ds = κ−1 h ηht 2h−1 ∫ t 0 s1/2−h(t − s)−1/2−hui (s)ds. (2.9) the process qi depends continuously on ui and therefore, the discrete observations of ui doesnot allow one to obtain the discrete observations of qi . the process qi can be approximated by q̃i (n) = κ −1 h ηhn 2h−1 n−1∑ j=0 j1/2−h(n− j)−1/2−hui (j). (2.10) https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 10 it is easy to show that q̃i (n) → qi (t) almost surely as n →∞, see tudor and viens [26].define a new partition 0 ≤ r1 < r2 < r3 < · · · < rmk = tk, k = 1, 2, · · · ,n. define q̃i (tk) = κ −1 h ηht 2h−1 k mk∑ j=1 r 1/2−h j (rmk − rj) −1/2−hui (rj)(rj − rj−1), (2.11) k = 1, 2, · · · ,n.it is easy to show that q̃i (tk) → qi (t) almost surely as mk →∞ for each k = 1, 2, · · · ,n.we use this approximate observation in the calculation of our estimators. thus our observationsare ui (t) ≈ ∫ t 0 kh(t,s)dz̃i (s) where z̃i (t) = θ∫ t 0 q̃i (s)dvs + mht . (2.12)observed at poisson arrivals t1,t2, . . . ,tn. we observe just one such approximate fourier coefficient ui (t) which we denote by u(t) and the corresponding observations are denoted by ut1,ut2, . . . ,utnand let n →∞. ideally we are in a large time asymptotic framework.now we focus on the estimation methodology. define ρ := ρ(λ,θ) = λ λ−κ(θ) + β2m i . (2.13) the quasi likelihood estimator is the solution of the estimating equation: g∗n(θ) = 0 (2.14) where g∗n(θ) = β2νi λ(ρ(λ,θ)) 2 ρ(λ, 2θ) n∑ i=1 uti−1 ( (uti−1θρ(λ,θ)) 2 + λ )−1 (uti −ρ(λ,θ)uti−1 ) (2.15) we call the solution of the estimating equation the quasi likelihood estimator. there is no explicitsolution for this equation.the optimal estimating function for estimation of the unknown parameter θ is gn(θ) = β 2ν i n∑ i=1 uti−1 [uti −ρ(λ,θ)uti−1 ]. (2.16) the martingale estimation function (mef) estimator of ρ is the solution of gn(θ) = 0 and isgiven by ρ̂n := ∑n i=1 uti−1uti∑n i=1 u 2 ti−1 . (2.17) 3. main results we do the parameter estimation in two steps: the rate λ of the poisson process can be estimatedgiven the arrival times ti , therefore it is done at a first step. since we observe total number ofarrivals n of the poisson process over the t intervals of length one, the mle of λ is given by λ̂n := n t . (3.1) https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 11 theorem 3.1 we have λ̂n → λ a.s. as n →∞, √ n(λ̂n −λ) →d n(0, eλ(1 −e−λ)) as n →∞. proof. let vi be the number of arrivals in the interval (i − 1, i]. then vi, i = 1, 2, . . . ,n are i.i.d.poisson distributed with parameter λ. since φ is continuous, we have i{0}(vi ) = i{0}(u(ti )) a.s. i = 1, 2, . . . ,n. note that 1 n n∑ i=1 i{0}(uti ) → a.s. e(i{0}v1) = p (v1 = 0) = e −λ as n →∞. lln and clt and delta method applied to the sequence i{0}(uti ), i = 1, 2, . . . ,n give the results. the clt result above allows us to construct confidence interval for the jump rate λ. corollary 3.1 a 100(1 −α)% confidence interval for λ is given by[ n t −z1−α 2 √ 1 n − 1 t , n t + z1−α 2 √ 1 n − 1 t ] where z1−α 2 is the (1 − α 2 )-quantile of the standard normal distribution. we obtain the strong consistency and asymptotic normality of the mef estimator. theorem 3.2 we have ρ̂n → ρ a.s. as n →∞, √ n(ρ̂n −ρ) →d n(0, λ−i (1 −e−ρ)) as n →∞. proof: by using the fact that every stationary mixing process is ergodic, it is easy to show thatif ut is a stationary ergodic o-u process and ti is a process with nonnegative i.i.d. incrementswhich is independent of ut, then {uti, i ≥ 1} is a stationary ergodic process. hence {uti, i ≥ 1} isa stationary ergodic process.observe that uθi (t) := vi is stationary ergodic and vi ∼ n(0,σ2) where σ2 is the variance of u0. thus by slln for zero mean square integrable martingales, we have as n →∞, 1 n n∑ i=1 uti−1uti → a.s. e(ut0ut1 ) = ρe(u 2 t0 ) 1 n n∑ i=1 u2ti−1 → a.s. e(u2t0 ) thus ∑n i=1 uti−1uti∑n i=1 u 2 ti−1 →a.s. ρ. further, √ n(ρ̂n −ρ) = n−1/2 ∑n i=1 uti−1 (uti −θuti−1 ) n−1 ∑n i=1 u 2 ti−1 . https://doi.org/10.28924/ada/ma.2.15 eur. j. math. anal. 10.28924/ada/ma.2.15 12 since e(ut1ut2|ut1 ) = θu 2 t1it follows by lemma 3.1 in bibby and srensen [2] n−1/2 n∑ i=1 uti−1 (uti −θuti−1 ) converges in distribution to normal distribution with mean zero and variance equal to e[(ut1ut2 ) −e(ut1ut2|ut1 )] 2 = 1 −e2(θ−β1δ){2(β1 −θ)(βi + 1)}−1. applying delta method the result follows. in the next step, we use the estimator of λ to estimate θ.note that 1 ρ̂n = ∑n i=1 u 2 ti−1∑n i=1 uti−1uti . hence 1 + β2m1 −κ(θ) λ = ∑n i=1 u 2 ti−1∑n i=1 uti−1uti . thus β2m1 −κ(θ) λ = ∑n i=1 u 2 ti−1∑n i=1 uti−1uti − 1 = − ∑n i=1 uti−1 [uti −uti−1 ]∑n i=1 uti−1uti . now replace λ by its estimator mle λ̂n. β2m1 −κ(θ) = − ∑n i=1 uti−1 [uti −uti−1 ] t n ∑n i=1 uti−1uti . thus θ̂n = κ −1 ( β2m1 + ∑n i=1 uti−1 [uti −uti−1 ] t n ∑n i=1 uti−1uti ) . since the function κ−1(·) is a continuous function, by application of delta method, the followingresult is a consequence of theorem 3.2. theorem 3.3 θ̂n →a.s. θ as n →∞, √ n(θ̂n −θ) →d n(0, (κ′(θ))−2λ2(1 −e−2λ −1(κ(θ)−β2m1 ))) as n →∞. in the second stage, we plug-in λ by its estimator λ̂n. remark sub-fractional brownian motion, which has main properties of the fractional brownianmotion, excluding the stationarity of increments, 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https://doi.org/10.1214/009053606000001541 https://doi.org/10.1137/1010093 references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 13doi: 10.28924/ada/ma.2.13 on the semi-local convergence of a third order scheme for solving nonlinear equations samundra regmi1, ioannis k. argyros2,∗, santhosh george3, christopher argyros4 1learning commons, university of north texas at dallas, dallas, tx, usa samundra.regmi@untdallas.edu 2department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 3department of mathematical and computational sciences,national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 4department of computing and technology, cameron university, lawton, ok 73505, usa christopher.argyros@cameron.edu ∗correspondence: iargyros@cameron.edu abstract. the semi-local convergence analysis of a third order scheme for solving nonlinear equationin banach space has not been given under lipschitz continuity or other conditions. our goal isto extend the applicability of the cordero-torregrosa scheme in the semi-local convergence underconditions on the first fréchet derivative of the operator involved. majorizing sequences are used forproving our results. numerical experiments testing the convergence criteria are given in this study. 1. introduction cordero and torregrosa in [10] considered the third order scheme, defined for n = 0, 1, 2, . . . , by yn = xn −f ′(xn)−1f (xn) xn+1 = xn − 3m−1n f (xn), (1.1) for solving the nonlinear equation f (x) = 0, (1.2) where mn = 2f ′(3xn+yn4 )− f ′(xn+yn2 ) + 2f ′(xn+3yn4 ) . here f : d ⊂ e −→ e1 is an operatoracting between banach spaces e and e1 with d 6= ∅. in general a closed form solution for (1.2) isnot possible, so iterative schemes are used for approximating a solution x∗ of (1.2) (see [1–27]). received: 14 feb 2022. key words and phrases. semi-local convergence; cordero-torregrosa scheme; iterative schemes; banach space; con-vergence criterion. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 2 the local convergence of the this scheme in the special case when e = e1 = r was shown to beof order three using taylor expansion and assumptions on the fourth order derivative of f, whichis not on these schemes [10]. so, the assumptions on the fourth derivative reduce the applicabilityof these schemes [1–27].for example: let e = e1 = r, d = [−0.5, 1.5]. define λ on d by λ(t) = { t3 log t2 + t5 − t4 if t 6= 0 0 if t = 0. then, we get f (1) = 0, and λ′′′(t) = 6 log t2 + 60t2 − 24t + 22. obviously λ′′′(t) is not bounded on d. so, the convergence of scheme (1.1) is not guaranteed bythe previous analyses in [1–27].in this study we introduce a majorant sequence and use general continuity conditions to extendthe applicability of scheme (1.1). our analysis includes error bounds and results on uniqueness of x∗ based on computable lipschitz constants not given before in [1–27] and in other similar studiesusing taylor series. our idea is very general. so, it applies on other schemes too.the rest of the study is set up as follows: in section 2 we present results on majorizing sequences.sections 3,4 contain the semi-local and local convergence, respectively, where in section 4 thenumerical experiments are presented. concluding remarks are given in the last section 5. 2. majorizing sequences scalar sequences are developed that majorize scheme (1.1). let k0 > 0,k > 0 and η > 0 begiven constants. define sequences {tn},{sn} by t0 = 0, s0 = η tn+1 = sn + 2k(sn − tn)(tn+1 − tn) 9(1 −k0tn)(1 −pn) , sn+1 = tn+1 + k(tn+1 − tn + sn − tn)(tn+1 − tn) 2(1 −k0tn+1) , (2.1) where pn = 5k06 (sn + tn). notice that tn+1 is given implicitly in the first substep of sequence (2.1).it we solve for tn+1, we get its explicit form tn+1 = 9sn(1 −k0tn)(1 −pn) − 2tnk(sn − tn) 9(1 −k0tn)(1 −pn) − 2k(sn − tn) . but for the convergence analysis in theorem 3.1 we prefer tn+1 in its implicit form.next, we present sufficient conditions for the convergence scheme (1.1). https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 3 lemma 2.1. suppose that 5(tn + sn) < 6 k0 . (2.2) for all n = 0, 1, 2, . . . . then, sequences {tn} is nondecreasing and bounded from above by t∗ = 35k0 and as such it converge to its unique least upper t ∈ [0,t∗]. proof. it follows from the definition (2.1) of sequences {tn} and (2.2) that this sequence isnondecreasing and bounded from above by t∗, and as such it converges to t. �the next result shows the convergence of sequence {tn}, under stronger but easier to verifyconditions than (2.2). but first we need to introduce some functions and parameters. definefunctions g1 and g2 on the interval (0, 1) by g1(t) = 4k(1 + t)t − 4k(1 + t) + 9k0t, and g2(t) = k(2 + t)(1 + t)t −k(2 + t)(1 + t) + 2k0t3.then, we get g1(0) = −4k, g1(1) = 9k0, g2(0) = −2k and g2(1) = 2k0.hence, functions g1 and g2 have roots in (0, 1). denote the minimal such roots by α1 and α2, re-spectively. set a = 2k(t1−t0) 9(1−k0t)(1−p0) , b = k(t1−t0+s0−t0)(t1−t0) 2η(1−k0t1) , c̄ = min{a,b}, c = max{a,b},α3 = min{α1,α2} and α = max{α1,α2}.then, we can show the second result on majorizing sequences for method (1.2). lemma 2.2. suppose 0 < c̄ ≤ c ≤ α3 ≤ α ≤ 1 − 10 3 k0η. (2.3) then, sequence {tn} is nondecreasing, bounded from above by t = η1−α and as such it converges to its unique least upper bound t∗ ∈ [0,t ]. proof. items 0 ≤ 2k(tk+1 − tk) 9(1 −k0tk)(1 −pk) ≤ α, (2.4) 0 ≤ k(tk+1 − tk + sk − tk)(tk+1 − tk) 2(1 −k0tk+1) ≤ α(sk − tk), (2.5) 0 ≤ 1 1 −pk ≤ 2 (2.6) and tk ≤ sk ≤ tk+1 (2.7)are shown using induction on k. these estimates are true for k = 0 by (2.3). suppose thesehold for all k smaller than n − 1. by induction hypotheses and (1.2), we have 0 ≤ sk − tk ≤ α(sk−1 − tk−1) ≤ . . . ≤ αkη, tk+1 − tk = (tk+1 − sk) + (sk − tk) ≤ (1 + α)(sk − tk) https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 4 and tk+1 ≤ (1 −αk+2)η 1 −α < t. evidently, (2.4) holds if 4k(1 + α)αk−1η 9(1 −k01−α k+1 1−α η ≤ α, (2.8) where we used (2.6). define recurrent polynomials f (1) k on the interval (0, 1) by f (1) n (t) = 4k(1 + t)t k−1η + 9k0(1 + t + . . . + t k−1)η − 9. (2.9) then, estimate (2.8) holds if f (1) n (t) ≤ 0 at t = α1. (2.10) we need a relationship between two consecutive polynomials f (1) k : f (1) k+1 (t) = 4k(1 + t)tkη + 3k0(1 + t + . . . + t k)η − 9 + f (1) k (t) −4k(1 + t)tk−1 + 3k0(1 + t + . . . + tk−1)η + 9 = f (1) k (t) + g1(t)t k−1η. (2.11) in particular, one gets f (1) k+1 (α1) = f (1) k (α1) since by the definition of α1 and g1, g1(α1) = 0.define function f (1)∞ (t) = lim k−→∞ f (1) k (t). (2.12) then, (2.10) holds if f (1)∞ (t) ≤ 0 at t = α1. (2.13)but by (2.9) and (2.12) one gets f (1)∞ (t) = 9k0η 1 − t − 9, (2.14) so (2.13) holds if f (1)∞ (t) ≤ 0 at t = α1 which is true by (2.3).similarly, (2.5) holds if k(2 + α)(1 + α)αkη 2(1 −k01−α k+2 1−α η) ≤ α. (2.15) define polynomials f (2) k (t) on the interval (0, 1) by f (2) k (t) = k(2 + t)(1 + t)tk−1η + 2k0(1 + t + . . . + t k+1)η − 2. (2.16) then, (2.15) holds if f (2) k (t) ≤ 0 at t = α2. (2.17) https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 5 we get f (2) k+1 (t) = k(2 + t)(1 + t)tkη + 2k0(1 + t + . . . + t k+2)η − 2 + f (2) k (t) −k(2 + t)(1 + t)tk−1η − 2k0(1 + t + . . . + tk+1)η + 2 = f (2) k (t) + g2(t)t k−1η, (2.18) and f (2) k+1 (α2) = f (2) k (α2). (2.19) define function f (2)∞ (t) = lim k−→∞ f (2) k (t). (2.20) then, (2.17) holds if f (2)∞ (t) ≤ 0 at t = α2. (2.21) by (2.16) and (2.20), we get f (2)∞ (t) = k0η 1 − t − 1, so (2.21) holds by (2.3). moreover, estimate (2.6) certainly holds if 2pk = 5k03 (sk +tk) < 5k03 ( η1−α + η 1−α) = 10k0η 3(1−α) < 1, which is true by (2.3). furthermore, estimate (2.7) holds by (2.4)-(2.6) and thedefinition of sequence {tk}. hence the induction for estimates (2.4)-(2.7) is completed. it followsthat sequence {tk} is nondecreasing and bounded from above by t∗, and such it converges to t. �if one desires iterates to be given explicitly in (2.1), then define instead sequence {tn} as follows t0 = 0, s0 = η tn+1 = sn + 2k(1 + k0tn)(sn − tn)2 3(1 −k0tn)(1 −pn) (2.22) sn+1 = tn+1 + 2k(tn+1 − tn + sn − tn)(tn+1 − tn) 2(1 −k0tn+1) . moreover, define recurrent polynomial on the interval [0, 1) by f (1) n (t) = 4k 3 tn−1η + 4kk0 3 tn−1(1 + t + . . . + tn)η2 + k0(1 + t + . . . + t n)η − 1. this time we have f (1) n+1(t) = f (1) n (t) + g (1) n (t)t n−1η, (2.23) where g (1) n (t) = 4kk0 3 tn+2η + 4kk0 3 tn+1η + 4 3 kt − 4k 3 (1 −k0η). https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 6 we get g(1)n (0) = −4k3 (1 −k0η) < 0 for k0η < 1, and g(1)1 (1) = 4kk0η > 0. denote by rn thesmallest solution of g(1)n (t),respectively. notice that these solutions are increasing as n increases,since g(1)n (t) ≤ g(1)n−1(t). hence, it follows by (2.23) that f (1) n+1(t) ≤ f (1) n (t) + g (1) 1 (t)t n−1η. in particular for α1 = r1, we get f (1) n+1(t) ≤ f (1) n (t) at t = α1. hence, f (1) n (t) ≤ 0 holds if f (1) 1 (t) ≤ 0 at t = α1.but f (1) 1 (t) = 4k 3 η + 4 3 kk0η 2 + k0η − 1. define b = 2k(s0−t0) 3 . then, we arrive at the following convergence results for majorizing sequence(2.2). lemma 2.3. suppose 5(tn + sn) < 6 k0 , where {tn} is the sequence defined by (2.22). then, the conclusions of lemma 2.2 hold for this sequence. lemma 2.4. suppose 0 < c̄ ≤ c ≤ α3 ≤ α ≤ 1 − 10k0 3 η (2.24) and ( 4k 3 + 4 3 k0kη + k0 ) η ≤ 1. (2.25) then, the conclusions of lemma 2.2 hold for sequence {tn} given by (2.22). remark 2.5. the solutions α1 and α2 in lemma 2.2 depend only on k0 and k. similarly α2 in lemma 2.4 depends on k0 and k1. but α1 depends k0,k and η. to avoid this dependence pick any γ ∈ (0, 1] and set γ = k0η. define functions ḡ (1) n (t) on [0, 1) by ḡ (1) n (t) = 4kγ 3 tn+1 + 4kγ 3 tn=1 + 4 3 kt − 4k 3 (1 −γ). then, according to the proof of lemma 2.2 we can set α1 = r̄1, where r̄1 is the smallest solution in (0, 1) of equation ḡ(1)1 (t) = 0 assured also to exist. finally, notice that the first condition shows implicitly and the second explicitly the smallness of η. https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 7 3. semi-local convergence the following sufficient convergence criteria (a) are used. suppose: (a1) there exist x0 ∈ d and η > 0 such that f ′(x0)−1 exists and ‖f ′(x0)−1f (x0)‖≤ η. (a2) ‖f ′(x0)−1(f ′(w) −f ′(x0))‖≤ k0‖w −x0‖for all w ∈ d. set d0 = d∩u(x0, 1k0 ).(a3) ‖f ′(x0)−1(f ′(w) −f ′(v)‖≤ k‖w −v‖for all v ∈ d0 and w = v −f ′(v)−1f (v). denote by l the constant, if (a3) holds for all u,v ∈ d0, and by l1 the constant for all u,v ∈ d. it follows that k ≤ l ≤ l1. in practicewe shall use whichever of k or l is easier to compute (see also the numerical section).(a4) hypotheses of lemma 2.1 or lemma 2.2 hold.and(a5) u[x0,t∗] ⊂ d (or u[x0,t ] ⊆ d).next, the semi-local convergence of scheme (1.1) is developed based on conditions (a) and theaforementioned notation. theorem 3.1. suppose conditions (a) hold. then, the following items hold {xn}∈ u(x0,t∗) (3.1) and ‖x∗ −xn‖≤ t∗ − tn, (3.2) where x∗ = limn−→∞xn ∈ u[x0,t∗] and f (x∗) = 0. proof. mathematical induction is used to show ‖yk −xk‖≤ sk − tk (3.3) and ‖xk+1 −yk‖≤ tk+1 − sk. (3.4)it follows from (a1) and (1.1) that ‖y0 −x0‖ = ‖f ′(x0)−1f (x0) ≤ η =≤ s0 − t0 = η ≤ t, (3.5) so y0 ∈ u(x0,t∗) and (3.3) hold for k = 0. let z ∈ u(x0,t∗). in view of (a2), one has ‖f ′(x0)−1(f ′(z) −f ′(x0)) ≤ k0‖z −x0‖≤ k0t∗ < 1, so f ′(z)−1 ∈ l(e1,e) and https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 8 ‖f ′(z)−1f ′(x0)‖≤ 1 1 −k0‖z −x0‖ . (3.6) by a result due to banach [14] on linear invertible operators. operator mk can be shown to beinvertible. indeed, by the definition of operator mk, (2.2) and (a2) we obtain ‖(3f ′(x0))−1(mk − 3f ′(x0))‖ ≤ 1 3 [2‖f ′(x0)−1 ( f ′ ( 3xk + yk 4 ) −f ′(x0))‖ +‖f ′(x0)−1 ( f ′ ( xk + yk 2 ) −f ′(x0) ) ‖ +2‖f ′(x0)−1 ( f ′ ( xk + 3yk 4 ) −f ′(x0) ) ≤ 1 3 (2k0‖ 3xk + yk 4 −x0‖ + k0‖ xk + yk 2 −x0‖ +2k0‖ xk + 3yk 4 −x0‖) ≤ 1 3 (2k0 3tk + sk 4 + k0 sk + tk 2 + 2k0 tk + 3sk 4 ) = 5k0 6 (tk + sk) = pk < 1, so mk is invertible and ‖m−1 k f ′(x0)‖≤ 1 3(1 −pk) , (3.7) and xk+1 is well defined by the second substep of method (1.1). then, we can write by method(1.1) that xk+1 = xk −f ′(xk)−1f (xk) + (f ′(xk)−1 − 3m−1k )f (xk) = yk − 1 3 f ′(xk) −1(mk − 3f ′(xk))m−1k (xk+1 −xk). (3.8) we need the estimate, mk − 3f ′(xk) = 2f ′ ( 3xk + yk 4 ) −f ′ ( xk + yk 2 ) +2f ′ ( xk + 3yk 4 ) − 3f ′(xk) = ( f ′ ( 3xk + yk 4 ) −f ′ ( xk + yk 2 )) + ( f ′ ( 3xk + yk 4 ) −f ′(xk) ) + 2 ( f ′ ( xk + 3yk 4 ) −f ′(xk) ) , https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 9 so by (a3) ‖f ′(x0)−1(mk − 3f ′(xk))‖ ≤ k‖ 3xk + yk 4 − 2xk + 2yk 4 ‖ k‖ 3xk + yk 4 − 4xk 4 ‖ + 2k‖ xk + 3yk 4 − 4xk 4 ‖ = 2k‖yk −xk‖≤ 2k(sk − tk). (3.9) using (1.1), (3.6) (for z = xk) and (3.7)-(3.9) ‖xk+1 −yk‖ ≤ 2k(sk − tk)(tk+1 − tk) 9(1 −k0tk)(1 −pk) = tk+1 − sk. (3.10) we also have ‖xk+1 −x0‖≤‖xk+1 −yk‖ + ‖yk −x0‖≤ tk+1 − sk + sk − t0 = tk+1 ≤ t∗, (3.11) so xk+1 ∈ u(x0,t∗). we can write by method (1.1) f (xk+1) = f (xk+1) −f (xk) − 1 3 mk(xk+1 −xk) = ∫ 1 0 (f ′(xk + θ(xk+1 −xk))dθ− 1 3 mk)(xk+1 −xk). (3.12) one can obtain the estimate∫ 1 0 (f ′(xk + θ(xk+1 −xk))dθ− 2 3 f ′ ( 3xk + yk 4 ) + 1 3 f ′ ( xk + yk 2 ) − 2 3 f ′ ( xk + 4yk 4 ) = ∫ 1 0 f ′(xk + θ(xk+1 −xk))dθ−f ′(xk)) + 2 3 (f ′(xk) −f ′ ( 3xk + yk 4 ) ) + 1 3 (f ′(xk) −f ′ ( xk + 3yk 4 ) + 1 3 (f ′ ( xk + yk 2 ) −f ′ ( xk + 3yk 4 ) ), (3.13) so ‖f ′(x0)−1 ∫ 1 0 (f ′(xk + θ(xk+1 −xk))dθ− 1 3 mk)‖ ≤ k [ ‖xk+1 −xk‖ 2 + ‖yk −xk‖ 6 + ‖yk −xk‖ 4 + ‖yk −xk‖ 12 ] ≤ k( tk+1 − tk 2 + sk − tk 6 + sk − tk 4 + sk − tk 12 ) = k 2 (tk+1 − tk + sk − tk). (3.14) it follows from method (1.1), (3.6) (for z = xk+1), (3.11) and (2.10) that https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 10 ‖yk+1 −xk+1‖ ≤ ‖(f ′(xk+1)−1f ′(x0)f ′(x0)−1f (xk+1)‖ ≤ k(tk+1 − tk + sk − tk)(tk+1 − tk) 2(1 −k0tk+1) = sk+1 − tk+1, (3.15) showing (3.3). moreover, we get ‖yk+1 −x0‖ ≤ ‖yk+1 −xk+1‖ + ‖xk+1 −x0‖ ≤ sk+1 − tk+1 + tk+1 − t0 = sk+1 ≤ t∗, (3.16) so yk+1 ∈ u(x0,t∗). the induction for (3.3) and (3.6) is completed. it follows from(3.3), (3.6), (3.10)and (3.16) that sequence {xn} is fundamental in banach space e, and as such it converges to x∗ ∈ u[x0,t∗]. using (3.9) and letting k −→ ∞ in ‖f ′(x0)−1f (xk+1)‖ ≤ k2 (tk+1 − tk + sk − tk),we obtain f (x∗) = 0. �next, a uniqueness of the solution x∗ result is presented. proposition 3.2. suppose: (1) the element x∗ ∈ u(x∗,s∗) is a simple solution of (1.2), and (a2) holds. (2) there exists δ ≥ s∗ so that k0(s ∗ + δ) < 2. (3.17) set d1 = d∩u[x∗,δ]. then, x∗ is the unique solution of equation (1.2) in the domain d1. proof. let q ∈ d1 with f (q) = 0. define s = ∫10 f ′(q + θ(x∗ −q))dθ. using (h2) and (3.17)one obtains ‖f ′(x0)−1(s −f ′(x0))‖ ≤ k0 ∫ 1 0 ((1 −θ)‖q −x0‖ + θ‖x∗ −x0‖)dθ ≤ k0 2 (s∗ + δ) < 1, so q = x∗, follows from the invertability of s and the identity s(q−x∗) = f (q)−f (x∗) = 0−0 = 0. � remark 3.3. (i) point t given in closed form can repalce t∗ in theorem 3.1. (ii) we used majorizing sequence {tn} given by (2.1) and lemma 2.2 to prove theorem 3.1. but we can also use majorizing sequence {tn} given by (2.22) and lemma 2.3 to arrive at the conclusions of the theorem 3.1. simply notice that in the proof of this theorem we got using the second substep of https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 11 scheme (1.1), (3.8) and (3.9) estimate (3.10) leading to the definition of the first substep of sequence (2.1). but we can use the first substep of scheme (1.1) to write instead of (3.8) that xk+1 = yk −f ′(xk)−1(mk − 3f ′(xk))m−1k f (xk)(yk −xk) leading to ‖xk+1 −yk‖≤ 2k(1 + k0tk)(sk − tk)2 3(1 −k0tk)(1 −pk) = tk+1 − sk, where, we also used ‖f ′(x0)−1f (xk)‖ = ‖f ′(x0)−1((f ′(xk) −f (x0)) + f ′(x0))‖ ≤ 1 + k0‖xk −x0‖≤ 1 + k0tk. hence, we arrive at the second semi-local convergence rsult for scheme (1.1). theorem 3.4. suppose:conditions (a) hold with (a4) replaced by (a4)’ hypotheses of lemma 2.3 or lemma 2.4 hold. then, the conclusions of theorem 3.1 hold with (2.22) replacing (2.1). in practice we shall use the theorem providing the best results. 4. numerical experiments lipschitz parameters are determinded and convegence criteria are tested for some numericalexperiments. example 4.1. define scalar function ζ(t) = ξ0t + ξ1 + ξ2 sin ξ3t, x0 = 0, where ξj, j = 0, 1, 2, 3 are parameters. then, clearly for ξ3 large and ξ2 small, k0l1 can be small (arbitrarily). in particular, notice that k l1 −→ 0. example 4.2. let e = e1 = c[0, 1] and d = u[0, 1]. it is well known that the boundary value problem [12]. ς(0) = 0,ς(1) = 1, ς′′ = −ς −σς2 can be given as a hammerstein-like nonlinear integral equation ς(s) = s + ∫ 1 0 q(s,t)(ς3(t) + σς2(t))dt where σ is a parameter. then, define f : d −→ e1 by [f (x)](s) = x(s) − s − ∫ 1 0 q(s,t)(x3(t) + σx2(t))dt. https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 12 choose ς0(s) = s and d = u(ς0,ρ0). then, clearly u(ς0,ρ0) ⊂ u(0,ρ0 + 1), since ‖ς0‖ = 1. suppose 2σ < 5. then, conditions (a) are satisfied for k0 = 2σ + 3ρ0 + 6 8 , l = σ + 6ρ0 + 3 4 , and η = 1+σ 5−2σ. notice that k0 < l. example 4.3. let us consider a scalar function ψ defined on the set d = u[x0, 1 − q] for q ∈ (0, 1 2 ), by ψ(x) = x3 −q. choose x0 = 1. then, we obtain the estiamtes |ψ′(x0)−1(ψ′(x) −ψ′(x0))| = |x2 −x20 | ≤ |x + x0||x −x0| ≤ (|x −x0| + 2|x0|)|x −x0| = (1 −q + 2)|x −x0| = (3 −q)|x −x0|, for all x ∈ d, so k0 = 3 −q, d0 = u(x0, 1k0 ) ∩d = u(x0, 1 k0 ), |ψ′(x0)−1(ψ′(y) −ψ′(x)| = |y2 −x2| ≤ |y + x||y −x| ≤ (|y −x0 + x −x0 + 2x0)||y −x| = (|y −x0| + |x −x0| + 2|x0|)|y −x| ≤ ( 1 k0 + 1 k0 + 2)|y −x| = 2(1 + 1 k0 )|y −x|, for all x,y ∈ d0, so l = 2(1 + 1k0 ), |ψ′(x0)−1(ψ′(y) −ψ′(x)| = (|y −x0| + |x −x0| + 2|x0|)|y −x| ≤ (1 −q + 1 −q + 2)|y −x| = 2(2 −q)|y −x|, for all x,y ∈ d and l1 = 2(2 −q). notice that for all q ∈ (0, 12) k0 < l < l1. next, set y = x −ψ′(x)−1ψ(x), x ∈ d. then, we have y + x = x −ψ′(x)−1ψ(x) + x = 5x3 + q 3x2 . define fundtion ψ̄ on the interval d = [q, 2 −q] by ψ̄(x) = 5x3 + q 3x2 . https://doi.org/10.28924/ada/ma.2.13 eur. j. math. anal. 10.28924/ada/ma.2.13 13 then, we get by this definition that ψ̄′(x) = 15x4 − 6xq 9x4 = 5(x −q)(x2 + xq + q2) 3x3 , where p = 3 √ 2q 5 is the critical point of function ψ̄. notice that q < p < 2 − q. it follows that this function is decreasing on the interval (q,p) and increasing on the interval (q, 2 − q), since x2 + xq + q2 > 0 and x3 > 0. so, we can set k1 = 5(2 −q)2 + q 9(2 −q)2 ,η = 1 −q 3 and k1 < k0. but if x ∈ d0 = [1 − 1k0 , 1 + 1 k0 ], then k = 5%3 + q 9%2 , where % = 4−q 3−q and k < k1 for all q ∈ (0, 1 2 ). next, we verify conditions (2.2), (2.3), (2.24) and (2.25). then for q = 0.95, 6 k0 = 2.9268 and n 1 2 3 4 5 tn 0.1683 0.1694 0.1694 0.1694 0.1694 α1 = 0.1643 = α3, α2 = 0.6588 = α,a = 0.0030 = c̄,b = 0.0136 = c, 1 − 10k0η 3 = 0.8861, and (4k 3 + 4 3 k0kη + k0)η = 0.0521 < 1. hence, conditions (2.2),(2.3), (2.24) and (2.25) hold. 5. conclusion the semi-local convergence of scheme (1.1) with order three is extended using general conditionson f ′ and recurrent majorizing sequences. references [1] i.k. argyros, on the newton kantorovich hypothesis for solving equations, j. comput. math. 169 (2004) 315-332. https://doi.org/10.1016/j.cam.2004.01.029[2] i.k. argyros, computational theory of iterative schemes. series: studies in computational mathematics, 15, editors:c.k.chui and l. wuytack, elsevier publ. co. new york, u.s.a, 2007.[3] i.k. argyros, convergence and applications of newton-type iterations, springer verlag, berlin, germany, (2008).[4] i.k. argyros, s. hilout, weaker conditions for the convergence of newton’s scheme, j. complex. 28 (2012) 364–387. https://doi.org/10.1016/j.jco.2011.12.003.[5] i.k. argyros, s. hilout, on an improved convergence analysis of newton’s scheme, appl. math. comput. 225 (2013)372-386. https://doi.org/10.1016/j.amc.2013.09.049 https://doi.org/10.28924/ada/ma.2.13 https://doi.org/10.1016/j.cam.2004.01.029 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(1964).[27] r. verma, new trends in fractional programming, nova science publisher, new york, usa, (2019). https://doi.org/10.28924/ada/ma.2.13 https://doi.org/10.1007/s13226-020-0409-5 https://doi.org/10.1090/s0025-5718-04-01646-1 https://doi.org/10.1016/j.amc.2007.01.062 https://doi.org/10.1016/j.amc.2011.08.011 https://doi.org/10.1007/ s10910-018-0856-y https://doi.org/10.1007/ s10910-018-0856-y https://doi.org/10.1016/j.cam.2013.11.019 https://doi.org/10.1016/j.jco.2008.05.006 https://doi.org/10.1016/j.jco.2009.05.001 https://doi.org/10.1016/j.amc.2003.12.025 https://doi.org/10.1007/s11075-012-9585-7 1. introduction 2. majorizing sequences 3. semi-local convergence 4. numerical experiments 5. conclusion references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 16doi: 10.28924/ada/ma.2.16 a new approximate birkhoff orthogonality type chuanjiang zhou, qi liu, yongjin li∗ department of mathematics, sun yat-sen university, guangzhou, 510275, p. r. china 1090871744@qq.com, liuq325@mail2.sysu.edu.cn, stslyj@mail.sysu.edu.cn ∗correspondence: stslyj@mail.sysu.edu.cn abstract. in this note, we introduce a new approximate birkhoff orthogonality type and give a char-acterization for inner product spaces using the approximate orthogonality. we show some generalproperties of the approximate birkhoff orthogonality type as well as applications. in particular, westudy the relationship between the new approximate birkhoff orthogonality type and other approx-imate orthogonality types that have been defined before. furthermore we study the approximatepreserving mapping and give some properties. 1. introduction one of the important ideas playing a fundamental role in geometry of normed spaces is the con-cept of orthogonality. many mathematicians have introduced different types of orthogonality for thenormed linear spaces, cf. [2, 20, 24]. in 1934 [23], the first orthogonality type:roberts orthogonalitywas introduced by roberts. after that in 1935 [5], birkhoff introduced one of the most importantorthogonality types: x is said to be birhoff orthogonal to y (x ⊥b y) if ‖x +ty‖≥‖x‖ for all t ∈r.then james in 1945 [15] introduced the pythagorean orthogonality and isosceles orthogonality: x is said to be isosceles orthogonal to y (x ⊥i y) if ‖x + y‖ = ‖x − y‖. there are also otherorthognality types related to norm limit such as ρ-orthogonality and g-orthogonality [10, 18].let x be inner product spaces (x,〈·|·〉), all the orthogonality types are equivalent to x ⊥ y orequivalently, 〈x|y〉 = 0. in inner product spaces a natural way to generalize orthogonality is todefine the approximate orthogonality by: x ⊥� y if and only if |〈x|y〉| ≤ �‖x‖‖y‖, x,y ∈ x [9, 26].inspired by the approximate orthogonality, dragomir [13] gave the definition of the approximatebirkhoff orthogonality x� ⊥b y : ‖x + ty‖ ≥ (1 − �)‖x‖ for all t ∈ r. it is easy to see thatthis type of approximate orthogonality is equivalent to ⊥� in inner product spaces [13]. after thatjacek chmieliński [21] introduced the approximate birkhoff orthogonality x ⊥�b y : ‖x + ty‖2 ≥ ‖x‖2− 2�‖x‖‖ty‖ for all t ∈r, the approximate isosceles orthogonality [11] x ⊥�i y : |‖x + y‖2− received: 20 feb 2022. key words and phrases. birkhoff orthogonality; isosceles-orthogonality; approximate orthogonality; orthogonalitypreserving mappings. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 2 ‖x − y‖2| ≤ 4�‖x‖‖y‖ for all t ∈ r, and x� ⊥i y : |‖x + y‖−‖x − y‖| ≤ �‖x + y‖‖x − y‖ forall t ∈r. many meaningful results have been found about approximate orthogonality through thetireless efforts of mathematicians, see [12, 14].in this paper we will introduce a new approximate birkhoff othogonality type and investigateits properties and its relationship with other approximate orthogonality types. moreover we give acharacterization of inner product spaces by approximate orthogonality types and some propertiesabout approximately orthogonality preserving mapping.throughout the paper we will only consider normed spaces with dimx ≥ 2, we use 〈·|·〉 denotingthe inner product and (·|·) denoting the angle between x and y , i,e, in inner product spaces (x,y) = ‖x+y‖2−‖x‖2−‖y‖2 2‖x‖‖y‖ . 2. approximate birkhoff orthogonality ⊥b� let � ∈ [0, 1) and x,y be elements of inner product spaces x, we have the vertical relationship: x ⊥ y ⇐⇒ |〈x|y〉| = 0. to generalize the orthogonality, it is natural to consider the approximateorthogonality (�-orthogonality: x ⊥� y) defined by: x ⊥� y ⇐⇒ |〈x|y〉| ≤ �‖x‖‖y‖ ⇐⇒ |cos(x,y)| ≤ �. now we consider normed spaces, many mathematicians have introduced different types of orthogo-nality to represent orthogonality such as birkhoff orthogonality [5] and isosceles orthogonality [15].as an extension for the orthogonality, approximately orthogonality such as approximate birkhofforthogonalty [13, 21]: x� ⊥b y ⇐⇒ ‖x + ty‖≥ (1 − �)‖x‖ t ∈r. x ⊥�b y ⇐⇒ ‖x + ty‖ 2 ≥‖x‖2 − 2�‖x‖‖ty‖ t ∈r, and approximate isosceles orthogonality [11]: x ⊥�i y ⇐⇒ |‖x + y‖ 2 −‖x −y‖2| ≤ 4�‖x‖‖y‖ t ∈r. x� ⊥i y ⇐⇒ |‖x + y‖−‖x −y‖|≤ �‖x + y‖‖x −y‖ t ∈r,have been defined and studied. notice that the definition of � ⊥b is quadratic while the definitionof ⊥�b is of first order, we give a new approximate birkhoff orthogonality type: x ⊥b� y ⇐⇒ ‖x + ty‖≥‖x‖− �‖ty‖, which is also of first order but different from � ⊥b. it is easy to see that the inequality is alwayscorrect if t ≥ ‖x‖ �‖y‖. example 2.1. let x = (r2,‖ · ‖1), assume that x = (1, 0), y = (z, 1 − z), z ∈ [0, 1). if we want x� ⊥b y, then the inequality ‖x + ty‖≥ (1 −�)‖x‖ should hold for all t ∈r, thus we have: ‖(1 + tz,t(1 −z))‖≥ 1 − � t ∈r. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 3 if t ≥ 0, the inequality is always correct. for t < 0, if 1 + tz ≥ 0, we have: 1 + tz − t + tz ≥ 1 − �. thus z ≤ 1 2 − � 2t . by 1 + tz ≥ 0, we get z ≤ 1 2−�. if 1 + tz < 0, similarly we need t ≤ �−2. since z ≤ 1 2−�, from 1 + tz < 0, we get t ≤ �− 2. thus x � ⊥b y iff z ≤ 12−�. on the other hand, if we want x ⊥b� y, the inequality ‖x + ty‖≥‖x‖− �‖ty‖ should hold for all t ∈r, thus we have: ‖(1 + tz,t(1 −z))‖≥ 1 − �|t| t ∈r. if t ≥ 0, the inequality is also always correct. for t < 0, if 1 + tz ≥ 0, we have: 1 + tz − t + tz ≥ 1 + �t. thus z ≤ 1+� 2 . similarly we can get ‖(1 + tz,t(1 −z))‖≥ 1 − �|t| for 1 + tz < 0 if z ≤ 1+� 2 . thus x ⊥b� y iff z ≤ 1+�2 . we have the result that ⊥b� is not always equivalent to � ⊥b in x. since the definition of approximate birkhoff orthogonality comes from the notion of approximateorthogonality ⊥� in inner product spaces, it is natural to require the equivalence: x ⊥b� y ⇐⇒ x ⊥� y in inner product spaces. now we give some basic properties about ⊥b� before prove theequivalence. proposition 2.2. let x be normed spaces, then ⊥b� is homogeneous., this is x ⊥b� y implies αx ⊥b� βy (x,y ∈ x,α,β ∈r). proof. since x ⊥b� y , we have ‖x + ty‖ ≥ ‖x‖− �‖ty‖ for any t ∈ r. if α = 0 ,αx ⊥b� βy is always correct; if α 6= 0, we have ‖αx + tβy‖ = |α|‖x + β α ty‖≥ |α|{‖x‖− �‖ β α ty‖} = ‖ax‖− �‖tβy‖. thus αx ⊥b� βy . � recall that the limits [16] : n±(x; y) = lim n→±∞ ‖nx + y‖−|nx‖ = lim h→0± ‖x + hy‖−‖x‖ h , exist and satisfy the weakened linearity condition [4]. x,y are said to be gateaux differentiable [1]at 0 if n−(x,y) = n+(x,y). moreover we have [16]: n±(x; rx + sy) = r‖x‖ + s ·n±(x; y), f or s ≥ 0 and all r. we then give a characterization of x ⊥b� y using the definition of n±(x,y). https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 4 proposition 2.3. let x be normed spaces, then x ⊥b� y if and only if n+(x,y) + �‖y‖≥ 0 ≥ n−(x,y) − �‖y‖. proof. let x ⊥b� y and t ∈r\{0} then ‖x + ty‖−‖x‖ |t| ≥ �‖y‖. let t → 0+, we have n+(x,y) ≥−�‖y‖. similarly, let t → 0−, we have n−(x,y) ≤ �‖y‖. to sum up, n+(x,y) + �‖y‖≥ 0 ≥ n−(x,y) − �‖y‖. conversely, if n+(x,y) ≥−�‖y‖, for ∀η > 0, there ∃δ such that if 0 < t ≤ δ, we have: ‖x + ty‖−‖x‖ t ≥−(� + η)‖y‖, or equivalently ‖x + ty‖−‖x‖≥−t(� + η)‖y‖ f or t ∈ (0,δ]. because of the convexity of ‖x + ty‖, we have ‖x + ty‖−‖x‖≥−t(� + η)‖y‖ f or t > 0. let δ → 0, we have ‖x + ty‖−‖x‖≥−�‖ty‖ f or t > 0. similarly, using n−(x,y) ≤ �‖y‖, we have:‖x + ty‖−‖x‖≥ t(�)‖y‖ f or t < 0. if t = 0,‖x + ty‖−‖x‖≥ �‖ty‖ is obvious. to conclude, we have: ‖x + ty‖−‖x‖≥ �‖ty‖ f or t ∈r. thus x ⊥b� y . � to verify the validity of the new approximate birkhoff orthogonality, we have the followingproposition: proposition 2.4. let x be normed spaces, we have: x ⊥b� y if and only if x ⊥� y. proof. since x ⊥b� y , for 0 < t ≤ ‖x‖ �‖y‖ we have ‖x + ty‖≥‖x‖− �‖ty‖. square both sides we get ‖x‖2 + t2‖y‖2 + 2t(x,y) ≥‖x‖2 + �2t2‖y‖2 − 2�‖x‖‖ty‖. thus (1−�2)t‖y‖≥−2‖x‖(� + cos(x,y)) when t tends to 0, (1−�2)t‖y‖ tends to 0, so we have � + cos(x,y) ≥ 0 ⇐⇒ cos(x,y) ≥−�. similarly for 0 > t ≥− ‖x‖ �‖y‖ , we have cos(x,y) ≤ �, thus x ⊥ � y . https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 5 conversely, if |cos(x,y)| ≤ �, we have ‖x + ty‖−‖x‖≥ �‖ty‖ f or |t| ∈ [0, ‖x‖ �‖y‖ ]. on the other hand, ‖x + ty‖−‖x‖≥ �‖ty‖ is always correct for |t| ≥ ‖x‖ �‖y‖ . to conclude, ‖x + ty‖−‖x‖≥ �‖ty‖ f or t ∈r. thus x ⊥b� y . � from n±(x; rx + sy) = r‖x‖ + s ·n±(x; y), f or s ≥ 0 and all r, we have the following: proposition 2.5. in the normed space x, if x ⊥b� y , then we have x ⊥b� rx + sy for s ≥ 0, r satisfying �(‖rx + sy‖− s‖y‖) ≥ r‖x‖≥ �(s‖y‖−‖rx + sy‖). proof. since x ⊥b� y, we have n+(x,y) ≥−�‖y‖ and n−(x,y) ≤ �‖y‖, so n+(x,rx + sy) ≥ r‖x‖ + (−s�‖y‖) ≥−�‖rx + sy‖. similarly we have n−(x,rx + sy) ≤ �‖rx + sy‖, thus x ⊥b� rx + sy. � let x be normed spaces, it is known that [16] for any x,y ∈ x there exists a real number asuch that x ⊥b ax + y, moreover, such a number satisfies |a| ≤ ‖y‖‖x‖. on this basis, chmieliński [9]discovered that x ⊥�b y if and only if there exists a real number |a| ≤ ‖y‖‖x‖� such that x ⊥b ax + y.in fact, in inner product spaces, it is easy to see that x ⊥� y if and only if there exists |a| ≤ ‖y‖‖x‖�such that x ⊥b ax + y by taking a = −〈x|y〉‖x‖2 x + y for x 6= 0.in the following we will prove that it is also true for ⊥b�, that is, in normed spaces, x ⊥b� y if and only if there exists |a| ≤ ‖y‖‖x‖� such that x ⊥b ax + y. before the proof, we need some lemma. lemma 2.6. [16] let x be normed spaces, n−(x,y) ≤ n+(x,y). lemma 2.7. [16] let x be normed spaces, a ≤ b, a, b ∈ x, if x ⊥b ax + y,x ⊥b bx + y , then x ⊥b cx + y f or c ∈ [a,b]. lemma 2.8. [16] let x be normed spaces, x ⊥b ax + y ⇐⇒ n−(x,y) ≤−a‖x‖≤ n+(x,y). https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 6 theorem 2.9. let x be normed spaces, x ⊥b� y if and only if there exists |a| ≤ ‖y‖ ‖x‖� such that x ⊥b ax + y. proof. if x ⊥b� y , first we have n−(x,y) ≤ �‖y‖ , n+(x,y) ≥−�‖y‖. on the other hand, from lemma 2.8, we have if − n+(x,y) ‖x‖ ≤ a ≤− n−(x,y) ‖x‖ , then x ⊥b ax + y. if there exists no |a| ≤ ‖y‖ ‖x‖� such that x ⊥b ax + y , then − n+(x,y) ‖x‖ > ‖y‖ ‖x‖ � or − n−(x,y) ‖x‖ < − ‖y‖ ‖x‖ �. thus n+(x,y) < ‖y‖ or n(x,y) > ‖y‖�. contradict to x ⊥b� y , so there must exists |a| ≤ ‖y‖ ‖x‖�, such that x ⊥b ax + y. conversely, if there exists |a| ≤ �‖y‖‖x‖ such that x ⊥b ax + y , we have: x ⊥b ax + y =⇒ −n+(x,y) ≤ a‖x‖≤−n−(x,y) =⇒ n+(x,y) + �‖y‖≥ 0 ≥ n−(x,y) − �‖y‖ =⇒ x ⊥b� y. to conclude, in normed spaces, x ⊥b� y if and only if there exists |a| ≤ ‖y‖ ‖x‖� such that x ⊥b ax +y. � since both x ⊥b� y and x ⊥b� y are equivalent to there exists |a| ≤ ‖y‖‖x‖� such that x ⊥b ax +y.we have x ⊥b� y if and only if x ⊥�b y in normed spaces. now we give a direct proof for this. theorem 2.10. let x be normed spaces, x ⊥b� y if and only if x ⊥�b y. proof.we can assume that |t| ≤ ‖x‖ �‖y‖ and x 6= 0. if x ⊥b� y , we have: ‖x + ty‖≥‖x‖− �‖ty‖≥ 0. take square on both sides, we get ‖x + ty‖2 ≥‖x‖2 − �2‖ty‖2 − 2�‖x‖‖ty‖. thus ‖x + ty‖2 ≥‖x‖2 − 2�‖x‖‖ty‖, which means that x ⊥�b y. conversely, if x ⊥�b y , we have ‖x + ty‖2 ≥‖x‖2 − 2�‖x‖‖ty‖≥ 0. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 7 let both sides be divided by ‖x + ty‖ + ‖x‖, we get: ‖x + ty‖−‖x‖≥ −2�‖x‖‖ty‖ ‖x + ty‖ + ‖x‖ . since ‖x + ty‖ tends to ‖x‖ when t → 0, for every 2‖x‖ > δ > 0, we can find η > 0, such that ‖x + ty‖ + ‖x‖≥ 2‖x‖−δ if |t| ≤ η. we then have ‖x + ty‖−‖x‖≥ −2‖x‖‖ty‖ 2‖x‖−δ . let δ → 0 we have ‖x + ty‖−‖x‖≥ −2‖x‖‖ty‖ 2‖x‖ when t → 0. thus n+(x,y) + �‖y‖≥ 0 ≥ n−(x,y) − �‖y‖ =⇒ x ⊥b� y. � recall that dragomir gave the following definition about approximate birkhoff orthogonality: x� ⊥b y ⇐⇒ ‖x + ty‖≥ (1 − �)‖x‖. it is known that [19] in normed spaces, x ⊥�b y implies xδ ⊥b, where δ = 1 −√1 − 4�. now wegive a more accurate estimate of δ as an application of the above proposition. proposition 2.11. let x be normed spaces, let x,y ∈ x, then: x ⊥�b y =⇒ x δ ⊥b y where δ = 2�. proof. let f (t) = ‖x + ty‖ and assume that f (t) attains its minimum at t0, hence ‖x + t0y + ty‖≥‖x + t0y‖ f or all t ∈r. choose t = −t0 we have ‖x‖≥‖x + t0y‖≥ |‖x‖−|t0|‖y‖|, thus we get |t0| ≤ 2‖x‖ ‖y‖ , then ‖x + ty‖≥‖x + t0y‖≥‖x‖− �|t0|‖y‖≥ (1 − 2�)‖x‖ f or all t ∈r. thus xδ ⊥b y , where δ = 2�. by the equivalence between ⊥b� and ⊥�b, we have the result that x ⊥�b y implies x δ ⊥b y. since 2� ≤ 1 − √ 1 − 4�, 2� can be seen as a more accurate estimate. � https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 8 3. approximate isosceles orthogonality and approximate birkhoff orthogonality in the following we will use the notion of approximate isosceles orthogonality [11], recall thatthe approximate isosceles orthogonality is defined by: x ⊥�i y : |‖x + y‖ 2 −‖x −y‖2| ≤ 4�‖x‖‖y‖. x� ⊥i y : |‖x + y‖−‖x −y‖|≤ �(‖x + y‖ + ‖x −y‖). it is easy to see that in inner product spaces we have: x ⊥�i y ⇐⇒ |cos(x,y)| ≤ � ⇐⇒ x ⊥b� y, and [11] x� ⊥i y ⇐⇒ |cos(x,y)| ≤ � 1 + �2 (‖x‖2 + ‖y‖2). in the following we give some simple properties about approxiamte isosceles orthogonality. proposition 3.1. let x be normed spaces, if there exists |a| ≤ ‖y‖‖x‖� such that x ⊥i ax + y , then x� ⊥i y . proof. since x ⊥i ax + y we have ‖x + ax + y‖ = ‖x −ax −y‖, then |‖x + y‖−‖x −y‖| = |‖x + ax + y −ax‖−‖x −ax −y + ax‖|. on the other hand, by trigonometric inequality we have: ‖x + ax + y‖−‖ax‖− (‖x −ax −y‖ + ‖ax‖) ≤‖x + ax + y −ax‖−‖x −ax −y + ax‖, and ‖x + ax + y −ax‖−‖x −ax −y + ax‖ ≤‖x + ax + y‖ + ‖ax‖− (‖x −ax −y‖−‖ax‖). thus |‖x + ax + y −ax‖−‖x −ax −y + ax‖|≤ 2‖ax‖, then |‖x + y‖−‖x −y‖|≤ 2‖ax‖≤ �(‖x + y‖ + ‖x −y‖), thus x� ⊥i y . proposition 3.2. let x be normed spaces, if for every ‖x‖ = ‖y‖ = 1, there is no 0 ≤ � < 1 such that x ⊥�i y , then x is a strictly convex space. proof. for any ‖x‖ = ‖y‖ = ‖x+y‖ 2 = 1, if x 6= y ,we have |‖x + y‖2 −‖x −y‖2| = |4 −‖x −y‖| < 4, https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 9 thus there must exist a 0 ≤ � < 1 such that |4 −‖x − y‖2| ≤ 4� which means that x ⊥�i y , contradict to the condition. so there must be x = y. from the equivalent characterization of strictly convex space [23]. we get the result that x must be a strictly convex space. it is known that in inner product spaces, different orthogonality types such as isosceles,pythagorean, and birkhoff orthogonality is equivalent [3]. using the notions of orthogonality innormed linear spaces it is possible to give different characterizations for inner product spaces. forinstance [17], if x ⊥i y =⇒ x ⊥b y in a normed space x, then x must be inner product spaces.inspired by this, now we give a characterization for inner product spaces using approximate or-thogonality. theorem 3.3. let x be normed spaces, then x is inner product spaces iff the following two conditions are satisfied. (1) if there exists |a| ≤ ‖y‖‖x‖� such that x ⊥i ax + y, then x ⊥ � i y. (2) x ⊥�i y impliesx ⊥b� y. proof. if x is an inner product space, we have x ⊥i y ⇐⇒ x ⊥b y and x ⊥�i y ⇐⇒ x ⊥b� y. thus (2) is satisfied. if there exists |a| ≤ ‖y‖‖x‖� such that x ⊥i ax + y , we have: x ⊥b ax + y, |a| ≤ ‖y‖ ‖x‖ �. thus x ⊥b� y which implies x ⊥�i y. thus both (1) and (2) are satisfied. conversely, assume that both (1) and (2) are satisfied, let x ⊥i y, x 6= 0. if |a| ≤ � ‖ax+y‖ ‖x‖ , let b = −a, then |b| ≤ �‖ax+y‖‖x‖ and x ⊥i bx + ax + y , thus form (1) we have x ⊥ � i ax + y. to conclude we have: x ⊥�i ax + y if |a| ≤ � ‖ax + y‖ ‖x‖ . now define: f (t) = ‖tx+y‖‖x‖ �, we have f (t) = ‖tx + y‖ ‖x‖ � ≤ ‖tx‖ + ‖y‖ ‖x‖ � = �|t| + ‖y‖ ‖x‖ �. since 0 ≤ � < 1, when |t| tends to infinite, f (t) < |t|. when t = 0, f (0) = ‖y‖‖x‖� > 0. by the convexity of f (t) we have x ⊥�i t1x + y, t1 < 0, ‖t1x + y‖ ‖x‖ � = −t1. x ⊥�i t2x + y, t2 > 0, ‖t2x + y‖ ‖x‖ � = t2. from (2) we have x ⊥b� t1x + y and x ⊥b� t2x + y . from proposition 2.9,there must exist |a1| ≤ ‖t1x + y‖ ‖x‖ � = −t1 and |a2| ≤ ‖t2x + y‖ ‖x‖ � = t2 https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 10 such that x ⊥b a1 + t1 + y, x ⊥b a2 + t2 + y . by a1 + t1 ≤ 0 , a2 + t2 ≥ 0 and lemma 2.7, we have x ⊥b y . thus we have x ⊥i y =⇒ x ⊥b y, which means that x is inner product spaces. example 3.4. let x = (r2,‖ · ‖∞), that is, ‖(x1,x2)‖ = max(|x1|, |x2|), assume that x = (1, 0),y = (z, 1), |z| < 1. in order to satisfy x ⊥b� y or equivalently ‖x + ty‖≥‖x‖− �‖ty‖, the following inequality shuold hold for all t ∈r: ‖(1 + tc,t)‖≥ 1 − �|t|. since ‖(1 + tc,t)‖ ≥ ‖t‖, we know the above inequality is always correct if |t| ≥ 1 1+� , then we may assume that |t| < 1 1+� ≤ 1. when 1 > t ≥ 0, if 1 + tc ≥ t which means that t ≤ 1 1−c , we have 1 + tc ≥ 1 − �t which implies that c ≥ −�. if 1 + tc < t or equivalently t > 1 1−c , we have t ≥ 1 − �t that is t ≥ 1 1+� . so there must be 1 1 −c ≥ 1 1 + � which implies that c ≥ −�. similarly when −1 < t ≤ 0, we can get c ≤ �. to conclude we have: (1, 0) ⊥b� (c, 1) if |c| ≤ �. in order to satisfy x ⊥i� y or equivalently |‖x +y‖2−‖x−y‖2| ≤ 4�‖x‖‖y‖, the following inequality shuold hold: |‖(1 + c, 1)‖2 −‖(1 −c,−1)‖2| ≤ 4�. if c ≥ 0, we have (1 + c)2 − 1 ≤ 4� =⇒ 0 ≤ c < −1 + √ 1 + 4�. if c < 0, we have (1 −c)2 − 1 ≤ 4� =⇒ 1 − √ 1 + 4� ≤ c < 0. to conclude we have (1, 0) ⊥�i (c, 1) if |c| ≤ −1 + √ 1 + 4�. since � ≤ −1 + √ 1 + 4�, it can be seen as an example that x ⊥�i y does not imply x ⊥b� y. example 3.5. let x = (r2,‖ ·‖∞), we assume that x = (1, 1),y = (−1 − √ 2 2 �, 1 − √ 2 2 �), z = (−1, 1). we have x ⊥i z, and z = − � √ 2 x + y, � ≤ ‖y‖ ‖x‖ �. thus |− �√ 2 | ≤ ‖y‖‖x‖�, which implies that there exists |a| ≤ ‖y‖ ‖x‖� such that x ⊥i ax + y. on the other hand, since ‖x + y‖ = ‖(− √ 2 2 �, 2 − √ 2 2 �)‖ = 2 − √ 2 2 �. ‖x −y‖ = ‖(2 + √ 2 2 �, √ 2 2 �)‖ = 2 + √ 2 2 �. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 11 we have |‖x + y‖2−‖x −y‖2| = 4 √ 2� ≥ 4 √ 2. so x 6⊥�i y , thus it can be seen as an example that condition (1) is not satisfied. a mapping t : h→k which satisfies the condtion x ⊥ y =⇒ t (x) ⊥ t (y). is called orthogonality preserving(o.p.) [7, 8], and t is said to be an isometry maping [22] if ‖tx‖ = ‖x‖. to promote the concept, jacek chmielinski [6] introduced the notion of approximately orthogo-nality preserving (a.o.p.) mapping and have studied the properties of mapping that is approximarelyisosceles orthogonality preserving(t: x ⊥i y =⇒ t (x) ⊥�i t (y)). after that many mathematicianshave show great interest in the a.o.p mapping [25], and aleksej turnšek [19] studied the mappingthat is approximately birkhoff orthogonality preserving(t : x ⊥b y =⇒ t (x) ⊥�b t (x))innormed spaces. now we try to study the approximarely orthogonality preserving mapping types: t : x ⊥i y =⇒ tx ⊥�b ty, and t : x ⊥i y =⇒ tx �x ⊥b ty. proposition 3.6. let t : x → y be a nontrivial linear mapping satisfying x ⊥i y =⇒ tx ⊥�b ty, x,y ∈ x. then t is a bounded and bounded from below, ‖tx‖≥ (1−�) 2 3−�2+2 √ 2−�2 . proof. take two arbitrary unit vectors x and y and note that x+y 2 ⊥i x−y2 , it follows that t (x + y) ⊥�b t (x −y), hence for all λ ∈r we have ‖t (x + y) + λt (x −y)‖2 ≥‖t (x + y)‖2 − 2�‖t (x + y)‖‖λt (x −y)‖, by the triangle inequality and the linearity of t it follows that ‖t (x + y)‖2 ≤‖(1 + λ)tx + (1 −λ)ty‖2 + 2�|λ|‖tx + ty‖2. on the other hand we have ‖t (x + y)‖2 ≥ (‖tx‖−‖ty‖)2, thus we get: ‖tx‖2+‖ty‖2−2‖tx‖‖ty‖≤ (1+λ)2‖tx‖2+(1−λ)2‖ty‖2+2(1−λ2)‖tx‖‖ty‖+2�|λ|‖tx+ty‖2. if tx = 0, then let y ∈ x such that ty 6= 0, substitute x,y into the above formula,we have 0 ≤ λ2 − 2λ + 2�|λ| f or all λ ∈r. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 12 it is impossible cause 0 ≤ � < 0, so we can divide both sides of the inequality by ‖tx‖2 and denote z = ‖ty‖‖tx‖ . we get: (z − 1)2λ2 + ( 2 − 2z2 + 2�(1 + z)2 ) λ + 4z ≥ 0 f or all λ ≥ 0. the inequality is satisfied when − b 2a ≤ 0 or ∆ = b2 − 4ac ≤ 0. ∆ = 4 ( (1 −z2)2 + �2(1 + z)4 + 2�(1 + z)2(1 −z2) − (4z)(z − 1)2 ) . (1 −z2)2 + �2(1 + z)4 + 2�(1 + z)2(1 −z2) − (4z)(z + 1)2 ≤ 0 =⇒ ∆ ≤ 0. thus we get z ≤ 3−� 2+2 √ 2−�2 (1−�)2 which implies ‖ty‖ ≤ 3−�2+2 √ 2−�2 (1−�)2 ‖tx‖. since x,y are arbitrary, t is bounded and ‖tx‖≥ (1 − �)2 3 − �2 + 2 √ 2 − �2 ‖t‖‖x‖. theorem 3.7. let t : x → y be a nontrivial linear mapping satisfying x ⊥i y =⇒ tx � ⊥b ty, x,y ∈ x. then t is a scalar mutiple of a isometric mapping, i.e., for some γ > 0, ‖tx‖ = γ‖x‖. proof. take two arbitrary unit vectors x and y and note that x+y 2 ⊥i x−y2 , it follows that t (x + y)� ⊥b t (x −y). hence for all λ ∈r we have ‖t (x + y) + λt (x −y)‖≥ (1 − �)‖t (x + y)‖, thus ( ‖(1 + λ)tx‖ + ‖(1 −λ)ty‖ )2 ≥ (1 − �)2‖tx + ty‖2 ≥ (1 − �)2(‖tx‖−‖ty‖)2. if tx = 0, then let y ∈ x such that ty 6= 0, substitute x,y into the above formula, we have ‖(1 −λ)ty‖2 ≥ (1 − �)2‖ty‖2 =⇒ (1 −λ)2 ≥ (1 − �)2 f or all λ ∈r, it is impossible, then we can divide both sides by ‖tx‖2 like before and denote z = ‖ty‖‖tx‖ , we get (1 −z)2λ2 + 2(1 −z2)λ + (z + 1)2 − (1 − �)2(1 −z)2 ≥ 0 f or all λ ∈r. if z 6= 1, then ∆ = 4 · (1 − z)4(1 − �)2 > 0. thus the inequality is satisfied only when z = 1, so z = 1 which means that ‖tx‖ = ‖ty‖, thus t must be a scalar mutiple of a isometric mapping. https://doi.org/10.28924/ada/ma.2.16 eur. j. math. anal. 10.28924/ada/ma.2.16 13 references [1] m. abbasi, a.y. kruger, m. théra, gateaux differentiability revisited, appl. math. optim. 84 (2021) 3499–3516. https://doi.org/10.1007/s00245-021-09754-y.[2] j. alonso, h. martini, s. wu, on birkhoff orthogonality and isosceles orthogonality in normed linear spaces, aequat.math. 83 (2011) 153–189. https://doi.org/10.1007/s00010-011-0092-z.[3] d. amir, characterizations of inner product spaces, birkhäuser basel, 1986. https://doi.org/10.1007/ 978-3-0348-5487-0.[4] g. ascoli, sugli spazi lineari metrici e le loro varietà lineari, ann. mat. 10 (1932) 203–232. https://doi.org/10. 1007/bf02417142.[5] g. birkhoff, orthogonality in linear metric spaces, duke math. j. 1 (1935) 169-172. https://doi.org/10.1215/ s0012-7094-35-00115-6.[6] j. chmieliński, linear mappings approximately preserving orthogonality, j. math. anal. appl. 304 (2005) 158–169. https://doi.org/10.1016/j.jmaa.2004.09.011.[7] j. chmieliński, stability of the orthogonality preserving property in finite-dimensional inner product spaces, j. math.anal. appl. 318 (2006) 433–443. https://doi.org/10.1016/j.jmaa.2005.06.016.[8] j. chmieliński, j. chmieliński, orthogonality preserving property and its ulam stability. in: rassias, t., brzdek, j.(eds) functional equations in mathematical analysis. springer optimization and its applications, vol 52. springer,new york, ny. 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https://doi.org/10.1016/j.jmaa.2007.03.016 https://doi.org/10.1007/s00010-013-0233-7 https://doi.org/10.1007/s00010-013-0233-7 1. introduction 2. approximate birkhoff orthogonality b 3. approximate isosceles orthogonality and approximate birkhoff orthogonality references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 3doi: 10.28924/ada/ma.2.3 on the ostrowski method for solving equations ioannis k. argyros1,∗, santhosh george2, christopher i. argyros3 1department of mathematical sciences, cameron university, lawton, ok 73505, usa iargyros@cameron.edu 2 department of mathematical and computational sciences,national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 3department of computing and technology, cameron university, lawton, ok 73505, usa christopher.argyros@cameron.edu ∗correspondence: iargyros@cameron.edu abstract. in this paper, we revisited the ostrowski’s method for solving banach space valued equa-tions. we developed a technique to determine a subset of the original convergence domain and usingthis new lipschitz constants derived. these constants are at least as tight as the earlier ones leadingto a finer convergence analysis in both the semi-local and the local convergence case. these tech-niques are very general, so they can be used to extend the applicability of other methods withoutadditional hypotheses. numerical experiments complete this study. 1. introduction one of the most challenging tasks in computational mathematics is the problem of determininga solution x∗ of equation f (x) = 0, (1.1) where f : ω ⊂ b −→ b1 is an operator acting between banach spaces b and b1 with ω 6= ∅. theclosed form derivation of x∗ is possible only in rare cases. this leads practitioners and researchersin developing solution methods that are iterative.in this work, we consider ostrowski’s method defined for x0 ∈ ω and each n = 0, 1, 2, . . . by yn = xn −f ′(xn)−1f (xn) xk+1 = yn −a−1n f (yn), (1.2) received: 7 nov 2021. key words and phrases. ostrowski’s method; banach space; convergence criterion.1 https://adac.ee https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 2 where an = 2[yn,xn; f ] −f ′(xn). the convergence order is four obtained under certain conditionson the initial data (ω,f,f ′,x0) and taylor expansion [17, 25]. so, the assumptions on the fourthderivative reduce the applicability of these schemes.for example: let b = b1 = r, ω = [−0.5, 1.5]. define λ on ω by λ(t) = { t3 log t2 + t5 − t4 if t 6= 0 0 if t = 0. then, we get t∗ = 1, and λ′′′(t) = 6 log t2 + 60t2 − 24t + 22. obviously λ′′′(t) is not bounded on ω. so, the convergence of scheme (1.2) is not guaranteed bythe analyses in [17, 24].we study two types of convergence called local and semi-local. in the first one based on thesolution x∗ we find the radii of the convergence balls. but in the second one based on the starter x0 we develop criteria that guarantee convergence of sequence {xn}. there is a plethora of thistypes of results [10, 15, 16, 22, 28, 38]. but what all these results have in common is that the regionof accessibility (or convergence region) is limited in general reducing the applicability of newton’sand other methods [8,20,26,28,31]. moreover, the error bounds on distances ‖xk+1−xk‖ or ‖xk−x∗‖are pessimistic. the same is true for the uniqueness ball of these methods. these problems becomemore difficult when studying methods of convergence order three or higher [8, 17, 19, 31–33]. wehave developed different techniques to addres these problems.in technique 1, we determine a subset ω of ω also containing the iterates. but in this set ω thelipschitz-like parameters (or functions) are at least as tight as the original ones, so the resultingconvergence is finer. this technique does not depend on the convergence order of the method. butwe shall demonstrate it in case of fourth order methods. these methods require the evaluation ofthe second order fréchet derivative of operator f. notice that for a system (nonlinear) of i equationswith i unknowns, the first derivative is a matrix with i2 entries (values), whereas the second fréchetderivative has i3 entries. that is why there is a need for avoiding f ′′.the rest of the paper is organized as follows: in section 2 we develop the second technique basedon majorizing sequences. the local convergence analysis results appear in section 3. numericalexamples can be found in section 4. the paper ends with some concluding remarks. 2. semi-local convergence we base our semi-local convergence analysis on scalar parameters and functions. let η ≥ 0,k0 > 0,k > 0,k1 > 0,k2 > 0,k3 > 0,l0 > 0 with k0 ≤ k, l0 ≤ 2k1 and k4 = k2 + k3.define polynomials g1 and g2 on the interval [0, 1) by g1(t) = k1t 5 + (2k1 + k3)t 4 + k1t 3 + ( k 2 −k3)t2 − k 2 (2.1) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 3 and g2(t) = l0t 4 + (l0 + k3)t 3 + k4t 2 −k3t −k4. (2.2) we have g1(0) = −k2 < 0,g1(1) = 4k1 > 0,g2(0) = −k4 < 0 and g2(1) = 2l0 > 0. it thenfollows from the intermediate value theorem that polynomials g1 and g2 have at least one root in (0, 1). denote by δ1 and δ2 the least such roots, respectively. moreover, it is convenient to definescalar sequences and parameters t0 = 0, s0 = η, t1 = s0 + k0 2 (s0 − t0)2 1 − 2k1s0 , sn+1 = tn+1 + (k3(tn+1 − sn) + k4(sn − tn))(tn+1 − sn) 1 −l0tn+1 tn+2 = sn+1 + k(sn+1 − tn+1)2 2(1 − (k1(sn+1 + tn+1) + k3(sn+1 − tn+1)) , (2.3) αn = k(sn − tn) 2(1 − (k1(sn+1 + tn+1) + k3(sn+1 − tn+1)) , γn = k3(tn − sn) + k4(sn − tn) 1 −l0tn+1 , for all n = 0, 1, 2, . . . , δn = max{αn,γn}, λ = min{δ1,δ2} and µ = max{δ1,δ2}. next, we present a convergence result for sequences {tn} and {sn}. lemma 2.1. suppose: there exists δ satisfying 0 ≤ δ0 ≤ λ ≤ δ ≤ µ < 1 −k1η. (2.4) then, sequences {tn},{sn} are well defined nondecreasing, bounded from above bt s∗∗ = η1−δ and as such they converge to their unique least upper bound s∗ ∈ [η,s∗∗]. moreover, the following error estimates hold for all n = 1, 2, . . . 0 ≤ tn+1 − sn ≤ δ(sn − tn) ≤ δ2n+1η, (2.5) 0 ≤ sn − tn ≤ δ(tn − sn−1) ≤ δ2nη (2.6) and tn ≤ sn ≤ tn+1. (2.7) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 4 proof. items (2.5)-(2.7) hold if 0 ≤ αm ≤ δ, (2.8) 0 ≤ γm ≤ δ (2.9) and tm ≤ sm ≤ tm+1 (2.10) are true for all m = 0, 1, 2, . . . . notice that by the definition of s0,t1 and (2.4), t1 ≥ 0. we alsohave (2.8) and (2.9) hold for m = 0. suppose (2.8)-(2.10) hold for m = 1, 2, . . . ,n. then, we canobtain in turn that sm ≤ tm + δ2mη ≤ sm−1 + δ2m−1η + δ2mη ≤ η + . . . + δη + . . . + δ2mη = 1 −δ2m+1 1 −δ η ≤ η 1 −δ = s∗∗, and tm+1 ≤ sm + δ2m+1η ≤ tm + δ2mη + δ2m+1η ≤ η + δη + . . . + δ2m+1η = 1 −δ2m+2 1 −δ η ≤ η 1 −δ . hence, by (2.7) and the induction hypotheses, we deduce that sequences {tm} and {sm} arenondecreasing. evidently, (2.8) holds if k 2 δ2nη + δk1 ( 1 −δ2n+3 1 −δ η ) +δk1 1 −δ2n+2 1 −δ η + k3δ 2(n+1)η −δ ≤ 0. (2.11) estimate (2.11) motivates us to introduce recurrent functions h(1)n (t) on the interval [0, 1) by h (1) n )t) = k 2 t2n−1η + k1(1 + t + . . . + t 2n+2)η +k1(1 + t + . . . + t 2n+1)η + k3t 2n+3η − 1. (2.12) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 5 we need a relationship between two consecutive functions f (1)n (t). by this definition, we have inturn that h (1) n+1(t) = k 2 t2n+1η + k1(1 + t + . . . + t 2n+4)η +k1(1 + t + . . . + t 2n+3)η + k3t 2n+3η − 1 − k 2 t2n−1η −k1(1 + t + . . . + t2n+2)η −k1(1 + t + . . . + t2n+1)η −k3t2n+1η + 1 + h (1) n (t) = h (1) n (t) + k 2 t2n+1η − k 2 t2n−1η +k1(t 2n+3 + t2n+4)η + k1(t 2n+2 + t2n+3)η + k3t 2n+3η −k3t2n+1η = h (1) n (t) + g1(t)t 2n−1η. (2.13) notice that by the definition of δ1 h (1) n+1(δ1) = h (1) n (δ1). by (2.11)-(2.13), estimate (2.11) shall be true if for 4k1 < k h (1) n (δ1) ≤ 0. (2.14) let h(1)∞ (t) = lim n−→∞ h (1) n (t). (2.15) but then h(1)∞ (δ) = 2k1η 1 −δ − 1. (2.16) hence, instead of (2.13) we can show h(1)∞ (δ) ≤ 0, (2.17) which is true by (2.4). if 4k1 ≥ k then f∞(t) ≥ fn(t), so again f∞(δ) ≤ 0 holds. similarly, (2.9)holds if k3δ 2n+1η + k4δ 2nη + δl0 1 −δ2n+2 1 −δ η −δ ≤ 0 (2.18) or h (2) n (δ) ≤ 0, (2.19) where h (2) n (t) = k3t 2nη + k4t 2n−1η + l0(1 + t + . . . + t 2n+1)η − 1. (2.20) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 6 this time we have h (2) n+1(t) = k3t 2n+2η + k4t 2n+1η + l0(1 + t + . . . + t 2n+3)η −1 −k3t2nη −k4t2n−1η −l0(1 + t + . . . + t2n+1)η + 1 + h (2) n (t) = h (2) n (t) + k3t 2n+2η −k3t2nη + k4t2n+1 −k4t2n+1η +l0(t 2n+2 + t2n+3)η = h (2) n (t) + [k3t 3 −k3t + k4t2 −k4 + l0t2 + l− 0t4]t2n−1δ = h (2) n (t) + g2(t)t 2n−1η. (2.21) by the definition of δ2 h (2) n+1(δ2) = h (2) n (δ).let h(2)∞ (t) = limn−→∞h(2)n (t). then, we get h(2)∞ (δ) = l0η 1 −δ − 1. hence, instead of (2.19), we can show h(2)∞ (δ) ≤ 0,which is true by (2.4). the induction for (2.8)-(2.10) is completed. therefore, sequences {tn},{sn}are nondecreasing, bounded from above by s∗∗ and as such they converge to s∗. �the semi-local convergence analysis shall be based on conditions (a).suppose: (a1) there exists x0 ∈ ω, η ≥ 0 such that f ′(x0)−1 ∈ l(e1,e) and ‖f ′(x0)−1f (x0)‖≤ η. (a2) for each x ∈ ω ‖f ′(x0)−1(f ′(x) −f ′(x0))‖≤ l0‖x −x0‖. set ω0 = u[x0, 1l0 ] ∩ ω.(a3) for each x,y ∈ ω0 ‖f ′(x0)−1(f ′(y) −f ′(x))‖≤ k‖y −x‖, ‖f ′(x0)−1([y,x; f ] −f ′(x0))‖≤ k1(‖y −x0‖ + ‖x −x0‖), ‖f ′(x0)−1([z,y; f ] − [y,x; f ])‖≤ k2(‖z −y‖ + ‖y −x‖)and ‖f ′(x0)−1([z,y; f ] −f ′(y))‖≤ k3‖z −y‖.(a4) u[x0,s∗] ⊂ ω and(a5) conditions of lemma 2.1 hold. https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 7 next, we present the semi-local convergence of method (1.2). theorem 2.2. suppose that conditions (a) hold. then, sequences {yn},{xn} generated by method (1.2) are well defined in u[x0,s∗], remain in u[x0,s∗] for each n = 0, 1, 2, . . . and converge to a solution x∗ ∈ u[x0,s∗] of equation f (x) = 0. moreover, the following assertion holds ‖xn −x∗‖≤ s∗ − tn. (2.22) proof. mathematical induction on m shall be used to show(im) ‖ym −xm‖≤ sm − tmand(iim) ‖xm+1 −ym‖≤ tm+1 − sm.by the first substep of method (1.2) we have ‖y0 −x0‖ = ‖f ′(x0)−1f (x0)‖≤ η = s0 − t0 = s0 ≤ s∗. so (i0) holds and y0 ∈ u[x0,s∗]. by the first substep of method (1.2) we can write f (y0) = f (y0) −f (x0) −f ′(x0)(y0 −x0). (2.23) using (a2) and (2.23), we have ‖f ′(x0)−1f (y0)‖≤ k0 2 ‖y0 −x0‖2 ≤ k0 2 (s0 − t0)2. (2.24) we need to show the invertability of linear operator a. we have by (a3) ‖f ′(x0)−1(a0 −f ′(x0))‖ ≤ 2‖f ′(x0)−1([y0,x0; f ] −f ′(x0))‖ ≤ 2k1(‖y0 −x0‖ + ‖x0 −x0‖) ≤ 2k1(s0 + t0) < 1, so ‖a−10 f ′(x0)‖≤ 1 1 − 2k1(s0 + t0) , (2.25) by the banach lemma on linear invertible operators [19, 25]. then, iterate x1 exists by the secondsubstep of method (1.2), and we can write x1 −y0 = (a−10 f ′(x0))(f ′(x0) −1f (y0)). (2.26) by (2.24)-(2.26), we get ‖x1 −y0‖ ≤ ‖a−10 f ′(x0)‖‖f ′(x0)−1f (y0)‖ ≤ k0(s0 − t0)2 2(1 − 2k1(s0 + t0)) = t1 − s0, https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 8 showing (ii0). moreover, we have ‖x1 −x0‖ ≤ ‖x1 −y0‖ + ‖y0 −x0‖ ≤ t1 − s0 + s0 − t0 = t1 ≤ s∗, so x1 ∈ u[x0,s∗]. suppose that (im) and (iim) hold, ym,xm+1 ∈ u[x0,s∗] and f ′(xm)−1,a−1m existfor each m = 1, 2, . . . ,n. we shall prove they hold for m = n + 1. using the second substep ofmethod (1.2), we get ‖f (x0)−1f (xn+1)‖ = ‖f ′(x0)−1(f (xn+1) −f (yn)) −an(xn+1 −yn))‖ = ‖f ′(x0)−1([xn+1,yn; f ] −an)(xn+1 −yn)‖ ≤ f ′(x0)−1([xn+1,yn; f ] − [yn,xn; f ])‖ +‖f ′(x0)−1([yn,xn; f ] −f ′(xn))‖ ≤ (k2(‖xn+1 −yn‖ + ‖yn −xn‖) + k3‖yn −xn‖)‖xn+1 −yn‖ (2.27) we need to show f ′(xn+1) is invertible. by (a2) and the induction hypotheses we obtain ‖f ′(x0)−1(f ′(xn+1) −f ′(x0))‖ ≤ l0‖xn+1 −x0‖ ≤ l0(tn+1 − t0) = l0tn=1 < 1, so ‖f ′(xn+1)−1f ′(x0)‖≤ 1 1 −l0tn+1 . (2.28) hence, we get by (2.27), (2.28) and the first substep of method (1.20 that ‖yn+1 −xn+1‖ = ‖f ′(xn+1)−1f (x0)‖‖f ′(x0)−1f (xn+1)‖ ≤ (k2(tn+1 − sn) + (sn − tn)) + k3(sn − tn))(tn+1 − sn) 1 −l0tn+1 = sn+1 − tn+1, (2.29) since k4 = k2 + k3, showing (im) for m = n + 1. then, we also have ‖yn+1 −x0‖ ≤ ‖yn+1 −xn+1‖ + ‖xn+1 −x0‖ ≤ sn+1 − tn+1 + tn+1 − s0 = sn+1 ≤ s∗, https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 9 so yn+1 ∈ u[x0,s∗]. operator a−1n+1 shall be shown to exist ‖f ′(x0)−1(an+1 −f ′(x0))‖ ≤ ‖f ′(x0)−1([yn+1,xn+1; f ] −f ′(x0))‖ +‖f ′(x0)−1([yn+1; xn+1; f ] −f ′(xn+1))‖ ≤ k1(‖yn+1 −x0‖ + ‖xn+1 −x0‖) + k3‖yn+1 −xn+1‖ ≤ k1(sn+1 + tn+1) + k3(sn+1 − tn+1) < 1, so ‖a−1n+1f ′(x0)‖≤ 1 1 − (k1(sn+1 + tn+1) + k3(sn+1 − tn+1)) . (2.30) by the first substep of method (1.2), we can write f (yn+1) = f (yn+1) −f (xn+1) −f ′(xn+1)(yn+1 −xn+1), so ‖f ′(x0)−1f (yn+1)‖≤ k 2 ‖yn+1 −xn+1‖2 ≤ k 2 (sn+1 − tn+1)2, so ‖xn+2 −yn+1‖ ≤ ‖a−1n+1f ′(x0)‖‖f ′(x0)−1f (yn+1)‖ ≤ k(sn+1 − tn+1)2 2(1 − (k1(sn+1 + tn+1) + k3(sn+1 − tn+1))) = tn+2 − sn+1, showing (iim) for m = n + 1. we can get ‖xn+2 −x0‖ ≤ ‖xn+2 −yn+1‖ + ‖yn+1 −x0‖ ≤ tn+2 − sn+1 + sn+1 − t0 = tn+2 ≤ s∗, so xn+2 ∈ u[x0,s∗]. furthermore, we obtain ‖xn+1 −xn‖ ≤ ‖xn+1 −yn‖ + ‖yn −xn‖ = tn+1 − sn + sn − tn = tn+1 − tn, so sequence {xn} is fundamental in a banach space b, so it converges to some x∗ ∈ u[x0,s∗]. byletting n −→∞ in (2.27), we obtain ‖f ′(x0)−1f (xk+1)‖ ≤ (k2(tn+1 − sn) + k4(sn − tn))(sn+1 − sn) −→ 0, so f (x∗) = 0 by the continuity of f. �a uniqueness of the solution result is given next. proposition 2.3. suppose: https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 10 (i) there exists a simple solution x∗ of equation f (x) = 0.(ii) there exists s̄ ≥ s∗ such that l0(s̄ + s∗) < 2. set ω1 = u[x0, s̄] ∩ ω. then, the only solution of equation f (x) = 0 in the region ω1 is x∗. proof. let x̄ ∈ ω1 with f (x̄) = 0. let m = ∫10 f ′(x̄ + θ(x∗ − x̄))dθ. then, in view of (a2) and(ii), we obtain ‖f ′(x0)−1(m −f ′(x0))‖ ≤ l0 ∫ 1 0 [(1 −θ)‖x̄ −x0‖ + θ‖x∗ −x0‖]dθ ≤ l0 2 (s̄ + s∗) < 1, so x̄ = x∗ since m−1 exists and m(x∗ − x̄) = f (x∗) −f (x̄) = 0 − 0 = 0. � remark 2.4. notice that s∗∗ given in closed form can repalce s∗ in the conditions of theorem 2.2. 3. local convergence as in section 2 we develop some functions and parameters. let li, i = 0, 1, 2, 3, 4 be givenparameters. define function ϕ1 on the interval t = [0, 1l0 ) by ϕ1(t) = lt 2(1 −l0t) . notice that parameter ra = 2 2l0 + l < 1 l0 (3.1) solves equation ϕ1(t) = 1. define functions on the interval t by q(t) = l0ϕ1(t)t − 1 and p(t) = (2l1(1 + ϕ1(t)) + l)t. suppose that these functions have smallest zeros rq and rp in (0, 1l0 ), respectively. let r1 = min{rq, rp} and t0 = [0, r1). define function ϕ2 on t0 by ϕ2(t) = [ lϕ1(t) 2(1 −l0ϕ1(t)t) + l4(l2 + l3)(1 + ϕ1(t))ϕ1(t) (1 −l0ϕ1(t)t)(1 −p(t)) ] t. suppose that function ϕ2(t) − 1 https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 11 has smallest zero r2 ∈ (0, r1). we shall show that parameter r = min{ra, r2} (3.2) is a convergence radius for method (1.2). let t1 = [0, r). then, it follows by these definitions thatfor each t ∈ t1 l0t < 1 (3.3) 0 ≤ ϕ1(t) < 1, (3.4) 0 ≤ ϕ1(t)t < 1 (3.5) 0 ≤ p(t) < 1 (3.6) and 0 ≤ ϕ2(t)t < 1 (3.7) hold.the conditions (h) to be used in the local convergence of method (1.2) are as follows.suppose: (h1) there exists a simple solution x∗ ∈ ω of equation f (x) = 0.(h2) for each x ∈ ω ‖f ′(x)−1(f ′(x) −f ′(x∗))‖≤ l0‖x −x∗‖. set ω0 = u[x∗, 1l0 ] ∩ ω.(h3) for each x,y ∈ ω0 ‖f ′(x∗)−1(f ′(y) −f ′(x))‖≤ l‖y −x‖, ‖f ′(x∗)−1(f ′(y) −f ′(x∗))‖≤ l1(‖y −x∗‖ + ‖x −x∗‖), ‖f ′(x∗)−1([y,x; f ] −f ′(x))‖≤ l2‖y −x‖, ‖f ′(x∗)−1([y,x; f ] −f ′(y))‖≤ l3‖y −x‖ and ‖f ′(x∗)−1f ′(x)‖≤ l4‖x −x∗‖. and(h4) u[x∗, r] ⊂ ω.in view of conditions (h) and the developed notation we can show the local convergence result formethod (1.2). theorem 3.1. under the conditions (h), further suppose that x0 ∈ u(x∗, r) −{x∗}. then, sequence {xk},{yn} generated by method (1.2) is well defined in u(x∗, r), remains in u(x∗, r) for each k = 0, 1, 2, . . . and converges to x∗. https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 12 proof. let u ∈ u(x∗, r) −{x∗}. by (h1) and (h2), we get in turn that ‖f ′(x∗)−1(f ′(u) −f ′(x∗))‖≤ l0‖u −x∗‖≤ l0r < 1, so f ′(u) is invertibale and ‖f ′(u)−1f ′(x∗)‖≤ 1 1 −l0‖u −x∗‖ . (3.8) iterate y0 is well defined by the first substep of method (1.2) and (3.8) for u = x0. then, we canwrite y0 −x∗ = x0 −x∗ −f ′(x0)−1f (x0) = (f ′(x0) −1f ′(x∗)) ×( ∫ 1 0 f ′(x∗) −1(f ′(x∗ + θ(x0 −x∗)) −f ′(x0))dθ(x0 −x∗). (3.9) by (3.2), (3.4), (h3), (3.8) and (3.9), we have in turn that ‖y0 −x∗‖ ≤ l0‖x0 −x∗‖2 2(1 −l0‖x0 −x∗‖ ≤ l‖x0 −x∗‖2 2(1 −l0‖x0 −x∗‖) ≤ ϕ1(‖x0 −x∗‖)‖x0 −x∗‖≤‖x0 −x∗‖ < r (3.10) so y0 ∈ u(x∗, r). next, we show linear operator a) is invertible. indeed, using (3.2), (3.6), (h3) and(3.10), we get in turn that ‖f ′(x∗)−1(a0 −f ′(x∗))‖ ≤ ‖f ′(x∗)−1([y0,x0; f ] −f ′(x∗))‖ +‖f ′(x∗)−1([y0,x0; f ] −f ′(x∗))‖ +‖f ′(x∗)−1(f ′(x0) −f ′(x∗))‖ ≤ 2l1(‖y0 −x∗‖ + ‖x0 −x∗‖) + l0‖x0 −x∗‖ ≤ 2l1(1 + ϕ1(‖x0 −x∗‖))‖x0 −x∗‖ + l‖x0 −x∗‖ ≤ p(‖x0 −x∗‖) ≤ p(r) < 1, so ‖a−10 f ′(x∗)‖≤ 1 1 −p(‖x0 −x∗‖) (3.11) and iterate x1 is well defined by the second substep of method (1.2) for n = 0. then, we can write x1 −x∗ = (y0 −x∗ −f ′(y0)−1f (y0)) + f ′(y0)−1(a) −f ′(y0))a −1 0 f (y0). (3.12) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 13 using (3.2), (3.7), (h3), (3.8) (for u = x0,y0) and (3.10)–(3.13), we obtain in turn that ‖x1 −x∗‖ ≤ ‖y0 −x∗ −f ′(y0)−1f (y0)‖ +‖f ′(y0)−1f ′(x∗)‖‖f ′(x∗)−1(a0 −f ′(y0))‖‖a−10 f ′(x∗)‖‖f ′(x∗)−1f (y0)‖ ≤ l‖y0 −x∗‖2 2(1 −l0‖x0 −x∗‖ + (l2 + l3)‖y0 −x0‖l4‖y0 −x∗‖ (1 −l0‖y0 −x∗‖)(1 −p(‖x0 −x∗‖)) ≤ ϕ2(‖x0 −x∗‖)‖x0 −x∗‖≤‖x0 −x∗‖ < r, so x1 ∈ u(x∗, r), where we also used ‖f ′(x∗)−1(a0 −f ′(y0))‖ ≤ ‖f ′(x∗)−1([y0,x0; f ] −f ′(x0))‖ +‖f ′(x∗)−1([y0,x0; f ] −f ′(y0))‖ ≤ (l2 + l3)‖y0 −x0‖≤ (l2 + l3)(‖y0 −x∗‖ + ‖x0 −x∗‖) ≤ (l2 + l3)(1 + ϕ1(‖x0 −x∗‖))‖x0 −x∗‖, and ‖f ′(x∗)−1f (y0)‖ = ‖ ∫ 1 0 f ′(x∗) −1f ′(x∗ + θ(y0 −x∗))dθ(y0 −x∗)‖ ≤ l4‖y0 −x∗‖2. so, far showed ‖y0 −x∗‖≤ ϕ1(‖x0 −x∗‖)‖x0 −x∗‖ < r and ‖x1 −x∗‖≤ ϕ2(‖x0 −x∗‖)‖x0 −x∗‖ < r. by simply replacing x0,y0,x1 by xm,ym,xm+1 in the preceding calculations, we get ‖ym −x∗‖≤ ϕ1(‖xm −x∗‖)‖xm −x∗‖ < r and ‖xm+1 −x∗‖≤ ϕ2(‖xm −x∗‖)‖xm −x∗‖ < r. then, from the estimation ‖xm+1 −x∗‖≤ α‖xm −x∗‖ < r, (3.13) where α = ϕ2(‖x0 −x∗‖) ∈ [0, 1), limm−→∞xm = x∗ and ym,xm+1 ∈ u(x∗, r). � remark 3.2. by the definition of r, we see that r ≤ ra. (3.14) https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 14 parameter ra was shown in [4] to be a convergence radius for newton’s method. notice the radius of convergence for newton’s method given independently by traub [35] and rheinbold [29] is rtr = 2 3m1 , where l1 is the lipschitz constant on ω. so, we have rtr ≤ ra, since l ≤ l1 and l0 ≤ m1. 4. numerical experiments we provide some examples, showing that the old convergence criteria are not verified but oursare. example 4.1. define function f (t) = θ0t + θ1 + θ2 sin θ3t, t0 = 0, where θj, j = 0, 1, 2, 3 are parameters. then, clearly for θ3 large and θ2 small, l0k can be small (arbitrarily). example 4.2. let b = b1 = u[0, 1] the domain of functions given on [0, 1] which are continuous. we consider the max-norm. choose ω = b(0,d), d > 1. define f on ω be f (x)(s) = x(s) −w(s) −ξ ∫ 1 0 k(s,t)x3(t)dt, (4.1) x ∈ b,s ∈ [0, 1],w ∈ b is given, ξ is a parameter and k is the green’s kernel given by k(s2,s1) = { (1 − s2)s1, s1 ≤ s2 s2(1 − s1), s2 ≤ s1. by (4.1), we have (f ′(x)(z))(s) = z(s) − 3ξ ∫ 1 0 k(s,t)x2(t)z(t)dt, t ∈ bs ∈ [0, 1]. consider x0(s) = w(s) = 1 and |ξ| < 83. we get ‖i −f ′(x0)‖ < 3 8 |ξ|, f ′(x0)−1 ∈ l(b1b), ‖f ′(x0)−1‖≤ 8 8 − 3|ξ| , η = |ξ| 8 − 3|ξ| , l0 = 12|ξ| 8 − 3|ξ| , k = 6d|ξ| 8−3|ξ|,k1 = l0 2 and k2 = k2 = k3. https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 15 example 4.3. let b = b1 = r3 and ω be as in the example 4.2. it is well known that the boundary value problem [16] ψ(0) = 0,ψ(1) = 1, ψ′′ = −ψ −τψ2 can be given as a hammerstein-like nonlinear integral equation ψ(s) = s + ∫ 1 0 k(s,t)(ψ3(t) + τψ2(t))dt where τ is a parameter. then, define f : ω −→ t2 by [f (x)](s) = x(s) − s − ∫ 1 0 k(s,t)(x3(t) + τx2(t))dt. choose x0(s) = s and ω = u(x0, r0). then, clearly u(x0, r0) ⊂ u(0, r0 + 1), since ‖x0‖ = 1. suppose 2τ < 5. then, by conditions (a) are satisfied for l0 = 2τ+3r0+68 , k = τ+6r0+3 4 , k1 = l0 2 and k2 = k2 = k3. and η = 1+τ 5−2τ . notice that l0 < k. the rest of the examples are given for the local convergence study of newton’s method. example 4.4. let b = b1 = r3, ω = u[0, 1] and x∗ = (0, 0, 0)tr. define mapping e on ω for λ = (λ1,λ2,λ3) tr as e(λ) = (eλ1 − 1, e − 1 2 λ22 + λ1,λ3) tr. then, conditions (h) hold provided that l0 = e − 1,l = e 1 l0 and m1 = e, since f ′(x∗)−1 = f ′(x∗) = diag{1, 1, 1,}. notice that l0 < l < m1, l1 = l0 2 , l4 = l, l2 = l3 = l 2 . rtr = 0.2453 < ra = 0.3827, r = 0.2124. hence, our radius of convergence is larger. example 4.5. let b = b1 and ω be as in example 4.2. define f on ω as f (ϕ1)(x) = ϕ1(x) − ∫ 1 0 xϕ1(j) 3dj. then, we obtain f ′(ϕ1(ψ1))(x) = ψ1(x) − 3 ∫ 1 0 xjϕ1(j) 2ψ1(j)dj for all ψ1 ∈ ω. so, we can choose l0 = 1.5,l = m1 = 3. l1 = l0 2 , l4 = l, l2 = l3 = l 2 . but then, we get again rtr = 0.2222 < ra = 0.3333, r = 0.2663. https://doi.org/10.28924/ada/ma.2.3 eur. j. math. anal. 10.28924/ada/ma.2.3 16 5. conclusion ostrowski’s method was revisited and its applicability was extended in both the semi-localand local convergence case. in particular, the benefits in the semi-local convergence case include:weaker sufficient convergence criteria (i.e. more starters x0 become available); 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(2019).[38] p.p. zabrejko, d.f. nguen, the majorant method in the theory of newton-kantorovich approximations and the ptákerror estimates, numer. funct. anal. optim. 9 (1987) 671-684. https://doi.org/10.1080/01630568708816254. https://doi.org/10.28924/ada/ma.2.3 https://doi.org/10.1016/j.amc.2013.05.078 https://doi.org/10.1016/j.amc.2013.05.078 https://doi.org/10.1007/s10910-018-0856-y https://doi.org/10.1007/s10910-018-0856-y https://doi.org/10.1016/j.cam.2013.11.019 https://doi.org/10.1007/bf01385696 https://doi.org/10.1016/j.jco.2008.05.006 https://doi.org/10.1016/j.jco.2009.05.001 https://doi.org/10.1016/j.amc.2003.12.025 https://doi.org/10.1007/s11075-012-9585-7 https://doi.org/10.1007/s11590-013-0617-6 https://doi.org/10.1007/bf01400355 https://doi.org/10.1080/01630568708816254 1. introduction 2. semi-local convergence 3. local convergence 4. numerical experiments 5. conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 23doi: 10.28924/ada/ma.3.23 strong continuity of composition semigroups on the generalized bloch spaces of the upper half plane k. a. wandera1, j. o. bonyo2,∗ , d. o. ambogo1 1department of pure and applied mathematics, maseno university, p.o. box 333-40105, maseno kenya kwandera1@gmail.com, ambogos@maseno.ac.ke 2department of mathematics, multimedia university of kenya, p.o. box 15653-00503, nairobi kenya jbonyo@mmu.ac.ke ∗correspondence author abstract. we investigate strong continuity of composition semigroups on the generalized blochspaces of the upper half plane. these composition semigroups are induced by automorphisms ofthe upper half plane as classified into three distinct groups in [3]. 1. introduction consider h(ω) as the fréchet space of analytic functions f : ω →c endowed with the topologyof uniform convergence on compact subsets of ω. a function f ∈h(d) is in the bloch space of theunit disc b(d) if ‖f‖b1(d) := sup z∈d (1 −|z|2)|f ′(z)| < ∞ and in the little bloch space of the unit disc b0(d) if lim |z|−→1 (1 −|z|2)|f ′(z)| = 0. for f ∈b(d), we define the norm on b(d) by ‖f‖b(d) := |f (0)| + ‖f‖b1(d), where ‖.‖b1(d) is a seminorm on b(d).bloch space of the upper half plane b(u) is a set of analytic functions f ∈h(u) such that ‖f‖b1(u) := sup ω∈u =(ω)|f ′(ω)| < ∞. for f ∈b(u), we define the norm on b(u) by ‖f‖b(u) := |f (i)| + ‖f‖b1(u), received: 5 jun 2023. key words and phrases. composition semigroup; analytic functions; self analytic maps; bloch spaces; unit disc; upperhalf plane; strong continuity; infinitesimal generator. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.23 https://orcid.org/0000-0002-6442-4211 where ‖.‖b1(u) is a seminorm on b(u).let α > 0 be a real number, we define the generalized bloch space of the unit disc, bα(d) asthe space of all functions f ∈h(d) such that ‖f‖bα1(d) := sup z∈d ( 1 −|z|2 )α |f ′(z)| < ∞. for f ∈bα(d), we define the norm on bα(d) by ‖f‖bα(d) := |f (0)| + ‖f‖bα1(d). (1) we also define the corresponding generalized little bloch space of the unit disc as the space of allfunctions f ∈h(d) for which lim |z|→1 ( 1 −|z|2 )α |f ′(z)| = 0, with the same norm given by (1). here, bα(d) and bα◦ (d) are both banach spaces with respectto the norm ‖.‖bα(d). the generalized little bloch space of the unit disc, bα◦ (d) is the closure ofthe set of polynomials in the norm topology of bα(d). for more details see [17, 18]. the space b(d) has been studied by many authors because of its intrinsic interest since its introduction[1, 4, 8, 10, 13, 14, 18]. in [17], the generalized bloch spaces of the open unit disc, bα(d) aredefined and proved to be banach spaces with respect to their norm. zhu [17] further establishedgeneralized little bloch spaces of the unit disc bα◦ (d), as closed, separable subspaces of bα(d).there is scanty literature on the properties of the generalized bloch spaces of the upper half plane bα(u), including whether they are banach spaces. composition semigroups on bloch spaces ofthe unit disc have been studied in literature, see for instance [2, 11, 12] and references therein. onstrong continuity of composition semigroups, siskakis [12] proved that no nontrivial compositionsemigroups are strongly continuous on the bloch space of the unit disc b(d). the correspondingstudy of composition semigroups defined on the bloch spaces of the upper half plane has not yetbeen exhausted. moreover, existing works on the half plane, see [7, 13], have neither exhausted theinvestigation of properties of these semigroups nor considered these generalizations. in this papertherefore, we investigate the properties of the generalized bloch spaces of the upper half plane asbanach spaces and extend the study of semigroups of composition operators to the setting of thegeneralized bloch spaces of the upper half plane. 2. preliminaries and definitions let c be the complex plane. the set d := {z ∈ c : |z| < 1} is called the open unit disc.on the other hand, the set u := {ω ∈ c : =(ω) > 0} denotes the upper half of the complexplane c, where =(ω) is the imaginary part of ω ∈ c. the function ψ(z)= i(1+z) 1−z is referred to asthe cayley transform and maps the unit disc d conformally onto the upper half-plane u, with theinverse ψ−1(ω) = ω−i ω+i mapping the upper half plane u, onto the unit disc, d. we refer to [16] for2 details. let α > 0 be a real number. a function f ∈h(u) belongs to the generalized bloch spaceof the upper half plane, bα(u) if ‖f‖bα1(u) := sup ω∈u =(ω)α |f ′(ω)| < ∞ with the norm given by ‖f‖bα(u) := |f (i)| + ‖f‖bα1(u).the corresponding generalized little bloch space of the upper half plane, bα0 (u)is defined as bα◦ (u) := {f ∈h(u) : lim =(ω)−→0 =(ω)α |f ′(ω)| = 0} having the same norm as bα(u). there is little literature on the properties of the generalized blochspaces of the upper half plane as banach spaces. let x be a banach space. a one-parameterfamily (tt)t≥0 is a semigroup of bounded linear operators on x, if(i) to = i (identity operator on x), and(ii) tt+s = tt ◦ts for every t,s,≥ 0 (semigroup property).a semigroup (tt)t≥0 of bounded linear operators on x is strongly continuous if lim t→0+ ‖ttx −x‖ = 0 for all x ∈ x. the infinitesimal generator denoted by γ of (tt)t≥0 is defined by γx := lim t→0+ ttx −x t = ∂ ∂t (ttx) ∣∣∣∣ t=0 for each x ∈ dom(γ), where dom(γ) denotes the domain of γ given by dom(γ) = {x ∈ x : lim t→0+ ttx −x t exists} . we define a group of bounded linear operators as (tt)t∈r = tt, t ≥ 0, t−t, t ≥ 0. if both (tt)t≥0 and (t−t)t≥0 are semigroups on x. for more details see [5,6,9]. suppose ϕ : ω → ωis a self analytic map. the composition operator induced by ϕ on h(ω) is defined as cϕ(f ) = f o ϕ, for all f ∈ h(ω). on the other hand, given t ≥ 0 we define a semigroup as a family (ϕt)t≥0 ofself analytic maps on ω satisfying the following properties (i) ϕ0(z) = z (identity map on ω).(ii) ϕt+s = ϕt ◦ϕs,∀t,s ≥ 0 (semigroup property).(iii) ϕt → ϕ0 uniformly on compact subsets of ω as t → 0.3 composition semigroup induced by ϕt on h(ω) is defined as cϕt (f ) = f o ϕt, for all f ∈h(ω). 3. generalized bloch spaces of the upper half plane in this section, we study properties of the generalized bloch spaces as banach spaces. we alsorelate functions in the generalized bloch space of the upper half plane u to their counterparts inthe unit disc d. following [17, 18], it’s well known that bα(d) and bα0 (d) are banach spaces withrespect to the norm ‖.‖bα(d). moreover the set of analytic polynomials c[z] := { ∞∑ n=0 an z n : z ∈c } is dense in bα0 (d). these results are not explicitly clear from the literature in the setting of theupper half plane u.in the following theorem, we establish the completeness of bα(u) with respect to the norm ‖.‖bα(u). theorem 3.1. bα(u) is a banach space with respect to the norm ‖.‖bα(u) proof. it’s clear that (bα(u),‖.‖bα(u)) is a normed space. now we prove that the space bα(u)is complete in ‖.‖bα(u). let (fk)k denote a cauchy sequence in bα(u). for � > 0, there exists n ∈ n such that ‖fk − fl‖bα(u) < �, ∀k, l > n. hence by the definition of the norm, we have forall ∀k, l > n, |fk(i) − fl(i)| + sup ω∈u =(ω)α |f ′k(ω) − f ′ l (ω)| < �, which means that |fk(i) − fl(i)| < � and (=(ω))α |f ′k(ω) − f ′l (ω)| < �, for ω ∈u. so, (fk(i))k∈n is cauchy in c. by the completeness of c, (fk(i))k converges to a limit,say u0. similarly, (f ′k(ω))k∈n is cauchy in c and therefore converges to a limit, say g.since |f ′k(ω) − f ′l (ω)| < �=(ω)α and f ′k(ω) → g uniformly on compact subsets of u, then g ∈h(u).now, take f such that f ′(ω) = g(ω)∀ω ∈u and f (i) = u0.thus, ∀� > 0, ∃n such that ∀k, l > n, =(ω)α |f ′k(ω) − f ′ l (ω)| < �, ∀ω ∈u. taking limits as l →∞, then ∀k > n, =(ω)α |f ′k(ω) − f ′(ω)| < �, ∀ω ∈u. it follows that ‖fk − f‖bα(u) = |fk(i) − f (i)| + sup ω∈u =(ω)α |f ′k(ω) − f ′(ω)| < � 4 and so ‖fk − f‖bα(u) → 0 as k →∞.now, it remains to show that f ∈bα(u). we have =(ω)α |f ′(ω)| = =(ω)α |f ′(ω) − f ′k(ω) + f ′ k(ω)| ≤ =(ω)α |f ′(ω) − f ′kω| + =(ω) α |f ′k(ω)| < � + =(ω)α |f ′k(ω)| < ∞ since (fk)k ⊂bα(u).now, taking supremum over all ω ∈u in the above equation, we have that sup ω∈u =(ω)α |f ′(ω)| < ∞ which implies that f ∈bα(u), as desired. � as an immediate consequence, we have corollary 3.2. b(u) is a banach space with respect to the norm ‖ . ‖b(u) proof. follows immediately by taking α = 1 in theorem 3.1. � under the norm ‖ . ‖bα(u), the space bα0 (u) also becomes a banach space as in the followingtheorem, theorem 3.3. bα0 (u) is a banach space with respect to the norm ‖ . ‖bα(u). proof. following theorem 3.1, we need to show that every sequence in bα0 (u) convergent in bα(u)has its limit in bα0 (u).let (fn) ⊂ bα0 (u) and g ∈ bα(u) be such that fn → g as n → ∞. we need to prove that g ∈ bα0 (u). since fn,g are holomorphic on compact subsets of u, and fn → g, we have f ′n → g′uniformly. now that fn ⊂bα0 (u), we have lim =(ω)→0 (=(ω))α |f ′n(ω)| = 0,∀n. (2) since limn→∞ f ′n = g′, we have lim =(ω)→0 (=(ω))α |g′(ω)| = lim =(ω)→0 (=(ω))α | lim n→∞ f ′n(ω)| which is equivalent to lim =(ω)→0 (=(ω))α |g′(ω)| = lim n→∞ ( lim =(ω)→0 (=(ω))α |f ′n(ω)| ) . following equation (2), we see that lim =(ω)→0 (=(ω))α |g′(ω)| = 0. so, g ∈bα0 (u), completing the proof. �5 as a consequence, we have the following, corollary 3.4. b0(u) is a banach space with respect to the norm ‖.‖b(u) proof. follows immediately by taking α = 1 in theorem 3.3. � in the next results, we generate a relationship between functions in the generalized bloch spaceof the upper half plane u and their counterparts in the unit disc d proposition 3.5. let f ∈ bα(u) and ψ be the cayley transform, then f ∈ bα(u) if and only if f ◦ψ ∈bα(d) proof. it suffices to prove that ‖f‖bα1(u) < ∞ if and only if ‖f ◦ψ‖bα1(d) < ∞. let f be a functionin bα(u). then by definition, ‖f‖bα1(u) = supω∈u=(ω) α|f ′(ω)| < ∞. now, by changing variables, let ω = ψ(z), where ψ is the cayley transform. then =(ω) = ω −ω 2i = ψ(z) −ψ(z) 2i . using ψ(z) = i(1+z) 1−z and ψ(z) = −i(1+z)1−z , we have =(ω) = i(1+z) 1−z − −i(1+z) 1−z 2i = i(1 + z)(1 −z) + i(1 + z)(1 −z) 2i(1 −z)(1 −z) = i(2 − 2zz) 2i(1 −z)(1 −z) = 1 −|z|2 |1 −z|2 . we get the absolute of ψ′(z) = 2i (1−z)2 as |ψ′(z)| = 2 |1 −z|2 . (3) now, by definition we have ‖f‖bα1(u) = sup z∈d ( 1 −|z|2 |1 −z|2 )α |f ′(ψ(z))|. from equation (3), we have |1 −z|2 = 2|ψ′(z)| , therefore ‖f‖bα1(u) = 1 2α sup z∈d (1 −|z|2)α|ψ′(z)|α|f ′(ψ(z))|. 6 since, (f ◦ψ)′(z) = f ′(ψ(z))ψ′(z), we have |ψ′(z)|α|f ′(ψ(z))| = |ψ′(z)(f ◦ψ)′(z)||ψ′(z)α−1| and hence ‖f‖bα1(u) = 1 2α sup z∈d (1 −|z|2)α|ψ′(z)(f ◦ψ)′(z)||ψ′(z)α−1| = 1 2α |ψ′(z)α−1|‖f ◦ψ‖bα1(d),which is finite if and only if ‖f ◦ψ‖bα1(d) is finite. this completes the proof. �an immediate consequence is the following, corollary 3.6. let f ∈b(u) and ψ be the cayley transform, then ‖f‖b1(u) = 1 2 ‖f ◦ψ‖b1(d) (4) in particular, a function f ∈b(u) if and only if f ◦ψ ∈b(d). proof. this follows immediately from proposition 3.5 by taking α = 1. � 4. composition semigroups on the generalized little bloch space of the upper half plane in [3], the non trivial automorphisms of the upper half plane u were classified according to thelocation of their fixed points into three distinct classes namely; scaling, translation and rotationgroups. in this section, we determine composition semigroups induced by these automorphismgroups of the upper half plane u, on the generalized bloch space of the upper half plane bα(u). wethen employ the theory of linear operators on banach spaces to investigate the semigroup propertiesof the induced composition semigroup. for any given semigroup ϕt, the induced operator semigroup cϕt is known to be strongly continuous on the little bloch space. on the other hand, no non trivialcomposition semigroup is strongly continuous on the bloch space, see [11]. therefore, we shalldetermine the composition semigroup induced by these automorphism groups on the generalizedlittle bloch space of the upper half plane, bα0 (u). further, we show that composition semigroupsinduced by scaling and translation groups are strongly continuous on bα0 (u). we also establishstrong continuity of composition semigroups induced by rotation group on bα0 (d). the infinitesimalgenerator is identified and its domain stated. 4.1. scaling group. the automorphisms of this group are of the form ϕt(z) = ktz, where z ∈ uand k,t ∈ r with k 6= 0. as noted in [3], the semigroup properties of the induced compositionoperators will differ significantly depending on whether 0 < k < 1 or k > 1. thus for 0 < k < 1,we consider without loss of generality, the analytic self maps ϕt : u−→u of the form ϕt(z) = e −tz, z ∈u. (5) the composition semigroup induced by equation (5) on bα0 (u) is given by cϕtf (z) = (f ◦ϕt) (z) = f ( e−tz ) 7 it can be easily proved that (cϕt )t∈r is a group on bα0 (u).in what follows, we prove that the composition semigroup given by equation (4.1) fails to be anisometry on bα0 (u). proposition 4.1. the operator cϕt fails to be an isometry on bα0 (u). proof. by the definition of the norm, we have for all f ∈bα0 (u) ‖cϕtf‖bα(u) = |cϕtf (i)| + sup ω∈u =(ω)α|(cϕtf ) ′ (ω)| = |f (e−ti)| + sup ω∈u =(ω)α|e−tf ′(e−tω)|. now by change of variables:let z = e−tω, then ω = etz, and =(ω) = et=(z). therefore, ‖cϕtf‖bα(u) = |f (e −ti)| + sup z∈u etα=(z)α|e−tf ′(z)| = |f (e−ti)| + e(α−1)t sup z∈u =(z)α|f ′(z)| 6= |f (i)| + sup z∈u =(z)α|f ′(z)| = ‖f‖bα(u), which completes the proof. � next, we prove that the operator cϕt given by (4.1) is strongly continuous on bα0 (u). theorem 4.2. (cϕt )t∈r is strongly continuous on b α 0 (u). proof. to prove strong continuity of (cϕt )t∈r, it suffices to show that ‖cϕtf −f‖bα(u) → 0 as t → 0.that is, |(cϕtf − f ) (i)|+‖cϕtf −f‖bα1(u) → 0 as t → 0. this is equivalent to |(cϕtf − f ) (i)|→ 0and ‖cϕtf − f‖bα1(u) → 0, as t → 0. for the former, we have |(cϕtf − f ) (i)| = |cϕtf (i) − f (i)| (6) = |f (ϕt(i)) − f (i)| = |f (e−ti) − f (i)|→ 0 as t → 0, as desired. we now prove that ‖cϕtf − f‖bα1(u) → 0 as t → 0. recall that ψ : d→u, ϕt : u→ uand ψ−1 : u → d. we can therefore have d ψ−→ u ϕt−→ u ψ−1−−→ d. now, let xt = ψ−1 ◦ϕt ◦ψ : d→d. if (ϕt)t≥0 is an automorphism of the upper half plane u, then (xt)t≥0 is an automorphismof the unit disc d. since xt = ψ−1 ◦ϕt ◦ψ, it follows that ‖cϕtf − f‖bα1(u) → 0 as t → 0 if andonly if ‖cxtf ∗ − f ∗‖bα(d) → 0 as t → 0 8 cayley transform is given by ψ(z) = i(1+z) 1−z . we therefore have ψ−1 ◦ϕ−t ◦ψ(z) = ψ−1 (ϕt (ψ(z))) . = ψ−1 ( ϕt ( i(1 + z) 1 −z )) = ψ−1 ( e−t ( i(1 + z) 1 −z )) . substituting ψ−1(z) = z−i z+i , we obtain ψ−1 ◦ϕ−t ◦ψ(z) = e−t( i(1+z) 1−z ) − i e−t( i(1+z) 1−z ) + i . simplifying the fraction, we have ψ−1 ◦ϕ−t ◦ψ(z) = z + e−tz − 1 + e−t −z + e−tz + 1 + e−t . now, by factorizing z and dividing both the numerator and denominator by (1 + e−t), we obtain ψ−1 ◦ϕ−t ◦ψ(z) = z − (1−e −t) (1+e−t) 1 − (1−e −t) 1+e−t z . let bt = 1−e−t1+e−t , and substitute to obtain ψ−1 ◦ϕ−t ◦ψ(z) = z −bt 1 −btz := xt(z). further, we apply density of polynomials in bα0 (d) to prove that for f ∗ ∈bα0 (d), we have ‖cxtf ∗− f ∗‖bα1(d) → 0 as t → 0.by the definition of the norm, we have lim t→0+ ‖cxtf ∗ − f ∗‖bα(d) = lim t→0+ |(cxtf ∗ − f ∗)(0)| + sup z∈d ( 1 −|z|2 )α |(cxtf ∗ − f ∗)′(z)|. let f ∗(z) = zn and z ∈d. we need to show that ‖(cxtf ∗ − f ∗)‖bα1(d) → 0, as t → 0.since cxtz n −zn = (xt(z))n −zn,n ≥ 1, differentiating (xt(z))n −zn with respect to z, we obtain (cxtf ∗ − f ∗)′(z) = n(xt(z))n−1x ′t(z) −nz n−1 = n[(xt(z))n−1x ′t(z) −z n−1]. substituting for xt(z) = z −bt 1 −btz9 and x ′t(z) = (1 −btz)1 − (z −bt)(−bt) (1 −btz)2 = (1 −b2t ) (1 −btz)2 , we obtain (cxtf ∗ − f ∗)′(z) = n [( z −bt 1 −btz )n−1 (1 −b2t ) (1 −btz)2 −zn−1 ] = n [ (z −bt)n−1(1 −b2t ) (1 −btz)n−1(1 −btz)2 −zn−1 ] = n [ (z −bt)n−1(1 −b2t ) −zn−1(1 −btz)n+1 (1 −btz)n+1 ] . it therefore follows that limt→0+ ‖cxtf ∗ − f ∗‖bα1(d) is equivalent to lim t→0+ ( (sup z∈d ( 1 −|z|2 )α ∣∣∣∣n[(z −bt)n−1(1 −b2t ) −zn−1(1 −btz)n+1(1 −btz)n+1 ]∣∣∣∣) . now, let bt → 0 as t → 0, we obtain lim t→0+ ‖cxtf ∗ − f ∗‖bα1(d) = sup z∈d (1 −|z|2)α ∣∣n[zn−1 −zn−1]∣∣ = 0. since limt→0+ ‖(cxtf ∗ − f ∗‖bα1(d) = 0, it follows that lim t→0+ ( ‖cϕtf − f‖bα1(u) ) = 0. therefore ‖cϕtf − f‖bα(u) = |ϕtf (i)) − f (i)| + ‖cϕtf − f‖bα1(u) → 0 as t → 0, as desired. � in the next proposition, we compute the infinitesimal generator and determine the domain of thecomposition semigroup in equation (4.1). proposition 4.3. the infinitesimal generator γ of (cϕt )t≥0 on b α 0 (u) is given by γf (z) = −zf ′(z) with the domain dom (γ) = {f ∈bα0 (u) : zf ′(z) ∈bα0 (u)}. proof. using the definition of the infinitesimal generator γ of (cϕt )t≥0, for f ∈bα0 (u) we have γf (z) = lim t→0+ cϕtf (z) − f (z) t = lim t→0+ f ( e−tz ) − f (z) t = ∂ ∂t f (e−tz) ∣∣∣∣ t=0 = −zf ′(z).10 this implies that γf (z) = −zf ′(z) and therefore dom(γ) ⊆{f ∈bα0 (u) : zf ′ ∈bα0 (u)}. to provereverse inclusion, we let f ∈bα0 (u) be such that zf ′ ∈bα0 (u). then for z ∈u, cϕtf (z) − f (z) t = 1 t ∫ t 0 ∂ ∂s (cϕsf (z))ds = 1 t ∫ t 0 (−e−szf ′(e−sz))ds = 1 t ∫ t 0 cϕsf (z)ds, where f (z) = −zf ′(z). since f (z) is a function in bα0 (u), it remains to show that the limit of f (z) exist in bα0 (u). thus lim t→0+ cϕsf (z) − f (z) t = lim t→0+ 1 t ∫ t 0 cϕsf (z)ds. by strong continuity of (cϕs )s≥0 we have 1 t ∫ t 0 ‖cϕsf −f‖ds → 0 as t → 0+. hence {f ∈bα0 (u) : zf ′ ∈bα0 (u)}⊆ dom(γ).this completes the proof. � 4.2. translation group. in this case the automorphisms are of the form ϕt(z) = z + kt, where z ∈u and k,t ∈r with k 6= 0. as noted in [3], we can consider the self analytic maps of u of theform ϕt(z) = z + t. (7)the composition semigroup induced by translation group on bα0 (u) is given by cϕtf (z) = f (z + t). (8) the proof of our results given in equation (8) as a group on bα0 (u) is basic, we therefore omit thedetails.we shall now prove that the composition semigroup in equation (8), fails to be an isometry on bα0 (u). proposition 4.4. the operator cϕt fails to be an isometry on bα0 (u). proof. by norm definition, we have ‖cϕtf‖bα(u) = |cϕtf (i)| + sup z∈u =(z)α|(cϕtf ) ′ (z)| = |f (i + t)| + sup z∈u =(z)α|f ′(z + t)|. now by change of variables: let z + t = ω then z = ω − t, and =(z) = =(ω). therefore, ‖cϕtf‖bα(u) = |f (i + t)| + sup ω∈u =(ω)α|f ′(ω)|. (9) 11 the right hand side of equation (9) is not equal to the norm ‖f‖bα(u) for any t > 0. this impliesthat (8) is not an isometry on bα0 (u). this completes the proof. � in the following results, we investigate the strong continuity of the composition semigroup inequation (8) on bα0 (u). proposition 4.5. the operator cϕt is strongly continuous on bα0 (u). proof. we need to show that ‖cϕtf − f‖bα(u) → 0 as t → 0. this approach is similar to (7). weomit the details. we compute the automorphism of the unit disc d, denoted by xt as follows xt(z) = ψ−1 (ϕt (ψ(z))) = ψ−1 ( ϕt ( i(1 + z) 1 −z )) = ψ−1 ( i(1 + z) 1 −z + t ) . since the inverse of cayley transform is given by ψ−1 = z−i z+i , we substitute to obtain xt = i(1+z) 1−z − t − i i(1+z) 1−z − t + i = i(1+z) 1−z − (t + i) i(1+z) 1−z + (i − t) . we simplify further by multiplying both the numerator and denominator by (1 −z) to obtain xt(z) = i(1 + z) + (t − i)(1 −z)) i(1 + z) + (t + i)(1 −z) = (2i − t)z − t (2i + t) − tz . by dividing both the numerator and denominator by 2i − t, we get xt = z + t 2i−t 2i+t 2i−t − t 2i−tz.letting kt = t2i−t and mt = 2i+t2i−t . we have xt = z + kt mt −ktz . next, we apply density of polynomials in bα0 (d) to prove that for f ∗ ∈bα0 (d), we have ‖cxtf ∗ − f ∗‖bα1(d) → 0 as t → 0. lim t→0+ ‖cxtf ∗ − f ∗‖bα1(d) = limt→0+ ( sup z∈d ( 1 −|z|2 )α |(cxtf ∗ − f ∗)′(z)|) . using density of polynomials in bα0 (d), let f ∗(z) = zn and z ∈d be such that cxtz n −zn = (xt(z))n −zn,n ≥ 1. (10)12 now, differentiating (xt(z))n −zn with respect to z, we get (cxtf ∗ − f ∗)′(z) = n(xt(z))n−1x ′t(z) −nz n−1 = n[(xt(z))n−1x ′t(z) −z n−1]. (11) we also differentiate xt = z+ktmt−ktz by quotient rule to obtain x ′t(z) = (mt −ktz)1 − (z + kt)(−kt) (mt −ktz)2 = mt + k 2 t (mt −ktz)2 . substituting for xt = z+ktmt+ktz and x ′t(z) = mt−k2t(mt−ktz)2 in equation (11) we have (cxtf ∗ − f ∗)′(z) = n[(xt(z))n−1x ′t(z) −z n−1] = n [ (z + kt) n−1(mt −ktz2) −zn−1(mt −ktz)n+1 (mt −ktz)n+1 ] . it therefore follows that as t → 0, we have ‖cxtf ∗ − f ∗‖bα(d) = (|(xt(0)) n − 0|) + (sup z∈d ( 1 −|z|2 )α ∣∣n[(xt(z))n−1x ′t(z) −zn−1]∣∣ = 0. therefore ‖cϕtf − f‖bα(u) = |ϕtf (i)) − f (i)| + ‖cϕtf − f‖bα1(u) → 0 as t → 0, as desired. thiscompletes the proof. � in the next theorem, we obtain the infinitesimal generator of the strongly continuous compositionsemigroup given in equation (8). theorem 4.6. the infinitesimal generator γ of (cϕt )t≥0 on b α 0 (u) is given by γf (z)=f ′(z) with the domain dom(γ) = {f ∈bα0 (u) : f ′(z) ∈bα0 (u)}. proof. using the definition of the infinitesimal generator γ, for f ∈bα0 (u), we have; γf (z) = lim t→0+ f (z + t) − f (z) t = ∂ ∂t f (z + t) ∣∣∣∣ t=0 = f ′(z). this means that dom(γ) ⊂{f ∈bα0 (u) : f ′(z) ∈bα0 (u)}.it remains to prove the reverse inclusion. let f ∈bα0 (u) be such that f ′(z) ∈bα0 (u).then for z ∈u, we have; cϕtf (z) − f (z) = ∫ t 0 ∂ ∂s f (z + s)ds = ∫ t 0 f ′(z)ds. 13 letting f (z) = f ′(z), we obtain cϕtf (z) − f (z) = ∫ t 0 f (z)ds. this implies that f (z) = f ′(z) is a function of bα0 (u). it remains to show that the limit of f (z)exists in bα0 (u). since cϕtf (z) − f (z) t = 1 t ∫ t 0 f (z)ds, we now take limits as t → 0+ and invoke strong continuity of (cϕs )s≥0 to obtain lim t→0+ 1 t ∫ t 0 ‖cϕsfds −f‖ = 0. hence dom(γ) ⊇{f ∈bα0 (u) : f ′(z) ∈bα0 (u)} which completes the proof. � 5. rotation group the induced composition semigroups for rotation group are defined on the analytic spaces of theunit disk. we shall therefore generate composition semigroups induced by rotation group on thegeneralized little bloch space of the disc. the results obtained can then be mapped onto the upperhalf plane by use of cayley transform. in this case, the self analytic maps of d are of the form ϕt(z) = e iktz. we consider the composition semigroup induced by the rotation group on bα0 (d)given by cϕtf (z) = (f ◦ϕt) (z) = f ( eitz ) , (12) for all f ∈bα0 (d).it can be easily shown that (cϕt )t≥0 and (cϕ−t)t≥0 are semigroups on bα0 (d) thus (cϕt )t∈rdefines a group on bα0 (d).moreover, this group is an isometry, as we prove in the next proposition. proposition 5.1. the operator cϕt given by (12) is an isometry on bα0 (d). proof. we shall prove that for each t ∈r, the group (cϕt )t∈r is an isometry on bα0 (d). it sufficesto prove that ‖cϕtf‖bα(d) = ‖f‖bα(d).it follows from the definition that ‖cϕtf‖bα(d) = |cϕtf (0)| + sup z∈d ( 1 −|z|2 )α |(cϕtf )′(z)| = |(eit)f (0)| + sup z∈d ( 1 −|z|2 )α |eitf ′(eitz)| = |f (0)| + sup z∈d ( 1 −|z|2 )α |f ′(eitz)|. 14 now, let ω = eitz so that z = e−itω. then; ‖cϕtf‖bα(d) = |f (0)| + sup ω∈d ( 1 −|e−itω|2 )α |f ′(ω)|) = |f (0)| + sup ω∈d (1 −|ω|2)α|f ′(ω)| = ‖f‖bα(d). � theorem 5.2. the operator cϕt given by (12) is strongly continuous on bα0 (d). proof. since polynomials are dense in bα0 (d), it suffices to show that (cϕt )t∈r is strongly contin-uous on bα0 (d) that is, for a polynomial (zn)n≥0 where z ∈d we obtain lim t→0+ ‖cϕtz n −zn‖bα(d) = 0. clearly, lim t→0+ ‖cϕtz n −zn‖bα(d) = lim t→0+ |cϕtf (0) − f (0)| + ( sup z∈d (1 −|z|2)α|(cϕtz n −zn)′|) ) . but cϕtz n −zn = (eint − 1)zn. so its derivative is given by (cϕtz n −zn)′ = n(eint − 1)zn−1, implying that lim t→0+ ‖cϕtz n −zn‖bα(d) = lim t→0+ |eitf (0) − f (0)| + ( sup z∈d (1 −|z|2)α|nzn−1||(eint − 1)|) ) . hence, lim t→0+ ‖cϕtz n −zn‖bα(d) = 0 as desired . � proposition 5.3. the infinitesimal generator γ of (cϕt ) is given by γf (z) = izf ′(z) with the domain dom(γ) = {f ∈bα0 (d) : zf ′(z) ∈bα0 (d)}. proof. we obtain the infinitesimal generator as follows γf (z) = lim t→0+ cϕt(z) − f (z) t = ∂ ∂t f (eitz) ∣∣∣∣ t=0 = izf ′(z). 15 it therefore follows that dom(γ) ⊆ {f ∈ bα0 (d)} : zf ′(z) ∈ bα0 (d)}. on the other hand, let f ∈ bα0 (d)} be such that zf ′(z) ∈ bα0 (d)}, then for z ∈ d we have by the fundamental theoremof calculus, cϕtf (z) − f (z) = ∫ t 0 ∂ ∂s (cϕsf (z))ds = ∫ t 0 ieiszf ′(eisz)ds = ∫ t 0 cϕsf (z)ds, where f (z) = izf ′(z) is a function in bα0 (d). thus lim t→0+ cϕtf − f t = lim t→0+ 1 t ∫ t 0 cϕsfds and strong continuity of (cϕs )s≥0 implies that ‖1t ∫ t0 cϕsfds−f‖≤ 1t ∫ t0 ‖cϕsf −f‖ds → 0+ as t → 0+. thus dom(γ) ⊇{f ∈bα0 (d) : zf ′(z) ∈bα0 (d)}, as desired. � references [1] r. f. allen and f. colonna, isometries and spectral of multiplication operators on the bloch space, bull. aust. math.soc, 79, (2009), 147-160. https://doi.org/10.48550/arxiv.0809.3278.[2] m. bagasa, weighted composition groups on the little bloch space, j. funct. spaces, 2020 (2020), 5480602. https://doi.org/10.1155/2020/5480602.[3] s. ballamoole, j.o. bonyo, t.l. miller, v. g. miller, cesaro-like operators on the hardy and bergman spaces of thehalf plane, 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https://doi.org/:10.4236/jhepgc.2018.43031 https://doi.org/10.1007/0-387-36619-9 https://doi.org/10.1007/0-387-36619-9 https://doi.org/10.48550/arxiv.1901.07780 https://doi.org/10.1216/rmjm/1181075473 https://doi.org/10.1007/978-1-4612-5561-1 https://doi.org/10.1016/j.amc.2011.10.004 https://doi.org/10.5186/aasfm.2013.3806 https://doi.org/10.1007/bf03322189 [13] s. stevic, a.k. sharma, composition operators between hardy and bloch-type spaces of the upper half-plane, bull.korean math. soc. 44 (2007), 475-482. https://doi.org/:10.4134/bkms.2007.44.3.475.[14] j. zang, x. fu, bloch type space on the upper half plane, bull. korean math. soc. 54 (2017), 1337-1346. https: //doi.org/10.4134/bkms.b160572.[15] y. cheng, t. zhang, y. jiang, composition operators and the bloch spaces, in: 2010 third international conferenceon information and computing, ieee, wuxi, tbd, china, 2010: pp. 297-300. https://doi.org/10.1109/icic. 2010.170.[16] k. zhu, operator theory in function spaces, marcel dekker, inc., new york and basel, 1990. https://doi.org/ 10.1090/surv/138.[17] k. zhu, bloch type spaces of analytic functions, rocky mountain j. math. 23 (1993), 1143-1177. https://www. jstor.org/stable/44237763.[18] k. zhu, spaces of holomorphic functions in the unit ball, springer-verlag, new york, 2006. https://doi.org/10. 1007/0-387-27539-8. 17 https://doi.org/:10.4134/bkms.2007.44.3.475 https://doi.org/10.4134/bkms.b160572 https://doi.org/10.4134/bkms.b160572 https://doi.org/10.1109/icic.2010.170 https://doi.org/10.1109/icic.2010.170 https://doi.org/10.1090/surv/138 https://doi.org/10.1090/surv/138 https://www.jstor.org/stable/44237763 https://www.jstor.org/stable/44237763 https://doi.org/10.1007/0-387-27539-8 https://doi.org/10.1007/0-387-27539-8 1. introduction 2. preliminaries and definitions 3. generalized bloch spaces of the upper half plane 4. composition semigroups on the generalized little bloch space of the upper half plane 4.1. scaling group 4.2. translation group 5. rotation group references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 19doi: 10.28924/ada/ma.3.19 efficient derivative-free class of seventh order method for non-differentiable equations ioannis k. argyros1,∗, samundra regmi2, jinny ann john3, jayakumar jayaraman3 1department of computing and mathematical sciences, cameron university, lawton, 73505, ok, usa iargyros@cameron.edu 2department of mathematics, university of houston, houston, 77204, tx, usa sregmi5@uh.edu 3department of mathematics, puducherry technological university, pondicherry 605014, india jinny3@pec.edu, jjayakumar@ptuniv.edu.in ∗correspondence: iargyros@cameron.edu abstract. many applications from a wide variety of disciplines in the natural sciences and also inengineering are reduced to solving of an equation or a system of equations in a correspondinglychosen abstract area. for most of these problems, the solutions are found iterative, because theiranalytic versions are difficult to find or impossible. this article encompasses efficient, derivatives-free,high-convergence iterative methods. convergence of two types: local and semi-local areas will beinvestigated under the conditions of the ϕ,ψ-continuity utilizing operators on the method. the newmethod can also be applied to other methods, using inverses of the linear operator or the matrix. 1. introduction in the area of applied science and technology, a great number of problems can be resolved byconverting them into nonlinear form equation g(x) = 0 (1) where g : b ⊂ u → u is differentiable as per fréchet, u denotes complete normed linear spaceand b is a non-empty, open and convex set.normally, the solutions to these non-linear equations can not be obtained in a closed-form.therefore, the most frequently used solving techniques are of iterative nature. newton’s methodis a well-known iterative method for handling non-linear equations. recently, with advances inscience and mathematics many new iterative methods of higher order have been discovered for thehandling of non-linear equations and are currently being used [1, 2, 4–8, 10–22]. the computationof derivatives of second and higher order is a great disadvantage for the iterative systems of higherorder and is not suitable for the practical application. because of the computation of g′′, the received: 3 may 2023. key words and phrases. steffensen-like methods; convergence; banach space; divided difference.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 2 cubically converging classical schemas are not appropriate with respect to the cost of calculations.we found that many such methods rely on taylor series extensions to prove convergence resultsand require the existence of derivative with at least an order of magnitude greater than that ofthe methodology [1, 2, 4, 10–19, 21, 22]. here we consider, for example, a three-step two-parameterfamily of derivative free methods with seventh-order of convergence for solving systems of nonlinearequations proposed in [18] and which may be expressed in the following formulation:for x0 ∈b and each n = 0, 1, 2, . . . wn = xn + ag(xn), sn = xn −ag(xn), an = [wn,sn; g], yn = xn −a−1n g(xn), zn = yn −a−1n g(yn), un = zn + bg(zn), vn = zn −bg(zn), qn = [un,vn; g], xn+1 = zn − (pi + a−1n qn(qi + a −1 n qn(ri + da −1 n qn)))a −1 n g(zn), (2) where a,b,p,q,r,d ∈ r, [·, ·; g] : b ×b → w (u), the space of bounded linear operators from u into u. the local convergence analysis of the method (2) is provided in [18] using the taylorseries expansion approach and conditions reaching the eighth derivative of the operator g. thesederivatives do not appear on the method (2). the convergence order is shown to be seven providedthat p = 17 4 ,q = −27 4 , r = 19 4 and d = −5 4 . the conditions on high order derivatives restrict theapplicability of the method (2) for solving equations where at least g(8) should exist. although,the method may converge. let us consider the toy example for b = [−1, 2] and g defined by g(t) = { t4 log t + 5t7 − 5t6, if t 6= 0 0, if t = 0 it follows by this definition that g(ξ) = g(1) = 0 but g(4) is not bounded on b. thus, the resultsin [18] cannot assure that limn→∞xn = ξ = 1. but, the method converges to 1.therefore, there is a need to weaken the conditions. in this article, we use only conditions onthe operators on the method (2). therefore, the method can be utilized to solve non-differentiableequations. furthermore, the results should also demonstrate the isolation of the solution and thebounds of error in advance. this is what is new and what motivates our article. this meansextending its applicability, taking advantage of weaker conditions for such methods. in addition,we are also discussing a more interesting case of semi-local convergence. it is obvious that theaforementioned goals can be easily achieved in a similar way for other iterative methods [1, 2, 4,10–17, 19, 21, 22]. furthermore, our bounds of error is more precise and our criteria for convergenceapply even if the assumptions referred to in the references above are infringed.the remainder of the article is organized as follows: analysis of local convergence is provided insection 2. majorizing sequences will be introduced and analyzed for the semi-local convergenceanalysis of 2 in section 3. results demonstrating isolation of the solution is discussed in section https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 3 4. numeric experiments that use convergence results from the previous sections are described insection 5. the concluding remarks of section 6 bring this article to an end. 2. convergence 1: local let m = [0, +∞). the following conditions are used: (c1) there exist continuous and non-decreasing functions (cnf) ϕ0 : m×m → m, δ1 : m → m, δ2 : m → m, a solution ξ ∈b of the equation g(x) = 0 and a linear operator p such thatfor each w = x + ag(x), s = x −ag(x) and p−1 ∈ w (u) ‖p−1([w,s; g] − p)‖≤ ϕ0(‖w −ξ‖,‖s −ξ‖), ‖w −ξ‖≤ δ1(‖x −ξ‖) and ‖s −ξ‖≤ δ2(‖x −ξ‖). (c2) the equation ϕ0(δ1(t),δ2(t)) − 1 = 0 has a smallest positive solution denoted by ρ0. let m0 = [0,ρ0) and b0 = b∩s(ξ,ρ0). (c3) there exist cnf ϕ : m0 ×m0 ×m0 → m, δ3 : m0 → m, δ4 : m0 → m, ϕ1 : m0 ×m0 × m0×m0 → m, ϕ2 : m0 → m such that for each x,z ∈b0, u = z + bg(z), v = z −bg(z), ‖u −ξ‖≤ δ3(‖z −ξ‖), ‖v −ξ‖≤ δ4(‖z −ξ‖), ‖p−1([w,s; g] − [x,ξ; g])‖≤ ϕ(‖x −ξ‖,‖w −ξ‖,‖s −ξ‖), ‖p−1([w,s; g] − [u,v; g])‖≤ ϕ1(‖w −ξ‖,‖s −ξ‖,‖u −ξ‖,‖v −ξ‖) and ‖p−1([z,ξ; g] − p)‖≤ ϕ2(‖z −ξ‖). (c4) the equations hi(t)−1 = 0, i = 1, 2, 3 have smallest solutions ri ∈ m0−{0}, respectivelywhere the functions hi : m0 → m are defined by h1(t) = ϕ(t,δ1(t),δ2(t)) 1 −ϕ0(δ1(t),δ2(t)) , h2(t) = ϕ(h1(t)t,δ1(t),δ2(t))h1(t) 1 −ϕ0(δ1(t),δ2(t)) �(t) = ϕ1(δ1(t),δ2(t),δ3(h2(t)t),δ4(h2(t)t) 1 −ϕ0(δ1(t),δ2(t)) , λ(t) = |p + q + r + d − 1| + |p + 2r + 3d|�(t) + |r + 3d|�(t)2 + |d|�(t)3, h3(t) = [ ϕ(h2(t)t,δ1(t),δ2(t)) 1 −ϕ0(δ1(t),δ2(t)) + λ(t)(1 + ϕ2(h2(t)t)) 1 −ϕ0(δ1(t),δ2(t)) ] h2(t). https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 4 set r = min{ri}. let m1 = [0, r). it follows by these definitions that for each t ∈ m1 0 ≤ ϕ0(δ1(t),δ2(t)) < 1, 0 ≤ �(t), 0 ≤ λ(t) and 0 ≤ hi(t) < 1. notice that for x0 ∈ s(ξ,r) −{ξ} the conditions (c1)-(c2) and (c4) imply ‖p−1([w0,s0; g] − p)‖ϕ0(‖w0 −ξ‖,‖s0 −ξ‖) ≤ ϕ0(δ1(r),δ2(r)) < 1. thus a−10 ∈ w (u) by the banach lemma on invertible operators [3, 9, 10] and the firstiterate y0 is well-defined by the first sub-step of the method (2). (c5) s[ξ,r] ⊂b.the motivation for the development of the functions hi follows in turn by the estimates ‖a−1n p‖≤ 1 1 −ϕ0(‖wn −ξ‖,‖sn −ξ‖) ≤ 1 1 −ϕ0(δ1(‖xn −ξ‖),δ2(‖xn −ξ‖)) , yn −ξ = a−1n (an − [xn,ξ; g])(xn −ξ), ‖yn −ξ‖≤ ϕ(‖xn −ξ‖,‖wn −ξ‖,‖sn −ξ‖)‖xn −ξ‖ 1 −ϕ0(δ1(‖xn −ξ‖),δ2(‖xn −ξ‖)) ≤ h1(‖xn −ξ‖)‖xn −ξ‖≤‖xn −ξ‖ < r. similarly, ‖zn −ξ‖≤ ϕ(‖yn −ξ‖,‖wn −ξ‖,‖sn −ξ‖)‖yn −ξ‖ 1 −ϕ0(δ1(‖xn −ξ‖),δ2(‖xn −ξ‖)) ≤ h2(‖xn −ξ‖)‖xn −ξ‖≤‖xn −ξ‖, xn+1 −ξ = zn −ξ−a−1n g(zn) − [(p + q + r + d − 1)i + (q + 2r + 3d)(a −1 n qn − i) + (r + 3d)(a−1n qn − i) 2 + d(a−1n qn − i) 3]a−1n g(zn) which can be shortened for dn = a −1 n (qn −an), tn = (p + q + r + d − 1)i + (p + 2r + 3d)dn + (r + 3d)d2n + dd 3 n. thus xn+1 −ξ = a−1n (an − [zn,ξ; g])(zn −ξ) −tna −1 n g(zn). https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 5 but, ‖dn‖≤‖a−1n p‖‖p −1(qn −an)‖ ≤ ϕ1(‖wn −ξ‖,‖sn −ξ‖,‖un −ξ‖,‖vn −ξ‖) 1 −ϕ0(‖wn −ξ‖,‖sn −ξ‖) = �n, ‖tn‖≤ |p + q + r + d − 1| + |p + 2r + 3d|�n + |r + 3d|�2n + |d|� 3 n = λn, leading to ‖xn+1 −ξ‖≤ [ ϕ(‖zn −ξ‖,‖wn −ξ‖,‖sn −ξ‖) 1 −ϕ0(δ1(‖xn −ξ‖),δ2(‖xn −ξ‖) + λn(1 + ϕ2(‖zn −ξ‖)) 1 −ϕ0(δ1(‖xn −ξ‖),δ2(‖xn −ξ‖)) ] ‖zn −ξ‖ ≤ h3(‖xn −ξ‖)‖xn −ξ‖ < ‖xn −ξ‖. hence, the iterates {xn},{yn},{zn}⊂ s(ξ,r) and there exists c = h3(‖x0−ξ‖) ∈ [0, 1) such that ‖xn+1 −ξ‖≤ c‖xn −ξ‖ < r, from which it follows that limn→∞xn = ξ.therefore, we achieve the following local convergence result for the method (2). theorem 2.1. under the assumptions (c1)-(c5), {xn} ⊂ s(ξ,r) and limn→+∞xn = ξ provided that x0 ∈ s(ξ,r) −{ξ}. remark 2.2. the functions δj , j = 1, 2, 3, 4 are left uncluttered in the theorem 2.1. a possible choice for the first function δ1 is motivated by the estimate w −ξ = x −ξ + af (x) = (i + a[x,ξ; f ])(x −ξ) = (i + app−1([x,ξ; g] − p + p))(x −ξ), = [(i + ap) + app−1([x,ξ; g] − p)](x −ξ), ‖w −ξ‖≤ [‖i + ap‖ + |a|‖p‖ϕ0(‖x −ξ‖)]‖x −ξ‖. thus, we can choose δ1(t) = [‖i + ap‖ + |a|‖p‖ϕ0(t)]t. similarly, we can choose δ2(t) = [‖i −ap‖ + |a|‖p‖ϕ0(t)]t, δ3(t) = [‖i + bp‖ + |b|‖p‖ϕ0(h2(t)t)]h2(t)t and δ4(t) = [‖i −bp‖ + |b|‖p‖ϕ0(h2(t)t)]h2(t)t. two possible choices for the linear operator p are: the differentiable option : p = g′(ξ) and the non-differentiable option : p = [x0,x−1; g]. other choices are possible [18]. https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 6 3. convergence 2: semi-local the role of ξ, “ϕ” is replaced by x0, “ψ” as follows. assume:(h1) there exist cnf ψ0 : m × m → m, x0 ∈ b, g1 : m → m,g2 : m → m and a linearoperator p such that for x ∈b w = x + ag(x), s = x −ag(x), ‖w −x0‖≤ g1(‖x −x0‖), ‖s −x0‖≤ g2(‖x −x0‖) ‖p−1([w,s; g] − p)‖≤ ψ0(‖w −x0‖,‖s −x0‖). (h2) the equation ψ0(g1(t),g2(t)) − 1 = 0 has a smallest positive solution denoted by ρ.let m2 = [0,ρ) and b1 = b∩s(x0,ρ).notice that ‖p−1([w0,s0; g] − p)‖≤ ψ(0, 0) < 1.thus, a−10 ∈ w (u) and the iterate y0 is well-defined by the first sub-step of the method(2). (h3) there exists cnf g3 : m2 → m, g4 : m2 → m, ψ1,ψ2 : m2 ×m2 ×m2 ×m2 → m suchthat for each x,y ∈b1 ‖u −x0‖≤ g3(‖z −x0‖, ‖v −x0‖≤ g4(‖z −x0‖) ‖p−1([y,x; g] − [w,s; g])‖≤ ψ1(‖x −x0‖,‖y −x0‖,‖w −x0‖,‖s −x0‖) and ‖p−1([w,s; g] − [u,v; g])‖≤ ψ2(‖w −x0‖,‖s −x0‖,‖u −x0‖,‖v −x0‖). define the real sequence {αn} for α0 = 0,β0 ≥‖a−10 g(x0)‖, and each n = 0, 1, 2, . . . by γn = βn + ψ1(αn,βn,g1(αn),g2(αn))(βn −αn) 1 −ψ0(g1(αn),g2(αn)) , �n,1 = ψ2(g1(αn),g2(αn),g3(γn),g4(γn)) 1 −ψ0(g1(αn),g2(αn)) , λn,1 = |p + q + r + d| + |p + 2r + 3d|�n,1 + |r + 3d|�2n,1 + |d|� 3 n,1, αn+1 = γn + ψ1(βn,γn,g1(αn),g2(αn))(γn −βn)λn,1 1 −ψ0(g1(αn),g2(αn) , δn+1 = ψ1(αn,αn+1,g1(αn),g2(αn))(αn+1 −αn) + (1 + ψ0(g1(αn),g2(αn))(αn+1 −βn) βn+1 = αn+1 + δn+1 1 −ψ0(g1(αn+1,g2(αn+1) . (3) a convergence set of conditions for the sequence {αn} is given for each n = 0, 1, 2, . . .. (h4) ψ0(g1(αn),g2(αn)) < 1 and αn ≤ α < ρ.it follows by this condition and (3) that 0 ≤ αn ≤ βn ≤ γn ≤ αn+1 and there exists α∗ ∈ [0,α] such that limn→∞αn = α∗. https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 7 and (h5) s[x0,α∗] ⊂b.as in the local case the motivation for the introduction of the sequence {αn} follows in turn to formthe estimates: zn −yn = −a−1n g(yn),but g(yn) = g(yn) −g(xn) −an(yn −xn) = ([yn,xn; g] −an)(yn −xn), so ‖zn −yn‖≤ ψ1(‖xn −x0‖,‖yn −x0‖,‖wn −x0‖,‖sn −x0‖)‖yn −xn‖ 1 −ψ0(‖wn −x0‖,‖sn −x0‖) ≤ γn −βn, ‖zn −x0‖≤‖zn −yn‖ + ‖yn −x0‖≤ γn −βn + βn −α0 = γn < a∗, xn+1 −zn = −tna−1n g(zn), ‖xn+1 −zn‖≤ λn,1ψ1(‖yn −x0‖,‖zn −x0‖,‖wn −x0‖,‖sn −x0‖)‖zn −yn‖ 1 −ψ0(‖wn −x0‖,‖sn −x0‖) ≤ αn+1 −γn, since tn,1 = (p + q + r + d)i + (q + 2r + 3d)dn + (r + 3d)d 2 n + dd 3 n, ‖dn‖≤ ψ2(‖wn −x0‖,‖sn −x0‖,‖un −x0‖,‖vn −x0‖) 1 −ψ0(‖wn −x0‖,‖sn −x0‖) , ‖tn,1‖≤ λn,1 and ‖xn+1 −x0‖≤‖xn+1 −zn‖ + ‖zn −x0‖≤ αn+1 −γn + γn −α0 = αn+1 < α ∗. also, g(xn+1) = g(xn+1) −g(xn) −an(yn −xn) = g(xn+1) −g(xn) −an(xn+1 −xn) + an(xn+1 −yn), ‖p−1g(xn+1)‖≤ ψ1(‖xn −x0‖,‖xn+1 −x0‖,‖wn −x0‖,‖sn −x0‖)‖xn+1 −xn‖ + (1 + ψ0(‖wn −x0‖,‖sn −x0‖))‖xn+1 −yn‖ = δ̄n+1 ≤ δn+1, ‖yn+1 −xn+1‖≤‖a−1n+1p‖‖p −1g(xn+1‖ ≤ δ̄n+1 1 −ψ0(‖wn+1 −x0‖,‖sn+1 −x0‖) ≤ βn+1 −αn+1 (4) https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 8and ‖yn+1 −x0‖≤‖yn+1 −xn+1‖ + ‖xn+1 −x0‖≤ βn+1 −αn+1 + αn+1 −α0 = βn+1 < α ∗.therefore, the sequence {xn} is complete in banach space u. hence, there exists ξ = limn→∞xnand by (4) g(ξ) = 0.then, we achieve the following semi-local convergence result for the method (2). theorem 3.1. under the conditions (h1)-(h5) the sequence {xn} converges to a solution ξ ∈ s[x0,a ∗] of the equation g(x) = 0. remark 3.2. a possible choice for the functions gj , j = 1, 2, 3, 4 follows as in the local case. we have in turn w −x0 = x −x0 + a(g(x) −g(x0) + g(x0)) = [(i + ap) + app−1([x,x0; g] − p)](x −x0) + ag(x0), lead to the choice g1(t) = [‖i + ap‖ + |a|‖p‖ψ3(t)]t + |a|‖g(x0)‖ provided that for some cnf ψ3 : m1 → m, x ∈b ‖p−1([x,x0; g] − p)‖≤ ψ3(‖x −x0‖). similarly, we define g2(t) = [‖i −ap‖ + |a|‖p‖ψ3(t)]t + |a|‖g(x0)‖, g3(t) = [‖i + bp‖ + |a|‖p‖ψ3(t)]t + |b|‖g(x0)‖, and g4(t) = [‖i −bp‖ + |b|‖p‖ψ3(t)]t + |b|‖g(x0)‖. the options for p are: p = g′(x0) or p = [x0,x−1; g]. other options exist [10]. 4. isolation of a solution we first present the uniqueness result for the local convergence case. proposition 4.1. there exists a solution v∗ ∈ s(ξ,ρ2) of the equation g(x) = 0 for some ρ2 > 0; the last condition in (c3) holds in the ball s(ξ,ρ2) and there exists ρ3 ≥ ρ2 such that ψ2(ρ3) < 1. (5) https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 9 set b3 = b∩s[ξ,ρ3]. then, ξ is the only solution of the equation g(x)=0 in the set b3. proof. let v∗ 6= ξ. then, the divided difference v = [ξ,v∗; g] is well-defined. using the lastcondition in (c3) and (5), we obtain in turn that ‖p−1(v − p)‖≤ ψ2(‖v∗ −ξ‖) ≤ ψ2(ρ3) < 1, so, v −1 ∈ w (u) and from the approximation v∗ −ξ = v −1(g(v∗) −g(ξ)) = v −1(0) = 0, we deduce v∗ = ξ. � proposition 4.2. assume: there exists a solution v∗ ∈ s(x0,ρ4) of the equation g(x) = 0 for some ρ4 > 0; the condition (h1) holds on the ball s(x0,ρ4) and there exist ρ5 ≥ ρ4 such that ϕ0(ρ4,ρ5) < 1. (6) set b4 = b∩s[x0,ρ5]. then, v∗ is the only solution of the equation g(x) = 0 in the set b4. proof. let z∗ ∈b4 with g(z∗) = 0 and z∗ 6= v∗. define the linear operator f = [v∗,z∗; g]. then,by the condition (h1) and (6) ‖p−1(f − p)‖≤ ϕ0(‖v∗ −x0‖,‖z∗ −x0‖) ≤ ϕ0(ρ4,ρ5) < 1, thus, again v∗ = z∗. � remark 4.3. (i) the limit point α∗ can be replaced by ρ in the condition (h5). (ii) under all the assumptions (h1)-(h5), let v∗ = ξ and ρ4 = α∗ in proposition 4.2. 5. experiments example 5.1. consider the system of differential equations with g′1(w1) = e w1, g′2(w2) = (e − 1)w2 + 1, g ′ 3(w3) = 1 subject to g1(0) = g2(0) = g3(0) = 0. let g = (g1,g2,g3). let u = r3 and b = u[0, 1]. then ξ = (0, 0, 0)t is a root. let function g on b for w = (w1,w2,w3)t be g(w) = (ew1 − 1, e − 1 2 w22 + w2,w3) t . this definition gives g′(w) =  ew1 0 0 0 (e − 1)w2 + 1 0 0 0 1  https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 10 thus, by the definition of g it follows that g′(ξ) = 1. let p = g′(ξ) and [x,y; g] = ∫1 0 g′(x + θ(y − x))dθ. then, for a = b = 1, the conditions (c1)-(c5) are validated by remark 2.2 provided that δ1(t) = (2 + 1 2 (e − 1)t)t, δ2(t) = 1 2 (e − 1)t2, ϕ0(t1,t2) = 1 2 (e − 1)(δ1(t1) + δ2(t2)) δ3(t) = (2 + 1 2 (e − 1)h2(t))h2(t)t, δ4(t) = 1 2 (e − 1)h2(t)2t2, ϕ(t1,t2,t3) = 1 2 (e − 1)(t1 + δ1(t2) + δ2(t3)) ϕ1(t1,t2,t3,t4) = 1 2 (e − 1)[δ1(t1) + δ2(t2) + δ3(t3) + δ4(t4)] and ϕ2(t) = 1 2 (e − 1)t. by solving, we get ρ0 = 0.426037 and hence m0 = [0,ρ0). the radii are obtained as r1 = 0.204146, r2 = 0.134409 and r3 = 0.126891. therefore, by the definition r = min{ri}, we get the radius of convergence, r = 0.126891. remark 5.2. a non-differentiable non-linear system is solved using the method (2), where the divided difference is defined by the 2×2 matrix given for t̄ = (t1,t2) ∈r×r, t̃ = (t3,t4) ∈r×r and g = (g1,g2) by [t̄, t̃; g]i,1 = gi(t3,t4) −gi(t1,t4) t3 − t1 , t3 6= t1 and [t̄, t̃; g]i,2 = gi(t1,t4) −gi(t1,t2) t4 − t2 , t4 6= t2. otherwise, we set [·, ·; g] = 0. the actual example is given below example 5.3. let us solve the non-linear and non-differentiable system given as 3t21t2 + t 2 2 − 1 + |t1 − 1| = 0 t41 + t1t 3 2 − 1 + |t2| = 0. https://doi.org/10.28924/ada/ma.3.19 eur. j. math. anal. 10.28924/ada/ma.3.19 11 then, we set g = (g1,g2), where g1(t1,t2) = 3t 2 1t2 + t 2 2 − 1 + |t1 − 1| g2(t1,t2) = t 4 1 + t1t 3 2 − 1 + |t2| choose the initial points (5, 5) and (1, 0). then, using the aforementioned divided difference and the method (2), we obtain the solution ξ = (x∗1,x ∗ 2) after three iterations with x∗1 = 0.894655074977661 and x∗2 = 0.327826643198819. example 5.4. we consider the system of 25 equations 25∑ j=1,j 6=i xj −e−xi = 0, 1 ≤ i ≤ 25, with initial point x0 = {1.5, 1.5, . . . , 1.5}t . then, applying method (2) we get the solution ξ = {0.04003162719010837 · · · , 0.04003162719010837 · · · , . . . , 0.04003162719010837 · · ·}t after 4 iterations. 6. conclusion a new procedure has been developed to demonstrate both local and semi-local convergenceanalysis of high-order convergence methods, using only derivatives that appear on the methodology.previous works have proven convergence based on the existence of high-order derivatives that maynot be present in the methodology. hence, it has been a limitation of their applicability. thisprocedure also offers error limits and uniqueness results that were not available before. moreover,this procedure is general in the sense that it is not dependent on the method itself. this is thereason why it may be used in the same way to broaden the scope of other methods of higher order,such as single and multi-step methods [1, 2, 4, 10–17, 19, 21, 22]. references [1] a. cordero, j. l. hueso, e. martínez, j. r. torregrosa, a modified newton-jarratt’s composition, numer. algorithms55 (2010) 87-99. https://doi.org/10.1007/s11075-009-9359-z.[2] a. m. ostrowski, solutions of equations and system of equations, academic press (1960) new york.[3] f. a. potra, v. ptak, nondiscrete induction and iterarive processes. pitman publishing (1984) boston.[4] h. ren, q. wu, w. bi, a class of two-step steffensen type methods with fourth-order convergence, appl. math.comput. 209 (2009) 206–210. https://doi.org/10.1016/j.amc.2008.12.039.[5] i. k. argyros, j. a. john, j. jayaraman, on the semi-local convergence of a sixth order method in banach space, j.numer. anal. approx. theory 51 (2022) 144–154. https://doi.org/10.33993/jnaat512-1284.[6] i. k. argyros, j. a. john, j. jayaraman, s. regmi, extended local convergence for the chebyshev method under themajorant condition, asian res. j. math. 18 (2022) 102–109. https://doi.org/10.9734/arjom/2022/v18i12629. https://doi.org/10.28924/ada/ma.3.19 https://doi.org/10.1007/s11075-009-9359-z https://doi.org/10.1016/j.amc.2008.12.039 https://doi.org/10.33993/jnaat512-1284 https://doi.org/10.9734/arjom/2022/v18i12629 eur. j. math. anal. 10.28924/ada/ma.3.19 12 [7] i. k. argyros, s. regmi, j. a. john, j. jayaraman, extended convergence for two sixth order methods under the sameweak conditions, foundations 3 (2023) 127–139. https://doi.org/10.3390/foundations3010012.[8] j. a. john, j. jayaraman, i. k. argyros, local convergence of an optimal method of order four for solving non-linearsystem, int. j. appl. comput. math. 8 (2022) 194. https://doi.org/10.1007/s40819-022-01404-3.[9] j. f. steffensen, remarks on iteration, scand. actuar. j. 16 (1933) 64–72. https://doi.org/10.1080/03461238. 1933.10419209.[10] j. f. traub, iterative methods for the solution of equations, prentice-hall, new jersey (1964).[11] j. m ortega, w. c rheinboldt, iterative solution of nonlinear equations in several variables, academic press, newyork, (1970). https://doi.org/10.1137/1.9780898719468.fm.[12] j. r. sharma, h. arora, an efficient derivative free iterative method for solving systems of nonlinear equations, appl.anal. discrete math. 7 (2013) 390–403. https://doi.org/10.2298/aadm130725016s.[13] j. r. sharma, h. arora, a novel derivative free algorithm with seventh order convergence for solving systems ofnonlinear equations, numer. algorithms 4 (2014) 917–933. https://doi.org/10.1007/s11075-014-9832-1.[14] j. r. sharma, h. arora, efficient derivative-free numerical methods for solving systems of nonlinear equations,comput. appl. math. 35 (2016) 269–284. https://doi.org/10.1007/s40314-014-0193-0.[15] j. r. sharma, p. gupta, efficient family of traub-steffensen-type methods for solving systems of nonlinear equations.adv. numer. anal. 2014 (2014) 152187. https://doi.org/10.1155/2014/152187.[16] m. grau-sánchez, à. grau, m. noguera, frozen divided difference scheme for solving systems of nonlinear equations,j. comput. appl. math. 235 (2011) 1739-1743. https://doi.org/10.1016/j.cam.2010.09.019.[17] m. grau-sánchez, m. noguera, s. amat, on the approximation of derivatives using divided difference operatorspreserving the local convergence order of iterative methods, j. comput. appl. math. 237 (2013) 363-372. https: //doi.org/10.1016/j.cam.2012.06.005.[18] m. narang, s. bhatia, v. kanwar, new efficient derivative free family of seventh-order methods for solving systemsof nonlinear equations, numer. algorithms 76 (2017) 283-307. https://doi.org/10.1007/s11075-016-0254-0.[19] q. zheng, p. zhao, f. huang, a family of fourth-order steffensen-type methods with the applications on solvingnonlinear odes, appl. math. comput. 217 (2011) 8196–8203. https://doi.org/10.1016/j.amc.2011.01.095.[20] s. regmi, i. k. argyros, j. a. john, j. jayaraman, extended convergence of two multi-step iterative methods, foun-dations 3 (2023) 140–153. https://doi.org/10.3390/foundations3010013.[21] x. wang, t. zhang, a family of steffensen type methods with seventh-order convergence, numer. algorithms 62(2013) 429–444. https://doi.org/10.1007/s11075-012-9597-3.[22] z. liu, q, zheng, p. zhao, a variant of steffensen’s method of fourth-order convergence and its applications, appl.math. comput. 216 (2010) 1978-1983. https://doi.org/10.1016/j.amc.2010.03.028. https://doi.org/10.28924/ada/ma.3.19 https://doi.org/10.3390/foundations3010012 https://doi.org/10.1007/s40819-022-01404-3 https://doi.org/10.1080/03461238.1933.10419209 https://doi.org/10.1080/03461238.1933.10419209 https://doi.org/10.1137/1.9780898719468.fm https://doi.org/10.2298/aadm130725016s https://doi.org/10.1007/s11075-014-9832-1 https://doi.org/10.1007/s40314-014-0193-0 https://doi.org/10.1155/2014/152187 https://doi.org/10.1016/j.cam.2010.09.019 https://doi.org/10.1016/j.cam.2012.06.005 https://doi.org/10.1016/j.cam.2012.06.005 https://doi.org/10.1007/s11075-016-0254-0 https://doi.org/10.1016/j.amc.2011.01.095 https://doi.org/10.3390/foundations3010013 https://doi.org/10.1007/s11075-012-9597-3 https://doi.org/10.1016/j.amc.2010.03.028 1. introduction 2. convergence 1: local 3. convergence 2: semi-local 4. isolation of a solution 5. experiments 6. conclusion references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 68-85doi: 10.28924/ada/ma.1.68 unified convergence analysis of two-step iterative methods for solving equations ioannis k. argyros department of mathematical sciences, cameron university, lawton, ok 73505, usa correspondence: iargyros@cameron.edu abstract. in this paper we consider unified convergence analysis of two-step iterative methods forsolving equations in the banach space setting. the convergence order four was shown using taylorexpansions requiring the existence of the fifth derivative not on this method. but these hypotheseslimit the utilization of it to functions which are at least five times differentiable although the methodmay converge. as far as we know no semi-local convergence has been given in this setting. ourgoal is to extend the applicability of this method in both the local and semi-local convergence caseand in the more general setting of banach space valued operators. moreover, we use our idea ofrecurrent functions and conditions only on the first derivative and divided differences which appearon the method. this idea can be used to extend other high convergence multipoint and multistepmethods. numerical experiments testing the convergence criteria complement this study. 1. introduction we consider the problem of approximating a solution x∗ of equation f (x) = 0, (1.1) where f : ω ⊂ b −→ b1 is a continuous operator acting between banach spaces b and b1 with ω 6= ∅. since a closed form solution is not possible in general, iterative methods are used forsolving (1.1). many iterative methods are studied for approximating x∗. in this paper, we considerthe iterative methods, defined for n = 0, 1, 2, . . . , by yn = xn −f ′(xn)−1f (xn) xn+1 = yn −anf ′(xn)−1f (yn), (1.2) an = a(xn,yn), a : ω×ω −→ l(b,b1), where a−1 ∈ l(b1,b). many methods are special casesof (1.2). for example: received: 31 aug 2021. key words and phrases. iterative methods; banach space; convergence criterion; continuous functions.68 https://adac.ee https://doi.org/10.28924/ada/ma.1.68 eur. j. math. anal. 1 (2021) 69 traub [35] yn = xn −f ′(xn)−1f (xn) xn+1 = yn −f ′(xn)−1f (yn), (1.3) newton [6] yn = xn −f ′(xn)−1f (xn) xn+1 = yn −f ′(yn)−1f (yn), (1.4) ostrowski [25] yn = xn −f ′(xn)−1f (xn) xn+1 = yn − (2[xn,yn; f ] −f ′(xn))−1f (yn), (1.5) kung-traub [35–37] yn = xn −f ′(xn)−1f (xn) xn+1 = yn − [xn,yn; f ]−1f ′(xn)[xn,yn; f ]−1f (yn), (1.6) ostrowski-type [25] yn = xn −f ′(xn)−1f (xn) xn+1 = yn − (2[xn,yn; f ]−1 −f ′(xn)−1)f (yn), (1.7) sharma type [32] yn = xn −f ′(xn)−1f (xn) xn+1 = yn −p (xn,yn)f ′(xn)−1f (yn). (1.8) to obtain all these special cases choose, an = i,an = f ′(yn)−1f ′(xn), an = (2[xn,yn; f ] − f ′(xn))f ′(xn), an = [xn,yn; f ] −1f ′(xn)[xn,yn; f ] −1f ′(xn), an = (2[xn,yn; f ] −1−f ′(xn)−1)f ′(xn), an = p (xn,yn), respectively, where [., .; f ] : ω × ω −→ l(b,b1) is a divided difference of orderone and p : ω×ω −→ l(b,b1) is weight operator [32] (see also [15,28,40] and reference therein).these special methods were shown to be of order four using taylor expansion and assumptions onthe fifth order derivative of f, which is not on these methods . so, the assumptions on the fifthderivative reduce the applicability of these methods [1–41].for example: let b = b1 = r, ω = [−0.5, 1.5]. define λ on ω by λ(t) = { t3 log t2 + t5 − t4 if t 6= 0 0 if t = 0. then, we get t∗ = 1, and λ′′′(t) = 6 log t2 + 60t2 − 24t + 22. eur. j. math. anal. 1 (2021) 70 obviously λ′′′(t) is not bounded on ω. so, the convergence of method (1.2) is not guaranteed bythe previous analyses in [1–41].in this paper we introduce a majorant sequence and use our idea of recurrent functions to extendthe applicability of method (1.2). our analysis includes error bounds and results on uniqueness of x∗ based on computable lipschitz constants not given before in [1–41] and in other similar studiesusing taylor series. our idea is very general. so, it applies on other methods too.the rest of the paper is set up as follows: in section 2 we present results on majorizing sequences.sections 3,4 contain the semi-local and local convergence, respectively, where in section 4 thenumerical experiments are presented. concluding remarks are given in the last section 5. 2. results on majorizing sequences we recall the definition followed by convergence results. definition 2.1. let {w̄n} be a sequence in a banach space. then, a nondecreasing scalar sequence {wn} is called majorizing for {w̄n} if ‖w̄n+1 − w̄n‖≤ wn+1 −wn for each n = 0, 1, 2, . . . . (2.1) sequence {wn} is used instead to study the convergence of {w̄n} [23–25]. set m = [0,∞).let η > 0, p0 : m −→ r, p : m −→ r, a : m × m × m −→ r, ā : m × m × m −→ rand b : m ×m ×m ×m −→ r be continuous and nondecreasing functions. set an = a(n) and ξn = b(n). define scalar sequences {sn}, {tn} for each n = 0, 1, 2, . . . by t0 = 0,s0 = η, tn+1 = sn + ᾱn(sn − tn) sn+1 = tn+1 + βn(tn+1 − sn), (2.2) where ᾱn = ān ∫ 10 p̄ ((1 −θ)(sn − tn))dθ and βn = ξn 1 −p0(tn+1) , ān = { ā, if n = 0 a, if n = 1, 2, . . . , p̄ = { p0, if n = 0 p, if n = 1, 2, . . .next, we present results on the convergence of sequence {sn},{tn}. lemma 2.2. suppose that there exists µ > 0 such that for each n = 0, 1, 2, . . . , tn ≤ µ (2.3) and p0(µ) < 1. (2.4) eur. j. math. anal. 1 (2021) 71 then, sequences {sn},{tn} converge to their unique least upper bound t∗ ∈ [η,µ] and tn ≤ sn ≤ tn+1. proof. it follows from (2.2)-(2.4) that these sequences are nondecreasing, bounded from aboveby µ, and as such they converge to t∗. � lemma 2.3. if function p0 is increasing then conditions (2.3) and (2.4) can be replaced by tn ≤ p−10 (1). (2.5) proof. set µ = p−10 (1) in lemma 2.2. � remark 2.4. conditions (2.3)-(2.5) are very general and can be verified only in special cases. that is why we present stronger conditions that are easier to verify. define functions f and g on the interval [0, 1) by f (t) = a( η 1 − t , η 1 − t ,t2η) ∫ 1 0 p ((1 −θ)t2η)dθ− t and g(t) = b( η 1 − t , η 1 − t ,t2η,t3η) + tp0( η 1 − t ) − t. suppose that these functions have minimal zeros λf and λg in (0, 1), respectively. set λ = min{λf ,λg} and λ0 = max{α0,β0}. then, we can show the third result on majorizing sequencefor method (1.2). lemma 2.5. suppose that µ0 ≤ λ0 ≤ λ. (2.6) then, sequences {sn},{tn} are nondecreasing, bounded from above by t∗∗ = η1−λ, and converge to t∗ ∈ [0,t∗∗]. moreover, the following estimates hold for each n = 1, 2, . . . 0 ≤ sn − tn ≤ λ(tn − sn−1) ≤ λ2nη, (2.7) 0 ≤ tn+1 − sn ≤ λ(sn − tn) ≤ λ2n+1η, (2.8) 0 ≤ sn ≤ 1 −λ2n+1 1 −λ η (2.9) and 0 ≤ tn+1 ≤ 1 −λ2n+1 1 −λ η. (2.10) eur. j. math. anal. 1 (2021) 72 proof. estimates (2.7)-(2.10) hold if 0 ≤ αm ≤ λ, (2.11) 0 ≤ βm ≤ λ, (2.12) and tm ≤ sm ≤ tm+1, (2.13) are true for m = 0, 1, 2, . . . . these estimates hold for m = 0 by (2.6). we suppose that (2.11)-(2.13) are true for m = 1, 2, . . .n. by induction hypotheses, (2.7) and (2.8), we have sm ≤ tm + λ2mη ≤ sm−1 + λ2m−1η + λ2mη ≤ η + λη + . . . + λ2mη = 1 −λ2m+1 1 −λ η < η 1 −λ = t∗∗, and tm+1 ≤ sm + λ2m+1η ≤ tm + λ2mη + λ2m+1η ≤ η + λη + . . . + λ2m+1η = 1 −λ2m+2 1 −λ η < η 1 −λ = t∗∗. therefore, by (2.13) and the induction hypotheses, we see that sequences {sm} and {tm} arenondecreasing. then, (2.11) shall be true if a(tm,sm,sm − tm) ∫ 1 0 ψ((1 −θ)(sm − tm))dθ ≤ λ or a( 1 −λ2m 1 −λ η, 1 −λ2m+1 1 −λ η,λ2mη) ∫ 1 0 p ((1 −θ)λ2mη)dθ ≤ λ or a( η 1 −λ , η 1 −λ ,λ2η) ∫ 1 0 ψ((1 −θ)λ2η)dθ ≤ λ or f (λ) ≤ 0, which is true by the definition of λf and λ. similarly, (2.12) shall be true if b( 1 −λ2m 1 −λ η, 1 −λ2m 1 −λ η,λ2mη,λ2m+1η) +λp0( 1 −λ2m+2 1 −λ η) ≤ λ, or b( η 1 −λ , η 1 −λ ,λ2η,λ3η) + λp0( η 1 −λ ) ≤ λ eur. j. math. anal. 1 (2021) 73 or g(λ) ≤ 0, which is also true by the definition of λg and λ. hence, we conclude (2.13) holds and limm−→∞ sm = limm−→∞ tm = t ∗. � 3. semi-local convergence let u(x0, r) = {x ∈ b : ‖x − x0‖ < r,r > 0} and u[x0, r] = {x ∈ b : ‖x − x0‖ ≤ r,r > 0}.we use some parameters and functions. consider m = [0,∞). suppose that there exists function p0 : m −→ m which is continuous and nondecreasing such that functions p0(t) − 1 = 0 has aminimal zero s ∈ (0,∞). set m0 = [0,s). suppose function p0 : m0 −→ m is continuous andnondecreasing. the following conditions (c) are needed: (c1) there exists x0 ∈ ω and η > 0 such that f ′(x0)−1 ∈ l(b1,b) and ‖f ′(x0)−1f (x0)‖≤ η. (c2) for each x ∈ ω ‖f ′(x0)−1(f ′(u) −f ′(x0))‖≤ p0(‖u −x0‖). set s0 = u(x0,s) ∩ ω.(c3) for each x,y ∈ s0 the following hold ‖f ′(x0)−1(f ′(y) −f ′(x))‖≤ p (‖y −x‖) (c4) for each n = 0, 1, 2, . . . ‖anf ′(xn)−1f ′(x0)‖≤ an f ′(x0) −1([y,x; f ] −f ′(x))‖≤ l2‖y −x‖ and ‖f ′(x0)−1hn‖≤ ξn, where hn = f ′(x0) −1 ∫ 1 0 (f ′(yn + θ(xn+1 −yn)) −f ′(xn)a−1n )dθ. (c5) conditions of lemma 2.2 or lemma 2.3 or lemma 2.5 hold.and(c6) u[x0,t∗] ⊂ ω.then, we can show the semi-local convergence of method (1.2) using the conditions (c) and thepreceding notation. eur. j. math. anal. 1 (2021) 74 theorem 3.1. under the conditions (c), sequences {yn},{xn} generated by method (1.2) are well defined in u[x0,t∗], remain in u[x0,t∗] for each n = 0, 1, 2, . . . and converge to a solution x∗ ∈ u[x0,t∗] of equation f (x) = 0. moreover, the following error estimates hold for each n = 0, 1, 2, . . . ‖x∗ −xn‖≤ t∗ − tn. proof. we shall show items (pm) ‖ym −xm‖≤ sm − tm(qm) ‖xm+1 −ym‖≤ tm+1 − smusing mathematical induction on integer m. by the first substep of method (1.2) for n = 0 and (c1),we have ‖y0 −x0‖ = ‖f ′(x0)−1f (x0)‖≤ η = s0 − t0 = s0 ≤ t∗,so y0 ∈ u[x0,t∗] and (p0) holds. we can write by the first sustep of method (1.2) that f (y0) = f (y0) −f (x0) −f ′(x0)(y0 −x0) = ∫ 1 0 (f ′(x0 + θ(y0 −x0)) −f ′(x0))(y0 −x0)dθ, leading by (c2) and (p0) to ‖f ′(x0)−1f (y0)‖ ≤ ∫ 1 0 p0(θ‖y0 −x0‖)dθ‖y0 −x0‖ ≤ ∫ 1 0 p̄ (θ(s0 − t0))dθ(s0 − t0). (3.1) let z ∈ u(x0,t∗). in view of (c2), we get ‖f ′(x0)−1(f ′(z) −f ′(x0))‖ ≤ p0(‖z −x0‖) ≤ p0(t∗) < 1, (3.2) so ‖f ′(z)−1f ′(x0)‖≤ 1 1 −p0(‖z −x0‖) (3.3) holds by a lemma on invertible linear operators due to banach [24] and (3.2). therefore, iterate x1is well defined and we can write in turn by (c3) and (3.3) (for z = x0,y0) ‖x1 −y0‖ = ‖a0f ′(x0)−1f (y0)‖ ≤ ‖a0f ′(x0)−1f ′(x0)‖‖ ∫ 1 0 f ′(x0) −1(f ′(x0 + θ(y0 −x0)) −f ′(x0))dθ(y0 −x0)‖ ≤ a0 ∫ 1 0 p̄ ((1 −θ)‖y0 −x0‖)dθ‖y0 −x0‖ 1 −p0(‖x0 −x0‖) ≤ a0 ∫ 1 0 p̄ ((1 −θ)(s0 − t0))dθ 1 −p0(0) (s0 − t0) = t1 − s0, (3.4) eur. j. math. anal. 1 (2021) 75 showing (q0). then, we have ‖x1 −x0‖≤‖x0 −y0‖ + ‖y0 −x0‖≤ t1 − s0 + s0 − t0 = t1 ≤ t∗, so x1 ∈ u[x0,t∗]. moreover, we can write f (x1) = f (x1) −f (y0) + f (y0) = f (x1) −f (y0) −f ′(x0)a−10 (x1 −y0) = ∫ 1 0 (f ′(y0 + θ(x1 −x0)) −f ′(x0)a−10 )dθ(x1 −y0) = h0(x1 −y0), (3.5) since by the second substep of method (1.2), we have f (y0) = −f ′(x0)a−10 (x1−y0). by (c3), (3.4)and (3.5), we obtain ‖f ′(x0)−1f (x1)‖ ≤ ‖f ′(x0)−1h0‖‖x1 −y0‖ ≤ ξ0(t1 − s0), (3.6) so ‖y1 −x1‖ ≤ ‖f ′(x1)−1f ′(x0)‖‖f ′(x0)−1f (x1)‖ ≤ ξ0(t1 − s0) 1 −p0(t1) = s1 − t1, (3.7) showing (p1) for m = 1. suppose (pm), (qm) hold ym and xm+1 ∈ u[x0,t∗]. then, by repeatingthese computations with xm,ym,xm+1 replacing x0,y0,x1, respectively, we complete the induction.moreover, sequence {xm} is complete in a banach space, so it converges to some x∗ ∈ u[x0,t∗].finally, by letting m −→∞ in the estimation ‖f ′(x0)−1f (xm+1)‖≤ ξm(tm+1 − sm) (3.8) and using the continuity of f, we conclude f (x∗) = 0. �next, we present a result for uniqueness of the solution x∗. proposition 3.2. suppose (a) x∗ is a solution of f (x) = 0 (b) there exists s̃ ≥ t∗ such that ∫ 1 0 p0((1 −θ)s̃ + θt∗)dθ < 1. (3.9) set s1 = u[x0, s̃] ∩ ω. then, the only solution of equation f (x) = 0 in the region s1 is x∗. eur. j. math. anal. 1 (2021) 76 proof. set t = ∫ 1 0 f ′(x̃ + θ(x∗ − x̃))dθ for some x̃ ∈ s1 with f (x̃) = 0. using (c2) and (3.9),we get ‖f ′(x0)−1(t −f ′(x0))‖ ≤ ∫ 1 0 p0(‖x̃ + θ(x∗ − x̃) −x0‖dθ ≤ ∫ 1 0 p0((1 −θ)‖x̃ −x0‖ + θ‖x∗ −x0‖)dθ ≤ ∫ 1 0 p0((1 −θ)s̃ + θt∗)dθ < 1, leading to x̃ = x∗, where we used the identity t (x∗ − x̃) = f (x∗) −f (x̃) = 0 − 0 = 0 and theinvertability of t. � remark 3.3. let us specialize operators an to see how sequences {sn},{tn},{an},{ξn},{αn} and {βn} are defined. choose the case of newton’s method (1.4). then, we have ‖anf ′(xn)−1f ′(x0)‖ = ‖f ′(yn)−1f ′(x0)‖ ≤ 1 1 −p0(‖yn −x0‖) , and ‖f ′(x0)−1hn‖ = ‖ ∫ 1 0 f ′(x0) −1(f ′(yn + θ(xn+1 −yn)) −f ′(xn)a−1n )dθ‖ = ‖ ∫ 1 0 f ′(x0) −1(f ′(yn + θ(xn+1 −yn)) −f ′(yn))dθ‖ ≤ ∫ 1 0 p̄ (θ‖xn+1 −yn‖)dθ ≤ ∫ 1 0 p̄ (θ(tn+1 − sn))dθ, so we can choose an = 1 1 −p0(sn) (3.10) and ξn = ∫ 1 0 p̄ (θ(tn+1 − sn))dθ. (3.11) in this case we can show another result on majorizing sequences which is weaker than lemma 2.5 for the interesting case p0(t) = l0t and p (t) = lt. we get in this special case that αn = l(sn − tn) 2(1 −l0sn) (3.12) and βn = l(tn+1 − sn) 2(1 −l0tn+1) . (3.13) eur. j. math. anal. 1 (2021) 77 define sequences of function {f (1)n },{f (2)n } on the interval [0, 1) by f (1) n (t) = l 2 t2n−1η + l0(1 + t + . . . + t 2n)η − 1, f (2) n (t) = l 2 t2nη + l0(1 + t + . . . + t 2n+1)η − 1, and polynomial ϕ by ϕ(t) = l0t 3 + (l0 + l 2 )t2 − l 2 . notice that ϕ(0) = −l 2 and ϕ(1) = 2l0. denote by ρ the smallest zero of polynomial ϕ in (0, 1)assured to exist by the intermediate value theorem. lemma 3.4. suppose that λ0 ≤ ρ < 1 −l0η. (3.14) then, the conclusions of lemma 2.5 hold for sequences {sn},{tn} with ρ replacing λ. proof. we must show this time 0 ≤ l(sm − tm) 2(1 −l0sm) ≤ ρ, (3.15) 0 ≤ l(tm+1 − sm) 2(1 −l0tm+1) ≤ ρ (3.16) and tm ≤ sm ≤ tm+1. (3.17)these estimates hold for m = 0 by (3.14) and the definition of these sequences. then, as in lemma2.5 we can show instead for (3.15) that l 2 ρ2mη + ρl0(1 + ρ + . . . + ρ 2m)η − 1 ≤ 0. (3.18) this estimate motivates us to define recurrent functions f (1)m by f (1) m (t) = l 2 t2m−1η + l0(1 + t + . . . + t 2m)η − 1. (3.19) we shall find a relationship between recurrent functions f (1)m+1 and f (1)m . by definition (3.9), we havein turn that f (1) m+1(t) = l 2 t2m+1η + l0(1 + t + . . . + t 2m+2)η − 1 − l 2 t2m−1η −l0(1 + t + . . . + t2m)η + 1 + f (1) m (t) = f (1) m (t) + ( l 2 t2 − l 2 + l0(t 2 + t3))t2m−1η = f (1) m (t) + p(t)t 2m−1η. (3.20) in particular, we have fm+1(ρ) = fm(ρ), (3.21) eur. j. math. anal. 1 (2021) 78 so evidently (3.8) holds if f (1) m (ρ) ≤ 0. (3.22)define f (1)∞ (t) = limm−→∞ f (1)m (t). then, we have f∞(t) = l0η 1 − t − 1. (3.23) then, (3.22) holds if f∞(ρ) ≤ 0, (3.24)which is true by (3.14). similarly, (3.16) holds if l 2 ρ2m+1η + ρl0(1 + ρ + . . . + ρ 2m+1)η −ρ ≤ 0 (3.25) or f (2) m (ρ) ≤ 0. (3.26)as in (3.20), we get in turn that f (2) m+1(t) = l 2 t2m+2η + l0(1 + t + . . . + t 2m+3)η − 1 − l 2 t2mη −l0(1 + t + . . . + t2m+1)η + 1 + f (2) m (t) = f (2) m (t) + ϕ(t)t 2mη. (3.27) define f (2)∞ (t) = lim(2)m−→∞(t). then, we get again f (2)∞ (t) = f (1) ∞ (t), so f (2)∞ (ρ) ≤ 0,can be shown instead of (3.26). but this is true by (3.14). the induction for items (3.15)-(3.17) iscompleted. the rest of the proof follows as in lemma 2.2. 4. local convergence we shall introduce real parameters and functions to be used in the convergence analysis. set m = [0,∞).suppose function(i) ψ0(t) − 1 = 0 has a smallest zero r0 ∈ m −{0}, where function ψ0 : m −→ m is continuousand nondecreasing. set m0 = [0,r0).(ii) ψ1(t)−1 = 0, has a smallest zero r1 ∈ m0−{0}, where function ψ : m0 −→ m is continuousand nondecreasing and ψ1 : m0 −→ m is defined by ψ1(t) = ∫ 1 0 ψ((1 −θ)t)dθ 1 −ψ0(t) . eur. j. math. anal. 1 (2021) 79 (iii) ψ0(ψ1(t)t)−1 has a smallest zero r̄1 ∈ m0−{0}. ser r̄2 = min{r0, r̄1} and m1 = [0, r̄2).(iv) ψ2(t) − 1 = 0 has a smallest zero r2 ∈ m1 −{0}, where ψ2(t) = [ψ1(ψ1(t)t) + (ψ0(t) + h(t,ψ1(t)t)) ∫ 1 0 ω(θψ1(t)t)dθ (1 −ψ0(t))(1 −ψ0(ψ1(t)t)) ]ψ1(t), where ω : m1 −→ m and h : m ×m1 −→ m are continuous and nondecreasing. we shall showthat r = min{r1,r2}, (4.1)is a convergence radius for method (1.2). set m2 = [0,r). these definitions, imply that for each t ∈ m2 0 ≤ ψ0(t) < 1, (4.2) 0 ≤ ψ0(ψ1(t)t) < 1, (4.3)and 0 ≤ ψi (t) < 1, i = 1, 2. (4.4)the conditions (h) shall be used provided that x∗ is a simple solution of equation f (x) = 0.suppose: (h1) for each x ∈ ω ‖f ′(x∗)−1(f ′(x) −f ′(x∗))‖≤ ψ0(‖x −x0‖). set ω0 = u(x∗,r0) ∩ ω.(h2) for each x,y ∈ ω0 ‖f ′(x∗)−1(f ′(y) −f ′(x))‖≤ ψ(‖y −x‖), ‖f ′(x∗)−1f ′(x)‖≤ ω(‖x −x∗‖), and ‖f ′(x∗)−1(f ′(x∗) −a(x,y))‖≤ h(‖x −x∗‖,‖y −x∗‖). (h3) u[x∗,r] ⊂ ω. next, we show the local convergence of method (1.2) based on the preceding notation and conditions(h).. theorem 4.1. under conditions (h) further suppose that x0 ∈ u(x∗,r) − {x∗}. then, we conclude limn−→∞xn = x∗. proof. let v ∈ u(x∗,r) −{x∗}. using (4.1), (4.2), and (h1) we obtain in turn that ‖f ′(x∗)−1(f ′(v) −f ′(x∗))‖≤ ψ0(‖v −x∗‖) ≤ ψ0(r) < 1, so ‖f ′(v)−1f ′(x∗)‖≤ 1 1 −ψ0(‖v −x∗‖) . (4.5) eur. j. math. anal. 1 (2021) 80 in particular, iterate is well defined for v = x0 and the first substep of method (1.2), from which wecan also write y0 −x∗ = x0 −x∗ −f ′(x0)−1f (x0) = (f ′(x0) −1f ′(x∗)) ×( ∫ 1 0 f ′(x∗)−1(f ′(x∗ + θ(x0 −x∗)) −f ′(x0))dθ(x0 −x∗). (4.6) by (4.1), (4.4) (for i = 1), (4.5) (for v = x0), (4.6) and (h2), we get in turn that ‖y0 −x∗‖ ≤ ∫ 1 0 ψ̄((1 −θ)‖x0 −x∗‖)dθ‖x0 −x∗‖ 1 −ψ0(‖x0 −x∗‖) ≤ ‖x0 −x∗‖ < r, (4.7) so y0 ∈ u(x∗,r). we also have that (4.5) holds for v = y0, and iterate x1 is well defined fromwhich we can write in turn that x1 −x∗ = y0 −x∗ −f ′(y0)−1f (x0) +(f ′(y0) −1 −a0f ′(x0)−1)f (y0) = y0 −x∗ −f ′(y0)−1f (y0) + f ′(y0)−1(f ′(x0) −a0)f ′(x0)1f (y0). (4.8) in view of (4.1), (4.4) (for i = 2), (4.5)(for v = x0,y0), (4.7), (4.8) and (h2), we obtain in turn ‖x1 −x∗‖ ≤ [ψ1(ψ1(‖x0 −x∗‖)) + (ψ0(‖x0 −x∗‖) + h(‖x0 −x∗‖,‖y0 −x∗‖)) ∫ 1 0 ω(θ‖y0 −x∗‖)dθ (1 −ψ0(‖y0 −x∗‖))(1 −ψ0(‖x0 −x∗‖)) ]‖y0 −x∗‖ ≤ ψ2(‖x0 −x∗‖)‖x0 −x∗‖≤‖x0 −x∗‖ < r, (4.9) so x1 ∈ u(x∗,r). simply, switch x0,y0,x1 by xm,ym,xm+1, respectively in the preceding calcula-tions to get ‖ym −x∗‖≤ ψ1(‖xm −x∗‖)‖xm −x∗‖≤‖xm −x∗‖ < r (4.10)and ‖xm+1 −x∗‖≤ ψ2(‖xm −x∗‖)‖xm −x∗‖≤‖xm −x∗‖. (4.11)then, by the estimation ‖xm+1 −x∗‖≤ d‖xm −x∗‖ < r, (4.12)where d = ψ2(‖x0 −x∗‖) ∈ [0, 1), we get limm−→∞xm = x∗ and xm+1 ∈ u(x∗,r). �next, we present a uniqueness result. eur. j. math. anal. 1 (2021) 81 proposition 4.2. suppose: (i) there exists a simple solution x∗ of equation f (x) = 0 (ii) there exists r∗ ≥ r such that ∫ 1 0 ψ0(θr ∗)dθ < 1. (4.13) set ω2 = ω ∩u[x∗,r∗]. then, the only solution of equation f (x) = 0 in the region ω2 is x∗. proof. consider x̃ ∈ ω1 with f (x̃) = 0. set t = ∫ 10 f ′(x∗ + θ(x̃−x∗))dθ. then, using (h1) and(4.13), we get in turn that ‖f ′(x∗)−1(t −f ′(x∗))‖ ≤ ∫ 1 0 ψ0(θ‖x̃ −x∗‖dθ ≤ ∫ 1 0 ψ0(θr ∗)dθ < 1, so x̃ = x∗, follows by t−1 ∈ l(b1,b) and t (x̃ −x∗) = f (x̃) −f (x∗) = 0 − 0 = 0. � 5. numerical experiments we provide some examples in this section. example 5.1. define function q(t) = ξ0t + ξ1 + ξ2 sin ξ3t, x0 = 0, where ξj, j = 0, 1, 2, 3 are parameters. choose p0(t) = l0t and p (t) = lt. notice that l0 and l are the center lipschitz and lipschitz constants, respectively. then, from the graph of q(t) clearly for ξ3 large and ξ2 small, l0l can be small (arbitrarily). notice that l0 l −→ 0. example 5.2. let b = b1 = c[0, 1] and ω = u[0, 1]. it is well known that the boundary value problem [16]. ς(0) = 0, (1) = 1, ς′′ = −ς −σς2 can be given as a hammerstein-like nonlinear integral equation ς(s) = s + ∫ 1 0 q(s,t)(ς3(t) + σς2(t))dt where σ is a parameter. then, define f : ω −→ b1 by [f (x)](s) = x(s) − s − ∫ 1 0 q(s,t)(x3(t) + σx2(t))dt. eur. j. math. anal. 1 (2021) 82 choose ς0(s) = s and ω = u(ς0,ρ0). then, clearly u(ς0,ρ0) ⊂ u(0,ρ0 + 1), since ‖ς0‖ = 1. suppose 2σ < 5. then, conditions (a) are satisfied for l0 = 2σ + 3ρ0 + 6 8 , l = σ + 6ρ0 + 3 4 , and η = 1+σ 5−2σ. notice that l0 < l. in the last two examples we consider traub’s method (1.3). so, we take a(x,y) = i and h(s,t) = 0. example 5.3. consider the motion system g′1(v1) = e v1, g′2(y) = (e − 1)v2 + 1, g ′ 3(v3) = 1 with g1(0) = g2(0) = g3(0) = 0. let g = (g1,g2,g3). let b = b1 = r3, ω = ū(0, 1),x∗ = (0, 0, 0)t . define function g on ω for v = (v1,v2,v3)t by g(v) = (ev1 − 1, e − 1 2 v22 + v2,v3) t . then, we get g′(v) =  ex 0 0 0 (e − 1)v2 + 1 0 0 0 1  , so ψ0(t) = (e−1)t, ψ(t) = e 1 e−1t, ω(t) = e 1 e−1 and k = e is the lipschitz constant on ω and ρt is given in [29, 35]. then, the radii: r1 = 0.3827 = ρa = 2 2(e − 1) + e 1 e−1 , r2 = 0.3061 = r, ρt = 2 3k = 0.2453. example 5.4. consider b = b1 = c[0, 1], ω = u(0, 1) and q : ω −→ b1 defined by q(ς)(x) = %(x) − 5 ∫ 1 0 xθς(θ)3dθ. (5.1) we obtain q′(ς(ξ))(x) = ξ(x) − 15 ∫ 1 0 xθς(θ)2ξ(θ)dθ, for each ξ ∈ d. then, since x∗ = 0, we set ψ0(t) = 7.5t, ψ(t) = 15t, ω(t) = 15 and k = 15. then, the radii: r1 = 0.0667 = ρa = 2 2(7.5) + 15 , r2 = 0.0290 = r, ρt = 2 3k = 0.0444. notice that in the last two examples ρa is the radius given by us in [1–7] and is the largest. eur. j. math. anal. 1 (2021) 83 6. conclusion we have provided sufficient convergence criterion for the semi-local and local convergence oftwo-step methods. upon specializing the parameters involved we show that although our majorizingsequence is more general than earlier ones: convergence criteria are weaker (i.e., the utility of themethods is extended); the upper error estimates are more accurate (i.e. at least as few iterates arerequired to achieve a predecided error tolerance) and we have an at least as large ball containingthe solution. these benefits are obtained without additional hypotheses. according to our newtechnique we locate a more accurate domain than before containing the iterates resulting to moreaccurate (at least as small) lipschitz condition.our theoretical results are further justified using numerical experiments. references [1] i.k. argyros, on the newton kantorovich hypothesis for solving equations, j. comput. math. 169 (2004), 315-332, https://doi.org/10.1016/j.cam.2004.01.029[2] i.k. argyros, computational theory of iterative methods. series: studies in computational mathematics, 15, editors:c.k. chui and l. wuytack, elsevier publ. co. new york, u.s.a, 2007.[3] i.k. argyros, convergence and applications of newton-type iterations, springer verlag, berlin, germany, (2008), https://doi.org/10.1007/978-0-387-72743-1.[4] i.k. argyros, s. hilout, weaker conditions for the convergence of newton’s method. j. complex. 28 (2012), 364–387, https://doi.org/10.1016/j.jco.2011.12.003.[5] i.k. argyros, s. hilout, on an improved convergence analysis of newton’s method, appl. math. comput. 225 (2013),372-386, https://doi.org/10.1016/j.amc.2013.09.049.[6] i.k. argyros, a.a. magréñan, iterative methods and their dynamics with applications, crc press, new york, usa,2017.[7] i.k. argyros, a.a. magréñan, a contemporary study of iterative methods, elsevier (academic press),new york, 2018, https://www.elsevier.com/books/a-contemporary-study-of-iterative-methods/ magrenan/978-0-12-809214-9.[8] r. behl, p. maroju, e. martinez, s. singh, a study of the local convergence of a fifth order iterative method, indianj. pure appl. math. 51 (2020), 439-455, https://doi.org/10.1007/s13226-020-0409-5.[9] e. cătinaş, the inexact, inexact perturbed, and quasi-newton methods are equivalent models, math. comp.74 (2005), 291-301, https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.96.1713&rep=rep1& type=pdf.[10] x. chen, t. yamamoto, convergence domains of certain iterative methods for solving nonlinear equations, numer.funct. anal. optim. 10 (1989), 37-48, https://doi.org/10.1080/01630568908816289.[11] j.e. dennis jr., on newton-like methods. numer. math. 11 (1968), 324–330, https://doi.org/10.1007/ bf02166685.[12] j.e. dennis jr., r.b. schnabel, numerical methods for unconstrained optimization and nonlinear equations, siam,philadelphia, 1996. first published by prentice-hall, englewood cliffs, new jersey, (1983), https://epubs. siam.org/doi/pdf/10.1137/1.9781611971200.fm.[13] p. deuflhard, g. heindl, affine invariant convergence theorems for newton’s method and extensions to relatedmethods. siam j. numer. anal. 16 (1979), 1-10, https://doi.org/10.1137/0716001. https://doi.org/10.1016/j.cam.2004.01.029 https://doi.org/10.1007/978-0-387-72743-1 https://doi.org/10.1016/j.jco.2011.12.003 https://doi.org/10.1016/j.amc.2013.09.049 https://www.elsevier.com/books/a-contemporary-study-of-iterative-methods/magrenan/978-0-12-809214-9 https://www.elsevier.com/books/a-contemporary-study-of-iterative-methods/magrenan/978-0-12-809214-9 https://doi.org/10.1007/s13226-020-0409-5 https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.96.1713&rep=rep1&type=pdf https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.96.1713&rep=rep1&type=pdf https://doi.org/10.1080/01630568908816289 https://doi.org/10.1007/bf02166685 https://doi.org/10.1007/bf02166685 https://epubs.siam.org/doi/pdf/10.1137/1.9781611971200.fm https://epubs.siam.org/doi/pdf/10.1137/1.9781611971200.fm https://doi.org/10.1137/0716001 eur. j. math. anal. 1 (2021) 84 [14] p. deuflhard, newton methods for nonlinear problems. affine invariance and adaptive algorithms, springer seriesin computational mathematics, 35, springer verlag, berlin. 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complexity and information, lezioni lince.[lincei lectures] cambridge university press,cambridge, 1998, xii+139 pp.[37] j.f. traub, wozniakowski, h, path integration on a quantum computer, quant. inf. process. 1 (2002), 356-388, https://arxiv.org/abs/quant-ph/0109113.[38] t. yamamoto, a convergence theorem for newton-like methods in banach spaces. numer. math. 51 (1987), 545-557, https://eudml.org/doc/133212.[39] r. verma, new trends in fractional programming, nova science publisher, new york, usa, (2019).[40] l. xu, y.-m. chu, s. rashid, a.a. el-deeb, k.s. nisar, on new unifed bounds for a family of functions via fractionalq-calculus theory, j. funct. space 2020 (2020), 4984612, https://doi.org/10.1155/2020/4984612.[41] p.p. zabrejko, d.f. nguen, the majorant method in the theory of newton-kantorovich approximations and the ptákerror estimates, numer. funct. anal. optim. 9 (1987), 671-684, https://doi.org/10.1080/01630568708816254. https://doi.org/10.1080/03461238.1933.10419209 https://doi.org/10.1080/03461238.1933.10419209 https://doi.org/10.1017/s0008439500028125 https://doi.org/10.1017/s0008439500028125 https://arxiv.org/abs/quant-ph/0109113 https://eudml.org/doc/133212 https://doi.org/10.1155/2020/4984612 https://doi.org/10.1080/01630568708816254 1. introduction 2. results on majorizing sequences 3. semi-local convergence 4. local convergence 5. numerical experiments 6. conclusion references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 133-150doi: 10.28924/ada/ma.1.133 lie group analysis of a nonlinear coupled system of korteweg-de vries equations joseph owuor owino∗, benard okelo department of pure and applied mathematics, jaramogi oginga odinga university of science and technology, box 210-40601, bondo, kenya bnyaare@yahoo.com, josephowuorowino@gmail.com ∗correspondence: josephowuorowino@gmail.com abstract. in this paper, we consider coupled korteweg-de vries equations that model the propagationof shallow water waves, ion-acoustic waves in plasmas, solitons, and nonlinear perturbations alonginternal surfaces between layers of different densities in stratified fluids, for example propagation ofsolitons of long internal waves in oceans. the method of lie group analysis is used to on the systemto obtain symmetry reductions. soliton solutions are constructed by use of a linear combination oftime and space translation symmetries. furthermore, we compute conservation laws in two waysthat is by multiplier method and by an application of new conservation theorem developed by nailibragimov. 1. introduction the dynamics of shallow-water waves, ion-acoustic waves in plasmas, and long internal waves inoceans can be described by coupled kdv equations. the equations are derived from the classicalkdv equation. this section extends the previous study of kdv equations to that of a couplednonlinear system. from the kortweg-de vries equation qt + αqqx + βqxxx = 0, (1) for α and β as constants, we let q(t,x) = u(t,x) + iv(t,x), (2) where i2 = −1. then substituting (2) into (1) and separating the real and imaginary parts, weobtain ∆1 ≡ ut + αuux −αvvx + βuxxx = 0, ∆2 ≡ vt + αuvx + αvux + βvxxx = 0, (3) received: 3 sep 2021. key words and phrases. coupled kdv equations; lie group analysis; group-invariant solutions; stationary solutions;symmetry reductions; soliton; multipliers; conservation laws.133 https://adac.ee https://doi.org/10.28924/ada/ma.1.133 eur. j. math. anal. 1 (2021) 134 which is a nonlinear system of coupled kdv equations. we perform lie symmetry analysis on (3),that is , we obtain lie point symmetries, invariant solutions and conservation laws of (3).this paperuses symmetry analysis method to construct exact solutions and conservation laws for a nonlinearcoupled kdv system (3). 2. preliminaries in this section, we outline preliminary concepts which are useful in the sequel. in euclideanspaces rn of x = x i independent variables and rm of u = uα dependent variables, we considerthe transformations t� : x̄ i = ϕi (x i,uα,�), ūα = ψα(x i,uα,�), (4) involving the continuous parameter � which ranges from a neighbourhood n ′ ⊂ n ⊂ r of � = 0where the functions ϕi and ψα differentiable and analytic in the parameter �. definition 2.1. the set g of transformations given by (4) is a local lie group if it holds true that (1) (i). (closure) given t�1,t�2 ∈ g, for �1,�2 ∈ n ′ ⊂ n , then t�1t�2 = t�3 ∈ g, �3 = φ(�1,�2) ∈n .(2) (ii). (identity) there exists a unique t0 ∈g if and only if � = 0 such that t�t0 = t0t� = t�.(3) (iii). (inverse) there exists a unique t�−1 ∈g for every transformation t� ∈g,where � ∈n ′ ⊂n and �−1 ∈n such that t�t�−1 = t�−1t� = t0. remark 2.2. associativity of the group g in (4) follows from (1). in the system, ∆α ( x i,uα,u(1), . . . ,u(π) ) = ∆α = 0, (5)the variables uα are dependent. the partial derivatives u(1) = {uαi },u(2) = {uαij}, . . . ,u(π) = {uαi1...iπ}, are of the first, second, . . . , up to the πth-orders.denoting di = ∂ ∂x i + uαi ∂ ∂uα + uαij ∂ ∂uα j + . . . , (6) the total differentiation operator with respect to the variables x i and δj i , the kronecker delta, wehave di (x j) = δ j i , ′, uαi = di (u α), uαij = dj(di (u α)), . . . , (7) where uαi defined in (7) are differential variables [7].consider the local lie group g given by the transformations x̄ i = ϕi (x i,uα,�), ϕi ∣∣∣ �=0 = x i, ūα = ψα(x i,uα,�), ψα ∣∣∣ �=0 = uα, (8) where the symbol ∣∣∣ �=0 means evaluated on � = 0. eur. j. math. anal. 1 (2021) 135 definition 2.3. the construction of the group g given by (8) is an equivalence of the computationof infinitesimal transformations x̄ i ≈ x i + ξi (x i,uα)�, ϕi ∣∣∣ �=0 = x i, ūα ≈ uα + ηα(x i,uα)�, ψα ∣∣∣ �=0 = uα, (9) obtained from (4) by a taylor series expansion of ϕi (x i,uα,�) and ψi (x i,uα,�) in � about � = 0and keeping only the terms linear in �, where ξi (x i,uα) = ∂ϕi (x i,uα,�) ∂� ∣∣∣ �=0 , ηα(x i,uα) = ∂ψα(x i,uα,�) ∂� ∣∣∣ �=0 . (10) remark 2.4. the symbol of infinitesimal transformations, x, is used to write (9) as x̄ i ≈ (1 + x)x i, ūα ≈ (1 + x)uα, (11) where x = ξi (x i,uα) ∂ ∂x i + ηα(x i,uα) ∂ ∂uα , (12) is the generator of the group g given by (8). remark 2.5. to obtain transformed derivatives from (4), we use a change of variable formulae di = di (ϕ j)d̄j, (13) where d̄j is the total differentiation in the variables x̄ i . this means that ūαi = d̄i (ū α), ūαij = d̄j(ū α i ) = d̄i (ū α j ). (14) if we apply the change of variable formula given in (13) on g given by (8), we get di (ψ α) = di (ϕ j), d̄j(ū α) = ūαj di (ϕ j). (15) expansion of (15) yields ( ∂ϕj ∂x i + u β i ∂ϕj ∂uβ ) ū β j = ∂ψα ∂x i + u β i ∂ψα ∂uβ . (16) the variables ūαi can be written as functions of x i,uα,u(1), that is ūαi = φ α(x i,uα,u(1),�), φ α ∣∣∣ �=0 = uαi . (17) definition 2.6. the transformations in the space of the variables x i,uα,u(1) given in (8) and (17)form the first prolongation group g[1]. definition 2.7. infinitesimal transformation of the first derivatives is ūαi ≈ u α i + ζ α i �, where ζαi = ζαi (x i,uα,u(1),�). (18) remark 2.8. in terms of infinitesimal transformations, the first prolongation group g[1] is given by(9) and (18). eur. j. math. anal. 1 (2021) 136 definition 2.9. by using the relation given in (15) on the first prolongation group g[1] given bydefinition 2.6, we obtain [5] di (x j + ξj�)(uαj + ζ α j �) = di (u α + ηα�), which gives uαi + ζαj � + uαj �diξj = uαi + diηα�,(19) and thus ζαi =di (η α) −uαj di (ξ j), (20) is the first prolongation formula. remark 2.10. similarly, we get higher order prolongations [8], ζαij = dj(ζ α i ) −u α iκdj(ξ κ), . . . , ζαi1,...,iκ = diκ(ζ α i1,...,iκ−1 ) −uαi1,i2,...,iκ−1jdiκ(ξ j). (21) remark 2.11. the prolonged generators of the prolongations g[1], . . . ,g[κ] of the group g are x[1] = x + ζαi ∂ ∂uα i , . . . ,x[κ] = x[κ−1] + ζαi1,...,iκ ∂ ∂ζα i1,...,iκ , κ ≥ 1, (22) where x is the group generator given by (12). definition 2.12. a function γ(x i,uα) is called an invariant of the group g of transformations givenby (4) if γ(x̄ i, ūα) = γ(x i,uα). (23) theorem 2.13. a function γ(x i,uα) is an invariant of the group g given by (4) if and only if it solves the following first-order linear pde: [5] xγ = ξi (x i,uα) ∂γ ∂x i + ηα(x i,uα) ∂γ ∂uα = 0. (24) from theorem (2.13), we have the following result. theorem 2.14. the local lie group g of transformations in rn given by (4) [7] has precisely n− 1 functionally independent invariants. one can take, as the basic invariants, the left-hand sides of the first integrals ψ1(x i,uα) = c1, . . . ,ψn−1(x i,uα) = cn−1, (25) of the characteristic equations for (24): dx i ξi (x i,uα) = duα ηα(x i,uα) . (26) eur. j. math. anal. 1 (2021) 137 definition 2.15. the vector field x (12) is a lie point symmetry of the pde system (5) if thedetermining equations x[π]∆α ∣∣∣ ∆α=0 = 0, α = 1, . . . ,m, π ≥ 1, (27) are satisfied, where ∣∣∣ ∆α=0 means evaluated on ∆α = 0 and x[π] is the π-th prolongation of x. definition 2.16. the lie group g is a symmetry group of the pde system given in (5) if the pdesystem (5) is form-invariant, that is ∆α ( x̄ i, ūα, ū(1), . . . , ū(π) ) = 0. (28) theorem 2.17. given the infinitesimal transformations in (8), the lie group g in (4) is found by integrating the lie equations dx̄ i d� = ξi (x̄ i, ūα), x̄ i ∣∣∣ �=0 = x i, dūα d� = ηα(x̄ i, ūα), ūα ∣∣∣ �=0 = uα. (29) definition 2.18. a vector space vr of operators [5] x (12) is a lie algebra if for any two operators, xi,xj ∈vr , their commutator [xi,xj] = xixj −xjxi, (30) is in vr for all i, j = 1, . . . , r . remark 2.19. the commutator satisfies the properties of bilinearity, skew symmetry and the jacobiidentity [5]. theorem 2.20. the set of solutions of the determining equation given by (27) forms a lie algebra [5]. the methods of (g’/g)-expansion method [20], extended jacobi elliptic function expansion [21]and kudryashov [22] are usually applied after symmetry reductions. let a system of πth-orderpdes be given by (5). definition 2.21. the euler-lagrange operator δ/δuα is δ δuα = ∂ ∂uα + ∑ κ≥1 (−1)κdi1, . . . ,diκ ∂ ∂uα i1i2...iκ , (31) and the liebäcklund operator in abbreviated form [5] is x = ξi ∂ ∂x i + ηα ∂ ∂uα + . . . . (32) remark 2.22. the liebäcklund operator (32) in its prolonged form is x = ξi ∂ ∂x i + ηα ∂ ∂uα + ∑ κ≥1 ζi1...iκ ∂ ∂uα i1i2...iκ , (33) eur. j. math. anal. 1 (2021) 138 where ζαi = di (w α) + ξjuαij , . . . ,ζ α i1...iκ = di1...iκ(w α) + ξjuαji1...iκ, j = 1, . . . ,n. (34) and the lie characteristic function is wα = ηα −ξjuαj . (35) remark 2.23. the characteristic form of liebäcklund operator (33) is x = ξidi + w α ∂ ∂uα + di1...iκ(w α) ∂ ∂uα i1i2...iκ . (36) remark 2.24. noether’s theorem is applicable to systems from variational problems definition 2.25. a function λα (x i,uα,u(1), . . .) = λα, is a multiplier of the pde system given by(5) if it satisfies the condition that [16] λα∆α = dit i, (37) where dit i is a divergence expression. definition 2.26. to find the multipliers λα, one solves the determining equations (38) [3], δ δuα (λα∆α) = 0. (38) the technique [9] enables one to construct conserved vectors associated with each lie pointsymmetry of the pde system given by (5). definition 2.27. the adjoint equations of the system given by (5) are ∆∗α ( x i,uα,vα, . . . ,u(π),v(π) ) ≡ δ δuα (vβ∆β) = 0, (39) where vα is the new dependent variable. definition 2.28. formal lagrangian l of the system (5) and its adjoint equations (39) is [9] l = vα∆α(x i,uα,u(1), . . . ,u(π)). (40) theorem 2.29. every infinitesimal symmetry xof the system given by (5) leads to conservation laws [9] dit i ∣∣∣ ∆α=0 = 0, (41) where the conserved vector t i = ξil + wα [ ∂l ∂uα i −dj ( ∂l ∂uα ij ) + djdk ( ∂l ∂uα ijk ) − . . . ] + dj(w α) [ ∂l ∂uα ij −dk ( ∂l ∂uα ijk ) + . . . ] + djdk(w α) [ ∂l ∂uα ijk − . . . ] . (42) eur. j. math. anal. 1 (2021) 139 3. main results we now present our results in this section. an illustrative example with a simple kdv equationcan be found in [6]. the infinitesimal transformations of the lie group with parameter � are t̄ = t + ξt(t,x,u,v)�, x̄ = x + ξx (t,x,u,v)�, ū = u + ηu(t,x,u,v)�, v̄ = v + ηv (t,x,u,v)�.(43) the vector field x = ξt(t,x,u,v) ∂ ∂t + ξx (t,x,u,v) ∂ ∂x + ηu(t,x,u,v) ∂ ∂u + ηv (t,x,u,v) ∂ ∂v , (44) is a lie point symmetry of (3) if x[3]∆1∣∣∣ ∆1=0, ∆2=0 = 0, x[3]∆2∣∣∣ ∆1=0, ∆2=0 = 0. (45) expanding (45) and and splitting on derivatives of v and u, we have an overdetermined system often pdes, namely, ξtu = 0, ξ t v = 0, ξ t x = 0, ξ x u = 0, ξ x v = 0, ξ t tt = 0, ξ x tt = 0, 3ξ x x −ξ t t = 0, 3ηv + 2ξttv = 0, 3αη u + 2αξttu − 3ξ x t = 0. (46) solving the system (46) yields ξt = a1 + 3a2t, ξ x = a2x + αa3t + a4, η u = −2a2u + a3, ηv = −2a2v, (47) for arbitrary constants a1,a2,a3,a4. hence from (47), the infinitesimal symmetries of the coupledkdv equations (3) is a lie algebra generated by the vector fields x1 = ∂ ∂t , x2 = ∂ ∂x , x3 = αt ∂ ∂x + ∂ ∂u , x4 = 3t ∂ ∂t + x ∂ ∂x − 2u ∂ ∂u − 2v ∂ ∂v . (48) the set of all infinitesimal symmetries of coupled kdv equations forms a lie algebra and yield thefollowing commutation relations in table 1. [xi,xj] x1 x2 x3 x4 x1 0 0 αx2 3x1 x2 0 0 0 x2 x3 -αx2 0 0 -2x3 x4 -3x1 -x2 2x3 0table 1: a commutator table for the lie algebra generated by the symmetries of coupled kdvequation. eur. j. math. anal. 1 (2021) 140 the following lie groups, for i = 1, 2, 3, 4, are obtained t�1 : t̄ = t + �1, x̄ = x, ū = u, v̄ = v, (49) t�2 : t̄ = t, x̄ = x + �2, ū = u, v̄ = v, (50) t�3 : t̄ = t, x̄ = x + α�3t, ū = u + �3, v̄ = v, (51) t�4 : t̄ = te 3�4, x̄ = xe�4, ū = ue−2�4, v̄ = ve−2�4. (52) the symmetries obtained yield the following symmetry reductions. x1 = ∂ ∂t . (53) solving the characteristic equations dt 1 = dx c = du 0 = dv 0 , (54) associated to the operator x1 gives the invariants j1 = x, j2 = u, j3 = v. (55) hence, we have u = ϕ(x), v = ψ(x), (56) for arbitrary functions ϕ and ψ. substituting the expressions for u and v given by (56) into thesystem (3), we get a system of third order ordinary des namely, α [ ϕ(x)ϕ′(x) −ψ(x)ψ′(x) ] + βϕ′′′(x) = 0, α (ϕ(x)ψ(x)) ′ + βψ′′′(x) = 0. (57) integration of the system (57) yields; α 2 [ ϕ(x)2 −ψ(x)2 ] + βϕ′′(x) = c1, (58) α [ϕ(x)ψ(x)] + βψ′′(x) = c2, (59) for arbitrary constants c1 and c2. if we take c1 = c2 = 0, (60) the system (58)-(59) becomes α 2 [ ϕ(x)2 −ψ(x)2 ] + βϕ′′(x) = 0, (61) α [ϕ(x)ψ(x)] + βψ′′(x) = 0. (62) eur. j. math. anal. 1 (2021) 141 to find more solutions of the system (61)-(62), we determine its lie point symmetries. using thelie’s algorithm for computing point symmetries, we see that the lie point symmetries of (61)-(62)are x∗1 = ∂ ∂x , x∗2 = x ∂ ∂x − 2ϕ ∂ ∂ϕ − 2ψ ∂ ∂ψ . (63) proceeding as above, we see that the symmetry x∗1 yields the trivial solution u = 0, v = 0. (64) the second symmetry x∗2 has the characteristic equations dx x = dϕ −2ϕ = dψ −2ψ , (65) which provides the invariants j1 = x 2ϕ, j2 = x 2ψ. (66) letting ϕ = λ x2 , ψ = µ x2 , (67) substituting the values of ϕ and ψ into (61)-(62) and solving the resulting equations yield: case one. taking µ = 0 (68) gives λ = 0 (69) or λ = − 12β α . (70) when λ = 0, and µ = 0, (71) we also get the trivial solution (64). one can easily see that if λ = − 12β α , and µ = 0, (72) then ϕ = − 12β αx2 , ψ = 0, (73) which is a solution of the system (61)-(62). hence u1(t,x) = − 12β αx2 , v1(t,x) = 0, (74) eur. j. math. anal. 1 (2021) 142 is a solution of the coupled kdv system (3). case two. taking λ = − 6β α (75) gives µ = ± 6βi α , (76) with i2 = −1. consequently, u2(t,x) = − 6β αx2 , v2(t,x) = 6iβ αx2 , (77) and u3(t,x) = − 6β αx2 , v3(t,x) = − 6iβ αx2 , (78) are solutions of the coupled kdv system . hence lie group analysis has given us three steady-statesolutions for the coupled kdv system under the time translation symmetry x1 = ∂∂t . x2 = ∂ ∂x . (79) solving the characteristic equations dt 0 = dx 1 = du 0 = dv 0 , (80) associated to x2 gives the invariants j1 = t, , j2 = u j3 = v. (81) therefore, the group-invariant solution is u = φ(t), v = h(t), (82) for arbitrary functions h and φ. substitution of the solutions from (82) into (3), we get a system offirst order ordinary des, namely, φ′(t) = 0, h′(t) = 0, (83) which is integrated once with respect to t to yield, φ(t) = c1, h(t) = c2, (84) for arbitrary constants c1 and c2. consequently, the space translation group-invariant solution ofthe system (3) is u(t,x) = c1, v(t,x) = c2. (85) x3 = αt ∂ ∂x + ∂ ∂u . (86) eur. j. math. anal. 1 (2021) 143 solving the characteristic equations dt 0 = dx αt = du 1 = dv 0 , (87) associated to galilean boost gives the invariants j1 = t, j2 = v, j3 = −u + x αt , t 6= 0. (88) thus the invariant solution of (3) is u = x αt −g(t), v = f (t), t 6= 0, (89) for arbitrary functions f and g. substitution of the values of u and v from (89) into the system (3),we get a nonlinear system of coupled first order ordinary des, namely, tg′(t) + g(t) = 0, tf ′(t) + f (t) = 0, (90) whose solutions are g(t) = c1 t f (t) = c2 t , (91) for arbitrary constants c1 and c2. hence the galilean boost group-invariant solution of the system(3) is u(t,x) = x + a αt , v(t,x) = c2 t (92) where a = −αc1 and t 6= 0. the scaling x4 = 3t ∂ ∂t + x ∂ ∂x − 2u ∂ ∂u − 2v ∂ ∂v (93) . by solving of the characteristic equations dt 3t = dx x = − du 2u = − dv 2v , (94) associated to this symmetry, we obtain the invariants j1 = x3 t , j2 = ux 2, j3 = vx 2. (95) generally, the group-invariant solution pair is u(t,x) = f (λ) x2 , v(t,x) = g(λ) x2 , where λ = x3 t , (96) and the functions f and g satisfy the system of third order nonlinear coupled ordinary des 2α(g2 − f 2) −λ2f ′ + 3αλ(f f ′ −gg′) + β(−24f + 24λf ′ + 27λ3f ′′′) =0, (97) −4αf g −λ2g′ + 3αλ(f g)′ + β(−24g + 24λg′ + 27λ3g′′′) =0. (98) x = x1 + cx2. (99) eur. j. math. anal. 1 (2021) 144 we consider a symmetry x, which is a linear combination of the time and space translationssymmetries, that is, x = ∂ ∂t + c ∂ ∂x , (100) for a constant c. the invariants associated to this symmetry x are j1 = x −ct, j2 = u, j3 = v. (101) hence, the invariant solution for the symmetry x is u = f (x −ct), v = g(x −ct), (102) for arbitrary functions f and g. substitution of u and v from (102) into the system (3) yields asystem of nonlinear third order ordinary des, namely −cf ′(ξ) + α { f (ξ)f ′(ξ) −g(ξ)g′(ξ) } + βf ′′′(ξ) = 0, −cg′(ξ) + α(f (ξ)g(ξ))′ + βg′′′(ξ) = 0,(103) which on integrating once with respect to ξ yields −cf + 1 2 α(f 2 −g2) + βf ′′ + c1 = 0, −cg + αf g + βg′′ + c2 = 0, (104) for arbitrary constants c1 and c2. remark 3.1. if we take the constants c1 = c2 = 0, then when the wave velocity c = 0, we canrecover the stationary solutions given in (3). remark 3.2. traveling wave solutions of the system (3) must satisfy the system (104). computation of conservation laws for the coupled kdv equations (3) is done using two meth-ods; the method of multipliers and a theorem due to ibragimov. we seek local conservation lawmultipliers for the system (3), whose determining equations are δ δu [ λ1∆1 + λ 2∆2 ] = 0, δ δv [ λ1∆1 + λ 2∆2 ] = 0, (105) where δ δu = ∂ ∂u −dt ∂ ∂ut −dx ∂ ∂ux + d2x ∂ ∂uxx −d3x ∂ ∂uxxx + . . . , (106) δ δv = ∂ ∂v −dt ∂ ∂vt −dx ∂ ∂vx + d2x ∂ ∂vxx −d3x ∂ ∂vxxx + · · · , (107) are the euler-lagrange operators and dt = ∂ ∂t + ut ∂ ∂u + vt ∂ ∂v + utx ∂ ∂ux + vtx ∂ ∂vx + utt ∂ ∂ut + vtt ∂ ∂vt + · · · , (108) dx = ∂ ∂x + ux ∂ ∂u + vx ∂ ∂v + uxx ∂ ∂ux + vxx ∂ ∂vx + utx ∂ ∂ut + vtx ∂ ∂vt + · · · , (109) eur. j. math. anal. 1 (2021) 145 are total derivatives operators. we look for second order multipliers, that is, λn = λn(t,x,u,ux,uxx,v,vx,vxx ), n = 1, 2. (110) the determining equations (105) become δ δu [ λ1{ut + αuux −αvvx + βuxxx} + λ2{vt + αuvx + αvux + βvxxx} ] = 0, (111) δ δv [ λ1{ut + αuux −αvvx + βuxxx} + λ2{vt + αuvx + αvux + βvxxx} ] = 0. (112) expanding (111)-(112) and splitting on derivatives of u and v yields an overdetermined system of22 pdes, namely λ1xx = 0, λ 2 xx = 0 λ 1 vx = 0, λ 2 vx = 0, λ 1 xvxx = 0, λ2xvxx = 0, βλ 1 vv −αλ 2 vxx = 0, βλ2vv + αλ 1 vvxx = 0, λ1vvxx = 0, λ 2 vvxx = 0, λ1vxxvxx = 0, λ 2 vxxvxx = 0, λ1u + λ 2 v = 0, λ1t + α ( λ2xv + λ 1 xu ) = 0, λ2t + α ( λ2xu − λ 1 xv ) = 0, λ2u − λ 1 v = 0, λ 1 ux = 0, λ2ux = 0, λ1uxx + λ 2 vxx = 0, λ2uxx − λ 1 vxx = 0, λ2vx = 0 λ 1 vx = 0.(113) calculations reveal the solution of the system (113) as λ1 = α 2β ( c3{u2 −v2} + 2c4uv ) + (c2t + c5)u + (c1t + c6)v + c3uxx + c4vxx + c7 − 1 α c2x, λ2 = α 2β ( c4{u2 −v2}− 2c3uv+ ) + (c1t + c6)u − (c2t + c5)v + c4uxx −c3vxx + c8 − 1 α c1x,(114) for arbitrary constants c1, . . . ,c8. remark 3.3. essentially, the nonlinear coupled system of kdv equations (3) has eight sets of localconservation law multipliers. solving (105)„ we obtain conserved vectors corresponding to each set of multipliers as shownbelow. (i) the multiplier ( λ11, λ 2 1 ) = ( tv,tu − x α ) , (115) has the conserved vectors tt1 = tuv − xv α , tx1 = β [ t{vuxx + uvxx −vxux} + 1 α {vx −xvxx} ] + α [ t ( u2v − v3 3 )] (116) −xuv. (117) (ii) the multiplier ( λ12, λ 2 2 ) = ( tu − x α ,−tv ) , (118) eur. j. math. anal. 1 (2021) 146 has the conserved vectors tt2 = t 2 {u2 −v2}− xu α , tx2 = β [ t ( uuxx −vvxx + 1 2 {v2x −u 2 x} ) + 1 α {ux −xuxx} ] + αt [ u3 3 −uv2 ] + x 2 {v2 −u2}. (119) (iii) the multiplier ( λ13, λ 2 3 ) = ( α 2β {u2 −v2} + uxx,−{ αuv β + vxx} ) , (120) has the conserved vectors tt3 = α 2β ( u3 3 −uv2 ) , tx3 = α 2 [ (u2 −v2)uxx −v2vxx ] −αuvvxx + (121) β 2 [ u2xx −v 2 xx ] + utux −vtvx + α2 4β [ 1 2 {u4 + v4}− 3u2v2 ] . (122) (iv) the multiplier ( λ14, λ 2 4 ) = ( { αuv β + vxx}, α[u2 −v2] 2β + uxx ) , (123) has the conserved vectors tt4 = α 2β ( u2v − v3 3 ) , (124) tx4 = α2 2β [ (u3v −uv3) ] + vtux + utvx + α 2 (u2 −v2)vxx + {αuv + βvxx}uxx. (125) (v) the multiplier ( λ15, λ 2 5 ) = (u,−v) , (126) has the conserved vectors tt5 = 1 2 {u2 −v2}, tx5 = β ( uuxx −vvxx + v2x −u2x 2 ) + α ( u3 3 −uv2 ) . (127) (vi) the multiplier ( λ16, λ 2 6 ) = (v,u) , (128) has the conserved vectors tt6 = uv, t x 6 = β (vuxx + uvxx −uxvx ) + α ( u2v − v3 3 ) . (129) (vii) the multiplier ( λ17, λ 2 7 ) = (1, 0) , (130) has the conserved vectors tt7 = u, t x 7 = α 2 {u2 −v2} + βuxx. (131) eur. j. math. anal. 1 (2021) 147 (viii) the multiplier has ( λ18, λ 2 8 ) = (0, 1) , (132) the conserved vectors tt8 = v, t x 8 = αuv + βvxx. (133) remark 3.4. it can be verified that dtt t i + dxt x i ∣∣∣ ∆1=0, ∆2=0 = 0, (134) for i = 1, . . . , 8. remark 3.5. the expressions in (134) are eight conservation laws for the coupled kdv system (3). remark 3.6. the presence of multipliers( λ17, λ 2 7 ) = (1, 0) , ( λ18, λ 2 8 ) = (0, 1) (135) manifest that the coupled kdv equations are themselves conservation laws. at this point, we derive conserved vectors for coupled kdv equations (3) by a new theorem due toibragimov. the adjoint equations for the nonlinear system coupled kdv equations (3) are ∆∗1 ≡ ft + α ufx + αvgx + βfxxx = 0, ∆ ∗ 2gt −αvfx + αugx + βgxxx = 0. (136) the formal lagrangian l for the nonlinear coupled system of the kdv equations (3) and its adjointequations (136) is given by l = f{ut + αuux −αvvx + βuxxx} + g{vt + αuvx + αvux + βvxxx}, (137) where f and g are new variables. we shall use the lie point symmetries of the system (3) ,namely x1 = ∂t, x2 = ∂x, x3 = αt∂x + ∂u, x4 = 3t∂t + x∂x − 2u∂u − 2v∂v, (138) to derive conserved vectors corresponding to each symmetry below. case (i) the symmetry x1 = ∂∂t , yields lie characteristic functions given by w 11 = −ut, w 2 1 = −vt. (139) hence by ibragimov’s theorem [9], the associated conserved vector is given by tt1 =α [f{uux −vvx} + g{vux + uvx}] + β{f uxxx + gvxxx}, tx1 =α [f{−uut + vvt}−g{vut + uvt}] + β{fxutx + gxvtx −utfxx −vtgxx − f utxx −gvtxx}. (140) eur. j. math. anal. 1 (2021) 148 case (ii) the symmetry x2 = ∂∂x , yields lie characteristic functions w 12 = −ux, w 2 2 = −vx. (141) therefore by ibragimov’s theorem [9], the associated conserved vector is tt2 = −uxf −vxg, t x 2 = f ut + gvt + β{−uxfxx −vxgxx + fxuxx + gxvxx}. (142) case (iii) the symmetry x3 = αt ∂ ∂x + ∂ ∂u (143) yields lie characteristic functions given by w 13 = 1 −αtux, w 2 3 = −αtvx. (144) hence by ibragimov’s theorem [9], the associated conserved vector is given by tt3 = f −αt{uxf + vxg} , tx3 = α [ f u + gv + t{utf + vtg} + βt{ fxx αt −uxfxx −vxgxx + fxuxx + gxvxx} ] . (145) case (iv) the symmetry x4 = 3t ∂ ∂t + x ∂ ∂x − 2u ∂ ∂u − 2v ∂ ∂v (146) yields the lie characteristic functions w 14 = −2u − 3tut −xux, w 2 4 = −2v − 3tvt −xvx. (147) consequently by ibragimov’s theorem [9], the corresponding conserved vector is given by tt4 = α [3t{f uux − f vvx + guvx + gvux}] + β [3t{f uxxx + gvxxx}] − 2{f u + gv}−x{f ux + gvx}, tx4 = x{f ut + gvt} + β [ 3 ( fxux + gxvx + t{fxutx + gxvtx} )] −α [ 2 ( f{u2 −v2} + 2guv ) + 3t ( f{uut −vvt} + g{vut + uvt} )] −β [x{uxfxx + vxgxx − fxuxx −gxvxx} + 2{ufxx + vgxx}] −β [3t{fxxut + gxxvt + f utxx + gvtxx} + 4{f uxx + gvxx}] . (148) remark 3.7. the appearance of arbitrary functions f (t,x) and g(t,x) in the conserved vectorsproves the existence of infinite conservation laws for coupled kdv system obtained by ibagimov’smethod. eur. j. math. anal. 1 (2021) 149 4. conclusion in this paper, lie group analysis was employed in studying a nonlinear coupled kdv system.a four-dimensional lie algebra of symmetries was found for the nonlinear coupled system kdvequations. this was spanned by space and time translations, galilean boost and scaling symmetrieswhere the scaling symmetry acts on four variables. associated to each symmetry, we obtainedsymmetry reductions that gave six nontrivial solutions for the coupled system. all the group-invariant solutions describe the various states of the system. the obtained solutions can be usedas a benchmark against numerical simulations. lastly, we constructed infinite conservation laws ofa nonlinear coupled kdv system by using multipliers and a theorem proposed by nail ibragimov. acknowledgement the first author acknowledges the financial support of aims-south africa and mastercard foun-dation. the authors are also grateful to the referees for their careful reading of the manuscript andvaluable comments. references [1] d. j. arigo, symmetry analysis of differential equations: an introduction, john wiley & sons, 2015.[2] g. bluman, s. anco, symmetry and integration methods for differential equations, springer science & businessmedia, 2008.[3] g. w. bluman, s. kumei, symmetries and differential equations, springer science & business media, 1989.[4] bluman, g. w., cheviakov, a. f., and anco, s. c, applications of symmetry methods to partial differential equations,springer, 2010.[5] n. h. ibragimov, elementary lie group analysis and ordinary differential equations, wiley, 1999.[6] j. owuor, m. khalique, lie group analysis of nonlinear partial differential equations, lambert academic publishers,2021[7] n. h. ibragimov, crc handbook of lie group analysis of differential equations, crc-press, 1994.[8] n. h. ibragimov, selected works, alga publications, blekinge institute of technology, selected works, 2009.[9] n. h. ibragimov, a new conservation theorem, j. math. anal. appl. 333(2007), 311-328. https://doi.org/10. 1016/j.jmaa.2006.10.078.[10] n. h. ibragimov, a practical course in differential equations and mathematical modelling: classical and newmethods. nonlinear mathematical models. symmetry and invariance principles, world scientific publishing com-pany, 2009.[11] c. m. khalique, s. a. abdallah, coupled burgers equations governing polydispersive sedimentation; a lie symmetryapproach. results phys. 16(2020), 76-90. https://doi.org/10.1016/j.rinp.2020.102967.[12] r. j. leveque, numerical methods for conservation laws, springer verlag, new york, 1992.[13] s lie, vorlesungen aber differentialgleichungen mit bekannten infinitesimalen transformationen. bg teubner, 1891.[14] i. mhlanga, c. khalique, travelling wave solutions and conservation laws of the korteweg-de vriesburgers equationwith power law nonlinearity. malays. j. math. sci. 11(2017), 1-8.[15] e. noether, invariant variations problem, nachr. konig. gissel. wissen, gottingen. math. phys. kl, 6(1918), 235-257.[16] p. j. olver, applications of lie groups to differential equations, springer science & business media, 1993. https://doi.org/10.1016/j.jmaa.2006.10.078 https://doi.org/10.1016/j.jmaa.2006.10.078 https://doi.org/10.1016/j.rinp.2020.102967 eur. j. math. anal. 1 (2021) 150 [17] l. ovsyannikov, lectures on the theory of group properties of differential equations, world scientific publishingcompany, 2013.[18] h. pie, symmetry methods for differential equations: a beginners guide, cambridge university pres, cambridge,2013.[19] a. m. wazwaz, partial differential equations and solitary waves theory, springer science & business media, 2010.[20] n. hasibun, l. abdullah, f. aini, the improved gg-expansion method to the (3 dimensional kadomstev-petviashviliequation, amer. j. appl. math. stat. 1(2013), 64-70.[21] b. hong, d. lu, f. sun, the extended jacobi elliptic functions expansion method and new exact solutions for thezakharov equations, world j. model. simul. 5(2009), 78-109.[22] s.m. ege, e. misirli, the modified kudryashov method for solving some fractional-order nonlinear equations, adv.difference equ. 2014 (2014), 135. https://doi.org/10.1186/1687-1847-2014-135. https://doi.org/10.1186/1687-1847-2014-135 1. introduction 2. preliminaries 3. main results 4. conclusion acknowledgement references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 164-181doi: 10.28924/ada/ma.1.164 nonlinear differential problem with p-laplacian and via phi-hilfer approach: solvability and stability analysis hamid beddani1,∗, moustafa beddani2, zoubir dahmani3 1laboratory of complex systems of the higher school of electrical and energy engineering of oran, 31000, algeria beddanihamid@gmail.com 2department of mathematics, university of sidi bel-abbès 22000, algeria beddani2004@yahoo.fr 3laboratory of pure and applied mathematics, abdelhamid bni badis university, 27000, algeria zzdahmani@yahoo.fr ∗correspondence: beddanihamid@gmail.com abstract. this paper we consider a study of a general class of nonlinear singular fractional des withp-laplacian for the existence and uniqueness solution and the hyers-ulam (hu) stability. result via ϕ−hilfer derivative is studied. then, an existence of one solution is investigated. some illustrativeexamples are discussed at the end. 1. introduction recently, fractional differential equations with boundary conditions are being studied by manyinterested people. this is because fractional differential equations describe many more real op-erations than classical differential equations. therefore, partial differential equations appearin many engineering and technological disciplines that include several sciences; see for exam-ple [1, 3–6, 8, 17, 18, 20, 22, 23, 31]. currently there are several different definitions of fractional integrals and derivatives, from themost famous of which are the riemann-liouville and caputo fractional derivatives to other less wellknown definitions. a generalization of the derivatives of both riemann-liouville and caputo wasgiven by r. hilfer in [11], known as the fractional hilfer derivative of order α and type β ∈ [0, 1].some properties and applications of the helfer derivative are given in [12, 13] and the referencesmentioned therein. prime value problems involving fractional hilfer derivatives have been studiedby several authors, see [9, 10, 26]. however, in the literature there are few papers on the boundary received: 8 oct 2021. key words and phrases. ϕ−hilfer derivative; existence of solution; fixed point; hyers-ulam stability.164 https://adac.ee https://doi.org/10.28924/ada/ma.1.164 eur. j. math. anal. 1 (2021) 165 value problems of the fractional hilfer derivatives. the authors set out in [2] non-local value prob-lems for derivatives of helfer’s fractions. for some recent work on boundary value problems withfractional hilfer derivatives, we refer to the papers in [28–30].some authors have worked on the eu of solutions for fractional des with p−laplacian operator.we cite, for example; li., wang., khan et al. [15, 19, 27] studieds a nonlinear fractional de withp-laplacian operator for the eu of solutions. h. khan, t. abdeljawad, m. aslam, r. a. khan and a. khan [16]. worked on the followingproposal for the existence of a positive solution (eps) and stability analysis:  dr1ψp [dr2 (u(t) −v1(t,u(t)))] = −a(t)v2(t,u(t −τ)), ψp [dr2 (u(t) −v1(t,u(t)))]|t=0 = ψp [ dr2 (u(t) −v1(t,u(t)))′ ]∣∣ t=0 = 0, u(0) = u(1) = 0,[ i2−r2 (u(t) −v1(t,u(t))) ]∣∣ t=0 = 0, where 0 < r1 < 1 < r2 < 2, and v1,v2 are continuous but singular at some points. thefractional derivatives dr1 and dr2 are taken in the caputo sense and in the riemann–liouvillesense, respectively, and ψp(z) = |z|p−2 z denotes the p−laplacian operator and satisfies 1p + 1q = 1, (ψp) −1 = ψq.a. devi, a. kumar, d. baleanu and a. khan [7]. worked on the eu and hu stability results, fornonliner fdes involving caputo fractional derivatives of distinct orders with ψp laplacian operator:  cdr1ψp [ cdr2 ( u(t) − ∑m i=1 vi (t) )] = −w(t,u(t)),t ∈ (0, 1] ψp [ cdr2 ( u(t) − ∑m i=1 vi (t) )]∣∣ t=0 = 0, u(0) = ∑m i=1 vi (0), u′(1) = ∑m i=1 v ′ i (1), uj(0) = ∑m i=1 v j i (0), for j = 2, 3, ...,n− 1, where 0 < r1 ≤ 1,n − 1 < r2 ≤ n,n ≥ 4, and vi,w are continuous functions. cdr1 and cdr2 denotes the derivative of fractional order r1 and r2 in caputo’s sense, respectively, and ψp(z) = |z|p−2 z denotes the p−laplacian operator and satisfies 1p + 1q = 1, (ψp)−1 = ψq. in the present research work, we study the existence and uniqueness of a solution (eps) andstability analysis which includes the ϕ−hilfer fractional-order of the form: eur. j. math. anal. 1 (2021) 166  hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t) = h(t,u(t),rldµ;ϕ a+ u(t)), t ∈ j = (a,b] u(a) = 0,u(b) = n∑ i=1 λiu (ζi ) , ψp ( hdα2,β2;ϕ a+ u ) (a) = 0,and ψp (hdα2,β2;ϕa+ u(b)) = iρ;ϕa+ u (ζ) , a < ζ,ζi < b, (1.1) here, we take hdα1,β;ϕ 0+ ,h dα2,β;ϕ 0+ , are the ϕ−hilfer fractional derivative of orders α1,α2, 1 < α1,α2 < 2 and β1,β2 two parameters 0 ≤ β1,β2 ≤ 1, rldκ;ϕa+ the ϕ-riemann-liouville fractionalderivative of order µ where µ < α2, and iρ;ϕ0+ the left-sided ϕ−riemann liouville fractional integralof order ρ, where ρ > 0, and ψp(z) = |z|p−2 z denotes the p−laplacian operator and satisfies 1 p + 1 q = 1, (ψp) −1 = ψq, and ϕ : j → r be an increasing function such that ϕ′(t) 6= 0, for all t ∈ j, and f : j ×r×r→r, is given function will be "well defined" later. 2. phi-hilfer derivatives calculus in this section, we introduce some notations and definitions of phi-hilfer derivatives calculusand present preliminary results needed in our proofs later, for details, see [17, 24, 25].let ϕ : [a,b] → r be an increasing function with ϕ′(t) 6= 0, for all t ∈ j, and let c([a,b] ,r) bethe banach space. for all υ > −1 and s,t ∈ [0,∞), (t ≥ s), we pose ϕυ(t,s) = (ϕ(t) −ϕ(s))υ. definition 1. let (a,b), (−∞≤ a < b ≤∞) be a finite or infinite interval of the half-axis (0,∞) and α > 0. in addition, let ϕ(t) be a positive increasing function on (a,b], which has a continuous derivative ϕ′(t) on (a,b). the ϕ−riemann–liouville fractional integral of a function u with respect to another function ϕ on [a,b] is defined by iα;ϕ a+ u(t) = 1 γ(α) t∫ a ϕ′(s)ϕα−1(t,s)u(s)ds, (2.1) where γ (.) is the gamma function. definition 2. let n ∈ n and let ϕ,u ∈ cn (j) be two functions such that ϕ is increasing and ϕ′(t) 6= 0, for all t ∈ (a,b]. the left-sided ϕ−riemann liouville fractional derivative of a function u of order α is defined by dα;ϕ a+ u(t) = ( 1 ϕ′(t) d dt )n in−α;ϕ a+ u(t) = 1 γ(n−α) ( 1 ϕ′(t) d dt )n t∫ a ϕ′(s)ϕn−α−1(t,s)u(s)ds, eur. j. math. anal. 1 (2021) 167 where n = [α] + 1, [α] represents the integer part of the real number α. definition 3. let n− 1 < α < n with n ∈n, [a,b] is the interval such that −∞≤ a < b ≤∞ and ϕ,u ∈ cn ([a,b] ,r) two functions such that ϕ is increasing and ϕ′(t) 6= 0, for all t ∈ [a,b]. the ϕ-hilfer fractional derivative of a function u of order a and type 0 ≤ β ≤ 1 is defined by hdα,β;ϕ a+ u(t) = iβ(n−α);ϕ a+ ( 1 ϕ′(t) d dt )n i(1−β)(n−α);ϕ a+ u(t) = iγ−α;ϕ a+ dγ;ϕ a+ u(t), where n = [α] + 1, γ −α = β (n−α) . 2.1. auxiliary lemma. lemma 1. let α,ρ > 0. then, we have the following semigroup property given by iα;ϕ a+ iρ;ϕ a+ u(t) = iα+ρ;ϕ a+ u(t), t > a. next, we present the ϕ-fractional integral and derivatives of a power function. proposition 1. let α ≥ 0,σ > 0 and t > a. then, ϕ-fractional integral and derivative of a power function are given by (1) iα,ϕ a+ ϕσ−1(t,a)(t) = γ(σ) γ(α+σ) ϕσ+α−1(t,a).(2) hdα,β;ϕ a+ ϕσ−1(t,a)(t) = γ(σ) γ(σ−α)ϕσ−α−1(t,a),n− 1 < α < n,σ > n. lemma 2. if u ∈ cn([a,b],r),n− 1 < α < n, 0 ≤ β ≤ 1 and γ = α + β(n−α). then iα,ϕ a+ (hdα,β;ϕ a+ u)(t) = u(t) − k=n∑ k=1 ϕγ−k(t,s) γ(γ −k + 1) ∇[n−k]ϕ i (1−β)(n−α);ϕ a+ u(a), t ∈ [a,b], where ∇[n]ϕ u(t) := ( 1 ψ′(t) d dt )n u(t). lemma 3. let u ∈ cn [a,b] and 0 < q < 1, we have∣∣iq;ϕ a+ u(t2) −i q;ϕ a+ u(t1) ∣∣ ≤ 2‖u‖ γ (q + 1) ϕq(t2,t1). lemma 4. ( [14]) for the p−laplacian operator ψp, the following conditions hold true: (1) if |δ1| , |δ2| ≥ ρ > 0, 1 < p ≤ 2,δ1δ2 > 0, then |ψp(δ1) −ψp(δ2)| ≤ (p− 1) ρp−2 |δ1 −δ2| . (2) if p > 2, |δ1| , |δ2| ≤ ρ∗ > 0, then |ψp(δ1) −ψp(δ2)| ≤ (p− 1) ρp−2∗ |δ1 −δ2| . lemma 5. [9] for nonnegative ai, i = 1, ...,k,( k∑ i=1 ai )q ≤ kq−1 ( k∑ i=1 a q i ) ,q ≥ 1. eur. j. math. anal. 1 (2021) 168 lemma 6. let a ≥ 0, 1 < α1,α2 < 2, 0 ≤ β1,β2 ≤ 1, and 2 −γ1 = (1 −β1) (2 −α1) , 2 −γ2 = (1 −β2) (2 −α2) . for f ∈ c(j,,r,r), the unique solution of the sequential hilfer fractional boundary value problem hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t) = f (t), t ∈ j = [a,b] , (2.2)  u(a) = 0,u(b) = n∑ i=1 λiu (ζi ) , ψp ( hdα2,β2;ϕ a+ u ) (a) = 0, and ψp ( hdα2,β2;ϕ a+ u(b) ) = iρ;ϕ a+ u (ζ) , a < ζ,ζi < b, (2.3) is given by u(t) = 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t,s)x(s,a)ds − ϕγ2−1 (t,a) γ(α2)ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t)x(t,a)dt + ϕγ2−1 (t,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) . where x(s,a) = ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)f (z)dz + ( iρ;ϕ 0+ u (ζ) −iα1;ϕ 0+ f (b) ) ϕγ1−1 (b,a) ϕγ1−1 (s,a)  iρ;ϕ 0+ u (ζ) = 1 γ(ρ) ζ∫ a ϕ′(s)ϕρ(ζ,s)u (s) ds, iα1;ϕ 0+ f (b) = 1 γ(α1) b∫ a ϕ′(s)ϕα1−1(b,s)f (s)ds. proof. assume that u is a solution of the sequential nonlocal boundary value problems (3.6) and(2.3). applying the two operators iα1;ϕ a+ , iα2;ϕ a+ to both sides of equation (3.6) and using lemma 2and proposition 1, we obtain ψp ( hdα2,β2;ϕ a+ u ) (t) = iα1;ϕ a+ f (t) + m0 γ (γ1 − 1) ϕγ1−2 (t,a) + m1 γ (γ1) ϕγ1−1 (t,a) , (2.4) where m0,m1 ∈r, and 2 −γ1 = (1 −β1) (2 −α1) . from the boundary condition ψp ( hdα2,β2;ϕ a+ u ) (a) = 0, and if t → a then ϕγ1−2 (t,a) →∞, we get m0 = 0. eur. j. math. anal. 1 (2021) 169 and by ψp (hdα2,β2;ϕa+ u) (b) = iρ;ϕa+ u (ζ) , we obtain m1 = γ (γ1) ϕγ1−1 (b,a) ( iρ;ϕ 0+ u (ζ) −iα1;ϕ 0+ f (b) ) . so hdα2,β2;ϕ a+ u(t) = ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t,a) ϕγ1−1 (b,a) ( iρ;ϕ 0+ u (ζ) −iα1;ϕ 0+ f (b) )) , by (2.4)we have u(t) = iα2;ϕ a+ [ ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t,a) ϕγ1−1 (b,a) ( iρ;ϕ 0+ u (ζ) −iα1;ϕ 0+ f (b) ))] + m2 γ (γ2 − 1) ϕγ2−2 (t,a) + m3 γ (γ2) ϕγ2−1 (t,a) , where m2,m3 ∈r, and 2 − (1 −β2) (2 −α2) = γ2.and if t → a then ϕγ2−2 (t,a) →∞, we getby conditions u(a) = 0, and lim t→0 tγ2−2 = ∞, we get m2 = 0. so u(t) = iα2;ϕ a+ [ ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t,a) ϕγ1−1 (b,a) ( iρ;ϕ 0+ u (ζ) −iα1;ϕ 0+ f (b) ))] + m3 γ (γ2) ϕγ2−1 (t,a) . by conditions u(b) = n∑ i=1 λiu (ζi ) , we get m3 = γ (γ2) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) − γ (γ2) ϕγ2−1 (b,a) iα2;ϕ a+ [ ψq ( iα1;ϕ a+ f (t) + ϕγ1−1 (t,a) ϕγ1−1 (b,a) ( iρ;ϕ 0+ u (ζ) −iα1;ϕ 0+ f (b) ))] t=b . then u(t) = iα2 ;ϕ a+ [ ψq ( iα1 ;ϕ a+ f (t) + ϕγ1−1 (t,a) ϕγ1−1 (b,a) ( iρ;ϕ 0+ u (ζ) −iα1 ;ϕ 0+ f (b) ))] + ϕγ2−1 (t,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) − ϕγ2−1 (t,a) ϕγ2−1 (b,a) iα2 ;ϕ a+ [ ψq ( iα1 ;ϕ a+ f (t) + ϕγ1−1 (t,a) ϕγ1−1 (b,a) ( iρ;ϕ 0+ u (ζ) −iα1 ;ϕ 0+ f (b) ))] t=b . this finishes the proof. � eur. j. math. anal. 1 (2021) 170 conjecture 1. rldµ;ϕ a+ u(t) = 1 γ(α2 −µ) t∫ a ϕ′(s)ϕα2−µ−1(t,s)x(s,a)ds + γ (γ2) γ (γ2 −µ) ϕγ2−µ−1 (t,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) − γ (γ2) γ(α2)γ (γ2 −µ) ϕγ2−µ−1 (t,a) ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t)x(t,a)dt. 3. main results in this section, we present to the reader our main results on the existence and stability for theabove problem. we begin by considering the space cµϕ = { u : u,rldµ;ϕ a+ u ∈ c ([a,b] ,r) } , with the norm ‖u‖cµϕ = ‖u‖c + ∥∥rldµ;ϕ a+ u ∥∥ c , such that ‖u‖c = sup t∈[a,b] |u(t)| , and ∥∥rldµ;ϕ a+ u ∥∥ c = sup t∈[a,b] ∣∣rldµ;ϕ a+ u(t) ∣∣ . 3.1. criteria for uniqueness solution. now, wee need to consider the following assumptions: h1) h is continuous function. h2) there exists a constant υ > 0, such that |h(t,u,v) −h(t,x,y)| ≤ υ (|u −x| + |v −y|) , with t ∈ [a,b] , (u,v,x,y) ∈r4. h3) there exists two continuous functions π1,π2 : [a,b] →r+, such that |h(t,u,v)| ≤ π1(t) |u(t)| + π2(t) |v(t)| , where π∗1 = sup t∈[a,b] |π1(t)| , and π∗2 = sup t∈[a,b] |π2(t)| . now, we define the following quantities: eur. j. math. anal. 1 (2021) 171 ϕq(b,a) = m q, ω = 3q−2 [( 2mα1 γ(α1 + 1) )q−1 ( (π∗1 ) q−1 + (π∗2 ) q−1 ) + ( mρ γ (ρ + 1) )q−1] λ1 = 2.ω.mα2 γ(α2 + 1) , λ2 = ( n∑ i=1 |λi| ) , λ3 = ω.mα2−µ γ(α2 −µ + 1) + ω.γ (γ2) m α2−µ γ(α2 + 1)γ (γ2 −µ) , λ4 = γ (γ2) m −µ γ (γ2 −µ) λ2. based on the above hypotheses, we present to the reader the following result. theorem 1. under h2 and h3 the equation (1.1) has a solution. proof. firstly: we begin this proof by defining the operator g : cµϕ → cµϕ by: (gu) (t) = 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t,s)xu(s,a)ds − ϕγ2−1 (t,a) γ(α2)ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t)xu(t,a)dt + ϕγ2−1 (t,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) . where xu(s,a) = ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)hu(z)dz + ( iρ;ϕ 0+ u (ζ) −iα1;ϕ 0+ hu(b) ) ϕγ1−1 (b,a) ϕγ1−1 (s,a)  , where hu(t) = h(t,u(t), rldµ;ϕ a+ u(t)). we consider the set ur = {u ∈ cµϕ : ‖u‖cµϕ ≤ r} , so that max { (2 (λ1 + λ3)) 1 2−q , 2 (λ2 + λ4) } ≤ r. eur. j. math. anal. 1 (2021) 172 we show that gur ⊂ur . for any u ∈ur , and by lemma 5 we have |xu(s,a)| = ∣∣∣∣∣∣ ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)hu(z)dz + ( iρ;ϕ 0+ u (ζ) −iα1 ;ϕ 0+ hu(b) ) ϕγ1−1 (b,a) ϕγ1−1 (s,a)  ∣∣∣∣∣∣ ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)hu(z)dz + i ρ;ϕ 0+ u (ζ) + iα1 ;ϕ 0+ hu(b) ∣∣∣∣∣∣ q−1 ≤ 3q−2 sup t∈[a,b]  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)hu(z)dz q−1 + (iρ;ϕ 0+ u (ζ) )q−1 + ( iα1 ;ϕ 0+ hu(b) )q−1 ≤ 3q−2 [ 2 ( π∗1m α1 γ(α1 + 1) ‖u‖c + π∗2m α1 γ(α1 + 1) ∥∥rldµ;ϕ a+ u ∥∥ c )q−1 + ( mρ γ (ρ + 1) )q−1 (‖u‖c) q−1 ] ≤ 3q−2 [( 2.π∗1m α1 γ(α1 + 1) )q−1 (‖u‖c) q−1 + ( 2.π∗2m α1 γ(α1 + 1) )q−1 (∥∥rldµ;ϕ a+ u ∥∥ c )q−1 + ( mρ γ (ρ + 1) )q−1 (‖u‖c) q−1 ] ≤ 3q−2 [( 2mα1 γ(α1 + 1) )q−1 ( (π∗1 ) q−1 + (π∗2 ) q−1 ) + ( mρ γ (ρ + 1) )q−1] rq−1 ≤ ω.rq−1. then sup t∈[a,b] |(gu) (t)| (3.1) ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1γ(α2) t∫ a ϕ′(s)ϕα2−1(t,s)xu(s,a)ds + ϕγ2−1 (t,a) γ(α2)ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t)xu(t,a)dt + ϕγ2−1 (t,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) ∣∣∣∣∣ ≤ 2mα2 γ(α2 + 1) |xu| + ( n∑ i=1 |λi| ) sup t∈[a,b] |u (t)| ≤ 2.ω.mα2 γ(α2 + 1) rq + ( n∑ i=1 |λi| ) r. ≤ λ1rq + λ2r. also, we have sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t) ∣∣ (3.2) ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1γ(α2 −µ) t∫ a ϕ′(s)ϕα2−µ−1(t,s)xu(s,a)ds + γ (γ2) γ (γ2 −µ) ϕγ2−µ−1 (t,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) eur. j. math. anal. 1 (2021) 173 − γ (γ2) γ(α2)γ (γ2 −µ) ϕγ2−µ−1 (t,a) ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t)xu(t,a)dt ∣∣∣∣∣∣ ≤ [ mα2−µ γ(α2 −µ + 1) + γ (γ2) m α2−µ γ(α2 + 1)γ (γ2 −µ) ] |xu| + γ (γ2) m −µ γ (γ2 −µ) ( n∑ i=1 |λi| ) sup t∈[a,b] |u (t)| ≤ [ ω.mα2−µ γ(α2 −µ + 1) + ω.γ (γ2) m α2−µ γ(α2 + 1)γ (γ2 −µ) ] rq−1 + γ (γ2) m −µ γ (γ2 −µ) ( n∑ i=1 |λi| ) r ≤ λ3rq−1 + λ4r. by (3.1) and (3.2), we find ‖u‖cµϕ = sup t∈[a,b] |(gu) (t)|c + sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t) ∣∣ c (3.3) ≤ (λ1 + λ3) rq−1 + (λ2 + λ4) r ≤ r. that is gur belongs to ur on [a,b]. next, we prove that g is completely continuous. for any u ∈ ur and t1,t2 ∈ [a; b] such that t1 < t2, by lemma 3, we have sup t∈[a,b] |(gu) (t2) − (gu) (t1)| ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1γ(α2) t2∫ a ϕ′(s)ϕα2−1(t2,s)xu(s,a)ds − 1 γ(α2) t1∫ a ϕ′(s)ϕα2−1(t1,s)xu(s,a)ds + ϕγ2−1 (t2,a) −ϕγ2−1 (t1,a) γ(α2)ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t)xu(t,a)dt + ϕγ2−1 (t2,a) −ϕγ2−1 (t1,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) ∣∣∣∣∣ ≤ ω.rq−1 γ(α2 + 1) ϕα2 (t2,t1) + ω.mα2.rq−1 + γ(α2 + 1)λ2r γ(α2 + 1)ϕγ2−1 (b,a) ϕγ2−1 (t2,t1) . hence, sup t∈[a,b] |(gu) (t2) − (gu) (t1)|→ 0, as t2 → t1. eur. j. math. anal. 1 (2021) 174 also, we can say that sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t2) − ( rldµ;ϕ a+ gu ) (t1) ∣∣ ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1γ(α2 −µ) t2∫ a ϕ′(s)ϕα2−µ−1(t2,s)xu(s,a)ds − 1 γ(α2 −µ) t1∫ a ϕ′(s)ϕα2−µ−1(t1,s)xu(s,a)ds + γ (γ2) γ (γ2 −µ) ϕγ2−µ−1 (t2,a) −ϕγ2−µ−1 (t1,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) + γ (γ2) γ(α2)γ (γ2 −µ) ϕγ2−µ−1 (t2,a) −ϕγ2−µ−1 (t1,a) ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t)xu(t,a)dt ∣∣∣∣∣∣ ≤ ω.rq−1 γ(α2 −µ + 1) ϕα2−µ(t2,t1) + ( γ (γ2) λ2r γ (γ2 −µ) ϕγ2−1 (b,a) + γ (γ2) .ω.r q−1.mα2 γ(α2 + 1)γ (γ2 −µ) ϕγ2−1 (b,a) ) ϕγ2−µ−1 (t2,t1) . hence, sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t2) − ( rldµ;ϕ a+ gu ) (t1) ∣∣ → 0, as t2 → t1. as a consequence of the above three steps and thanks to arzela–ascoli theorem, we conclude that g is completely continuous.the proof of theorem 1 is thus completely achieved. � 3.2. criteria for existence of a solution. theorem 2. assume that h2 and h3 are satisfied. suppose that υ1 + υ2 < 1, where υ1 = 2 (q − 1) ∆q−2mα2 γ(α2 + 1) ( 4υmα1 γ(α1 + 1) + mρ γ (ρ + 1) ) + λ2, and υ2 = (q − 1) ∆q−2 ( 4υmα1 γ(α1 + 1) + mρ γ (ρ + 1) )( mα2−µ γ(α2 −µ + 1) + γ (γ2) m α2−µ γ(α2 + 1)γ (γ2 −µ) ) + γ (γ2) λ2m −µ γ (γ2 −µ) . then, (1.1) has a uniqueness solution. eur. j. math. anal. 1 (2021) 175 proof. we pass to prove that g is a contraction. for any u,v ∈ur , we have the following estimate |xu(s,a) −xv (s,a)| = ∣∣∣∣∣∣ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)hu(z)dz + ( iρ;ϕ 0+ u (ζ) −iα1;ϕ 0+ hu(b) ) ϕγ1−1 (b,a) ϕγ1−1 (s,a)  − ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)hv (z)dz + ( iρ;ϕ 0+ v (ζ) −iα1;ϕ 0+ hv (b) ) ϕγ1−1 (b,a) ϕγ1−1 (s,a)  ∣∣∣∣∣∣ ≤ (q − 1) yq−2 ∣∣∣∣∣∣ 1γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)hu(z)dz − 1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)hv (z)dz + ϕγ1−1 (s,a) ϕγ1−1 (b,a) ( iρ;ϕ 0+ u (ζ) −iρ;ϕ 0+ v (ζ) ) + ϕγ1−1 (s,a) ϕγ1−1 (b,a) ( iα1;ϕ 0+ hu(b) −i α1;ϕ 0+ hv (b) )∣∣∣∣ ≤ (q − 1) yq−2 ( 2mα1 γ(α1 + 1) sup t∈[a,b] |hu(t) −hv (t)| + mρ γ (ρ + 1) sup t∈[a,b] |u(t) −v(t)| ) ≤ (q − 1) yq−2 (( 2υmα1 γ(α1 + 1) + mρ γ (ρ + 1) ) sup t∈[a,b] |u(t) −v(t)| + 2υmα1 γ(α1 + 1) sup t∈[a,b] ∣∣rldµ;ϕ a+ u(t) −rl dµ;ϕ a+ v(t) ∣∣) ≤ (q − 1) ∆q−2 ( 4υmα1 γ(α1 + 1) + mρ γ (ρ + 1) ) ‖u −v‖cµϕ . where  ∆ > 2υm α1 γ(α1+1) + m ρ γ(ρ+1) , if q > 2,ou 0 < ∆ ≤ 2υm α1 γ(α1+1) + m ρ γ(ρ+1) , if 1 < q ≤ 2. then sup t∈[a,b] |(gu) (t) − (gv) (t)| (3.4) ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1γ(α2) t∫ a ϕ′(s)ϕα2−1(t,s) (xu −xv ) (s,a)ds ∣∣∣∣∣∣ + sup t∈[a,b] ∣∣∣∣∣∣ ϕγ2−1 (t,a)γ(α2)ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t) (xu −xv ) (t,a)dt ∣∣∣∣∣∣ + sup t∈[a,b] ∣∣∣∣∣ϕγ2−1 (t,a)ϕγ2−1 (b,a) n∑ i=1 λi (u (ζi ) −v(ζi )) ∣∣∣∣∣ eur. j. math. anal. 1 (2021) 176 ≤ 2mα2 γ(α2 + 1) sup t∈[a,b] |xu(t,a) −xv (t,a)| + λ2 sup t∈[a,b] (|u (t) −v(t)|) ≤ ( 2 (q − 1) ∆q−2mα2 γ(α2 + 1) ( 4υmα1 γ(α1 + 1) + mρ γ (ρ + 1) ) + λ2 ) ‖u −v‖cµϕ ≤ υ1 ‖u −v‖cµϕ .also sup t∈[a,b] ∣∣(rldµ;ϕ a+ gu ) (t) − ( rldµ;ϕ a+ gv ) (t) ∣∣ (3.5) ≤ sup t∈[a,b] ∣∣∣∣∣∣ 1γ(α2 −µ) t∫ a ϕ′(s)ϕα2−µ−1(t,s) (xu −xv ) (s,a)ds + γ (γ2) γ (γ2 −µ) ϕγ2−µ−1 (t,a) ϕγ2−1 (b,a) n∑ i=1 λi (u (ζi ) −v(ζi )) + γ (γ2) γ(α2)γ (γ2 −µ) ϕγ2−µ−1 (t,a) ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t) (xu −xv ) (t,a)dt ∣∣∣∣∣∣ ≤ ( mα2−µ γ(α2 −µ + 1) + γ (γ2) m α2−µ γ(α2 + 1)γ (γ2 −µ) ) sup t∈[a,b] |xu −xv| + γ (γ2) λ2m −µ γ (γ2 −µ) sup t∈[a,b] (|u (t) −v(t)|) ≤ { (q − 1) ∆q−2 ( 4υmα1 γ(α1 + 1) + mρ γ (ρ + 1) )( mα2−µ γ(α2 −µ + 1) + γ (γ2) m α2−µ γ(α2 + 1)γ (γ2 −µ) ) + γ (γ2) λ2m −µ γ (γ2 −µ) } ‖u −v‖cµϕ ≤ υ2 ‖u −v‖cµϕ .by (3.4) and (3.5), yields the following inequality ‖gu −gv‖cµϕ ≤ (υ1 + υ2)‖u −v‖cµϕ .where υ1 + υ2 < 1. hence g is a contraction operator and the contraction mapping principleimplies that (1.1) has a unique solution. � 3.3. ulam type stability. we introduce the following two definitions definition 4. the problem (1.1) is ulam–hyers stable if ∃ λ ∈r∗+, such that for each ε > 0,t ∈ j, and for each u ∈ cµϕ solution of the following inequality∥∥∥hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t) −h(t,u(t),rldµ;ϕ a+ u(t)) ∥∥∥ c µ ϕ < ε, (3.6) ∃v ∈ cµϕ solution of (1.1), i.e. hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ v ) (t) = h(t,v(t),rldµ;ϕ a+ v(t)), (3.7) eur. j. math. anal. 1 (2021) 177 such that, the inequality ‖u −v‖cµϕ ≤ λε, holds. definition 5. the equation (1.1) has the ulam–hyers stability in the generalized sense if ∃ ϕ ∈ c (j,r+), such that for each ε > 0,t ∈ j, and for each u ∈ cµϕ solution of:∥∥∥hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t) −h(t,u(t),rldµ;ϕ a+ u(t)) ∥∥∥ c µ ϕ < ε, (3.8) ∃v ∈ cµϕ solution of (1.1) that satisfies ‖u(t) −v(t)‖cµϕ ≤ εϕ(t). in the light of the first definition and using the above existence and uniqueness theorem, wepresent to the reader the following result. theorem 3. if the assumptions (h2) are satisfied, then eq (1.1) is ulam–hyers stable under the condition that n1 + n2 < 1, where n1 = 2 (q − 1) ∆q−2mα2 γ(α2 + 1) ( 4υmα1 γ(α1 + 1) + mρ γ (ρ + 1) ) , and n2 = ( 4υ (q − 1) ∆q−2mα1 γ(α1 + 1) + (q − 1) ∆q−2mρ γ (ρ + 1) )( mα2−µ γ(α2 −µ + 1) + γ (γ2) m α2−µ γ(α2 + 1)γ (γ2 −µ) ) . proof. let u ∈ cµϕ be a solution of the inequality (3.6), i.e.∥∥∥hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ u ) (t) −h(t,u(t),rldµ;ϕ a+ u(t)) ∥∥∥ c µ ϕ < ε, ∀t ∈ j. (3.9) let v ∈ cµϕ be a unique solution of: hdα1,β1;ϕ a+ ψp ( hdα2,β2;ϕ a+ v ) (t) = h(t,v(t),rldµ;ϕ a+ v(t)), ∀t ∈ j, and  u(a) = v(a), u(b) = v(b)and ψp ( hdα2,β2;ϕ a+ u ) (a) = ψp ( hdα2,β2;ϕ a+ v ) (a), ψp ( hdα2,β2;ϕ a+ u ) (b) = ψp ( hdα2,β2;ϕ a+ v ) (b), eur. j. math. anal. 1 (2021) 178 by using proof of lemma 6 v(t) = 1 γ(α2) t∫ a ϕ′(s)ϕα2−1(t,s)xv (s,a)ds − ϕγ2−1 (t,a) γ(α2)ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t)xv (t,a)dt + ϕγ2−1 (t,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) , where xv (s,a) = ψq  1 γ(α1) s∫ a ϕ′(s)ϕα1−1(s,z)hv (z)dz + ( iρ;ϕ 0+ u (ζ) −iα1;ϕ 0+ hv (b) ) ϕγ1−1 (b,a) ϕγ1−1 (s,a)  . by integration of inequality (3.9), for any t ∈ j, we have∥∥∥∥∥∥u(t) − 1γ(α2) t∫ a ϕ′(s)ϕα2−1(t,s)xu(s,a)ds (3.10) + ϕγ2−1 (t,a) γ(α2)ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t)xu(t,a)dt − ϕγ2−1 (t,a) ϕγ2−1 (b,a) n∑ i=1 λiu (ζi ) ∥∥∥∥∥ c ≤ iα2;ϕ a+ ψq ( iα1;ϕ a+ ε ) = mq−1ϕα1+α2 (t,a) γ (α1 + α2 + 1) ε. on the other hand, for any u,v ∈ cµϕ, we have the following estimate ‖u(t) −v(t)‖c (3.11) < mq−1ϕα1+α2 (t,a) γ (α1 + α2 + 1) ε + sup t∈[a,b] ∣∣∣∣∣∣ 1γ(α2) t∫ a ϕ′(s)ϕα2−1(t,s) (xu −xv ) (s,a)ds ∣∣∣∣∣∣ + sup t∈[a,b] ∣∣∣∣∣∣ ϕγ2−1 (t,a)γ(α2)ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t) (xu −xv ) (t,a)dt ∣∣∣∣∣∣ < mα1+α2+q1 γ (α1 + α2 + 1) ε + 2 (q − 1) ∆q−2mα2 γ(α2 + 1) ( 4υmα1 γ(α1 + 1) + mρ γ (ρ + 1) ) ‖u −v‖cµϕ < mα1+α2+q−1 γ (α1 + α2 + 1) ε + n1 ‖u −v‖cµϕ . eur. j. math. anal. 1 (2021) 179 also, for any t ∈ j, we have∥∥rldµ;ϕ a+ (u(t) −v(t)) ∥∥ c (3.12) ≤ mq−1ϕα1 +α2−µ (t,a) γ (α1 + α2 −µ + 1) ε + sup t∈[a,b] ∣∣∣∣∣∣ 1γ(α2 −µ) t∫ a ϕ′(s)ϕα2−µ−1(t,s) (xu −xv ) (s,a)ds + γ (γ2) γ(α2)γ (γ2 −µ) ϕγ2−µ−1 (t,a) ϕγ2−1 (b,a) b∫ a ϕ′(t)ϕα2−1(b,t) (xu −xv ) (t,a)dt ∣∣∣∣∣∣ ≤ mα1 +α2 +q−µ−1 γ (α1 + α2 −µ + 1) ε + ( mα2−µ γ(α2 −µ + 1) + γ (γ2) m α2−µ γ(α2 + 1)γ (γ2 −µ) ) sup t∈[a,b] |xu −xv| ≤ mα1 +α2 +q−µ−1 γ (α1 + α2 −µ + 1) ε + ( 4υ (q − 1) ∆q−2mα1 γ(α1 + 1) + (q − 1) ∆q−2mρ γ (ρ + 1) )( mα2−µ γ(α2 −µ + 1) + γ (γ2) m α2−µ γ(α2 + 1)γ (γ2 −µ) ) ‖u −v‖cµϕ ≤ mα1 +α2 +q−µ−1 γ (α1 + α2 −µ + 1) ε + n2 ‖u −v‖cµϕ . so, by (3.11) and (3.12) we have ‖u −v‖cµϕ ≤ ε ( mα1+α2+q−1 γ (α1 + α2 + 1) + mα1+α2+q−µ−1 γ (α1 + α2 −µ + 1) ) + (n1 + n2)‖u −v‖cµϕ . therefore, we get ‖u −v‖cµϕ ≤ λε,such that λ = 1 1 − (n1 + n2) ( mα1+α2+q−1 γ (α1 + α2 + 1) + mα1+α2+q−µ−1 γ (α1 + α2 −µ + 1) ) , for any t ∈ j. this implies that the ulam-hyers stability condition is satisfied. � 3.4. illustrative exemple. consider the following problem hd 13 10 , 6 7 ;t2 0+ ψp (( hd 17 10 , 2 3 ;t2 0+ u )) (t) = h(t,u(t),rld 1 2 ;t2 a+ u(t)),t ∈ j = [0, 2] , (3.13) u(a) = 0,u(2) = n∑ i=1 ( 3i 11 ) u ( i 2 + i ) , ψp (( hd 17 10 , 2 3 ;t2 0+ u )) (0) = 0,ψp (( hd 17 10 , 2 3 ;t2 0+ u )) (2) = i 3 2 ;t2 a+ u ( 4 3 ) f (t,u(t),v(t)) = exp ( 1 7 (1 + t2) ) u(t) + v(t) (1 + et) , eur. j. math. anal. 1 (2021) 180 then assumptions (h1), (h2) and (h3) are satisfied with υ = π∗2 = 1 2 ,π∗1 = e 1 7 , and m = 4. we conclude that (3.13) has an unique solution. references [1] b. ahmad, a. alsaedi, s. k. ntouyas, and j. tariboon, 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(2020), 279. https://doi.org/10.1186/ s13662-020-02747-1.[30] a. wongcharoen, s. k. ntouyas, and j. tariboon, boundary value problems for hilfer fractional differential in-clusions with nonlocal integral boundary conditions, mathematics 8 (2020), 1905. https://doi.org/10.3390/ math8111905.[31] y. zhou, basic theory of fractional differential equations, world scientific, singapore, 2014. https://doi.org/10.1186/s13662-017-1172-8 https://doi.org/10.1515/math-2020-0122 https://doi.org/10.1016/j.amc.2015.05.144 https://doi.org/10.1155/2018/1462825 https://doi.org/10.1155/2018/1462825 https://doi.org/10.1155/2020/9606428 https://doi.org/10.1155/2020/9606428 https://doi.org/10.1186/s13662-020-02747-1 https://doi.org/10.1186/s13662-020-02747-1 https://doi.org/10.3390/math8111905 https://doi.org/10.3390/math8111905 1. introduction 2. phi-hilfer derivatives calculus 2.1. auxiliary lemma. 3. main results 3.1. criteria for uniqueness solution. 3.2. criteria for existence of a solution. 3.3. ulam type stability. 3.4. illustrative exemple. references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 8doi: 10.28924/ada/ma.3.8 numerical stabilities of vasicek and geometric brownian motion models o. c. badibi1,∗, i. ramadhani2, m. a. ndondo1, s. d. kumwimba1 1université de lubumbashi, faculté des sciences, département de mathématiques et informatique, democratic republic of the congo christopheromak2014@gmail.com, apondondo@gmail.com, didierkumwimba@gmail.com 2université de kinshasa, faculté des sciences et technologies, département de mathématiques, informatique et statistiques, democratic republic of the congo issaramadhani@gmail.com ∗correspondence: christopheromak2014@gmail.com abstract. stochastic differential equations (sdes) are very often used as models for a large numberof phenomena in the physical, economic and management sciences. they generalize the notion ofordinary differential equations, taking into account a white additive and multiplicative noise term, tomodel random trajectories such as stock market prices or particles movements, on the quantum scale,subject to diffusion phenomena. in rare cases, it is generally impossible to have explicit solutionto these equations. in this case, the numerical approach, presenting itself under various aspects, isthe only favorable outcome. however, the stability of numerical schemes for stochastic differentialequations solution is much more significant. in this paper, we establish and make a classical proofof the mean and mean-square stabilities of the numerical sdes schemes for vasicek and geometricbrownian motion models. 1. introduction stochastic differential equations (sdes) can be seen as ordinary differential equations, or asintegral equations in which integrals occur with respect to brownian motion. they were presentedby ito, with the aim of building continuous and strongly markovian processes whose generatorsare second-order differential operators called diffusions [6]. in general, solving explicitly stochasticdifferential equations (sdes), except for cases where the diffusion and drift coefficients are linears,seems difficult or impossible [8]. this is why the numerical approach is relevant because there arenumerical methods allowing to predict the qualitative behavior such as the stability of the solutions. received: 20 jun 2022. key words and phrases. brownian motion; stochastic differential equations; stabilities of sdes; numerical schemes;vasicek and geometric brownian motion. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 2 the choice of a suitable numerical scheme is based on the understanding and manipulation ofcertain qualitative properties as stability, consistency etc. the qualitative property like stability ofstochastic differential equations solutions, introduced by i.kats and n.krasovskii [2] and perfectedby i.i. gikhman, a.v. skorokhold [3] and a. friedman [4] plays a major role in the study of sdes andthe numerical schemes associated. thus, looking for numerical schemes that preserve qualitativeproperties as the stability of solutions constitutes and remains a very widespread problem innumerical analysis of sdes. in this article we establish and prove the conditions of numerical schemes stabilities in mean andmean-square. we apply the approach described by y.saito [5] to defined and demonstrate the sta-bilities of numericals sdes schemes as: euler-maruyama, milshtein and implicit euler-maruyamafor vasicek and geometric brownian motion models. to begin, let present some elementary notionsrelative to sdes and the numerical schemes adapted to the sdes. 2. preliminary notions 2.1. stochastic differential equation and stabilities. in this section,we present some definitions inconnection with stochastic differential equation and stabilities of solutions of sdes. definition 2.1. (stochastic differential equation (sde) [13]) let ( ω,f, (ft)t≥0 ,p ) be a filtered probability space, (bt)t≥0 a standard brownian motion on r d defines in a filtered probability space. a stochastic differential equation (sde) on rd with the drift coefficient: b (t,xt) ∈ [0,t ] ×rn −→rn and the diffusion: σ (t,xt) ∈ [0,t ] ×rn −→rn×d when xo is random variable independent of (bt)t≥0 is an equation of the form:{ dxt = b (t,xt) dt + σ (t,xt) dbt x (o) = xo (2.1) the white noise σ (t,xt) can be additive or multiplicative, depending on whether it does notinfluence or does influence the state of the system. theorem 2.1. (existence and uniqueness [14]) we assume that there is a positive constant k such that ∀ t ≥ 0, x,y ∈rd (1) lipschitz condition: |b (t,x) −b (t,y ) | + |σ (t,x) −σ (t,y ) | ≤ k|x −y | https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 3 (2) linear growth condition: |b (t,x) | ≤ k (1 + |x|) , |σ (t,x) | ≤ k (1 + |x|) so the sde (2.1) admits, for any initial condition xo of square integrable (e [|xo|2] < ∞) the strong solution (xt)t∈[0,t ],unique, almost surely continuous and satisfying the following condition: e ( sup 0≤t≤t |x2t | ) < ∞ definition 2.2. (asymptotic stability in probability in large sense [1], [24]) the solution is said to be asymptotically and stochastically stable in the large sense if ∀ xo ∈l2ft ([−t, 0] ,r n) , then p { lim t−→∞ x (t) = 0 } = 1. definition 2.3. (stability of pth moment [23], [25]) (1) let p ≥ 2 we say that a solution of (2.1) is stable in pth moment if ∀� > 0 it exists δ > 0 such as e [ sup t>0 |x (t) |p ] < � avec |xo| < δ (2) let p ≥ 2, we say that a solution of (2.1) is stable asymptoticaly in pth moment if it is stable from peme moment ∀ xo ∈l2fto ([−t, 0] ,r n) then we have : lim t−→∞ e [ sup t>t |x (t) |p ] = 0 2.2. stochastic numerical schemes. in this section we present three numerical schemes as euler-maruyama, implicit euler-maruyama and milshtein schemes. definition 2.4. ( euler-maruyama scheme [10] , [11]) let {xt} the diffusion solution of the sde(2.1). let consider the interval [0,t ] and a regular subdivision t0 = 0 < t1 < t2 < t0 < · · · < tk = t with step ∆t = t n = t k , the euler-maruyama scheme of (2.1) is defined like:{ xemk+1 = xk + b(tk,xk)(tk+1 − tk) + σ(tk,xk)(bk+1 −bk) x(0) = x0 (2.2) https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 4 definition 2.5. (implicit euler-maruyama scheme [10]) the implicit euler-maruyama scheme is a convergent scheme like the euler-maruyama scheme. to be reassured of the existence of the solutions of this scheme, only the term of the drift is implicit. for this fact: b(xk)∆tk which is in the euler-maruyama scheme is replaced by b(xk+1)∆tk and the diffusion term: σ(xk)∆bk remains unchanged. the implicit euler-maruyama scheme of the eds (2.1) has given by: xiemk+1 = xk + b(xk+1)∆tk + σ(xk)∆bk (2.3) definition 2.6. (milshtein scheme [7]) let consider the sde (2.1) and a regular subdivision of the intervalle et une subdivision of the interval [0,t ]: 0 = t0 < t1 < t2 < · · · < tn = t de [0,t ] the milshtein scheme is defined like: xmk+1 = xk + b(xk)∆tk + σ(xk)∆bk + 1 2 σ(xk)σ ′(xk)(∆bk − ∆tk) x(0) = x0 (2.4) remark 2.1. it should be noted that the euler-maruyama scheme converges strongly up to the order 1 2 while that of milshtein converges up to the order 1. 3. numerical stabilities of vasicek model 3.1. explicit solution. the vasicek model (1977) is one of the first stochastic interest rate models.it is a gaussian process generalizing the ornstein-unlenbeck model and explains the observedempirical mean reversion effect on interest rate curves [15], this model looks like:dxt = (θ1 −θ2xt)dt + θ3dbt x(0) = x0 ∀θ1,θ2 et θ3 > 0 (3.1) with xt: the instant interest rate; θ2: mean reversion rate; θ1: the long-term average and θ3: thevolatility.the analytical solution of (3.1) model is: xt = θ1 θ2 + ( x0 − θ1 θ2 ) e−θ2t + θ3 ∫ +∞ 0 e−θ2(t−u)dbu (3.2) the model (3.1) is equivalent to the model:dxt = θ(µ−xt)dt + σdbt x(0) = x0 (3.3) https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 5 the solution of (3.3) has given by: xt = µ + (x0 −µ) e−θt + θ ∫ t 0 e−θ2(t−u)dbu (3.4) considering the solution of (3.2), the mean and the mean-square give respectively: e[xt] = θ1 θ2 ∀ θ2 > 0 and v (xt) = θ23 2θ2 ∀ θ2 > 0 which means that the stochastic process xt 'n (θ1θ2 , θ232θ2 ) by using some properties of brownian motion, the solution of the model (3.2) can be written asfollows: xt = θ1 θ2 + θ3e −2θ2t √ 2θ2 b(e2θ2t) (3.5) now, we present some numerical stabilities conditions for the system (3.1) of some numericalschemes (euler-maruyama, implicit euler-maruyama and milshtein) and the proofs of these basedon the approach described in [5]. 3.2. euler-maruyama scheme stabilities. the euler-maruyama scheme associated to the system(3.1) is : xemk+1 = xk + (θ1 −θ2xk) ∆t + θ3∆bk xemk+1 = θ1∆t + (1 −θ2∆t)xk + θ3 √ ∆tzk (3.6) 3.2.1. mean stability of euler-maruyama scheme. theorem 3.1. (mean stability of euler-maruyama scheme) the euler-maruyama scheme (3.6) of the vasicek model (3.1) is mean asymptotically stable if: e [ xemk+1 ] = (1 −θ2∆t)k+1e[x0] + θ1∆t [ k+1∑ i=0 (1 −θ2∆t)i ] (3.7) with |1 −θ2∆t| < 1 and lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = θ1 θ2 https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 6 proof. to prove the theorem, we start by evaluating the mean of the (3.1) equation using theapproach defined in [5]. in effect, e [ xemk+1 ] = e [ θ1∆t + xk (1 −θ2∆t) + θ3 √ ∆tzk ] = e [θ1∆t] + e [xk (1 −θ2∆t)] + e [ θ3 √ ∆tzk ] = e [θ1∆t] + (1 −θ2∆t) e [xk] + 0] with zk 'n(0, 1) = θ1∆t + (1 −θ2∆t) e[xk] = θ1∆t + (1 −θ2∆t){(1 −θ2∆t) e[xk−1] + θ1∆t} = θ1∆t + θ1∆t(1 −θ2∆t) + (1 −θ2∆t)2e[xk−1] = θ1∆t(1 + (1 −θ2∆t)) + (1 −θ2∆t)2e[xk−1] = θ1∆t(1 + (1 −θ2∆t)) + (1 −θ2∆t)2 {(1 −θ2∆t) e[xk−2] + θ1∆t} = θ1∆t(1 + (1 −θ2∆t)) + θ1∆(1 −θ2∆t)2 + (1 −θ2∆t)3e[xk−2] = θ1∆t(1 + (1 −θ2∆t) + (1 −θ2∆t)2) + (1 −θ2∆t)3e[xk−2] = θ1∆t(1 + (1 −θ2∆t) + (1 −θ2∆t)2 + · · · + (1 −θ2∆t))k+1 + (1 −θ2∆t)k+1e[x0] = (1 −θ2∆t)k+1e[x0] + θ1∆t [ k+1∑ i=0 (1 −θ2∆t)i ] using the theory of geometric sequences and series, we get: e [ xemk+1 ] = (1 −θ2∆t)k+1e[x0] + θ1∆t ( (1 − (1 −θ2∆t)k+1) 1 − (1 −θ2∆t) ) (3.8) as the identity (3.8) represents a geometric sequence, we have that it converges if |1 −θ2∆t| < 1 by calculating the limit of the (3.8), for ∆t → 0 and k → +∞, we get: lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = θ1 θ2 � 3.2.2. mean-square stability of euler-maruyama scheme. theorem 3.2. (mean-square stability of euler-maruyama scheme) the euler-maruyama scheme(3.6) of the vasicek model (3.1) is mean-square asymptotically stable if: e [∣∣xemk+1∣∣2] = (1 −θ2∆t)2(k+1) e (|x0|2) + (θ23 + θ21 ∆t) ∆t k+1∑ i=0 (1 −θ2∆t)2i https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 7 and that the following two conditions are satisfied simultaneously: (1) |1 −θ2∆t| < 1 (2) lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]2) = θ23 2θ2 proof. as in the previous theorem, we start by calculating the expression: e [∣∣xemk+1∣∣2] of vasicek model of the equation (3.1). in effect, e [∣∣xemk+1∣∣2] = |θ1∆t|2 + e (|xk(1 −θ2∆t)|)2 + θ23 ∆t zk 'n(0, 1) = (1 −θ2∆t)2e ( |xk| 2 ) + (θ21 ∆t + θ 2 3 )∆t = (1 −θ2∆t)2e ( |xk| 2 ){ e ( |xk+1| 2 ) (1 −θ2∆t)2 + (θ23 + θ 2 1 ∆t)∆t } + (θ23 + θ 2 1 ∆t)∆t = (1 −θ2∆t)4e ( |xk−1| 2 ) + (θ23 + θ 2 1 ∆t)∆t [ (1 −θ2∆t)2 + 1 ] = (1 −θ2∆t)6e ( |xk−2| 2 ) + (θ23 + θ 2 1 ∆t)∆t[(1 −θ2∆t) 4 + (1 −θ2∆t)2 + 1] = (1 −θ2∆t)8e ( |xk−3| 2 ) + (θ23 + θ 2 1 ∆t)∆t [ (1 −θ2∆t) 6 + (1 −θ2∆t)4 + (1 −θ2∆t)2 + 1 ] = (1 −θ2∆t)2k+1e ( |x0|2 ) + ( θ23 + θ 2 1 ∆t ) ∆t [ (1 −θ2∆t)2k + ... + (1 −θ2∆t)4 +(1 −θ2∆t)2 + (1 −θ2∆t)0 ] = (1 −θ2∆t) 2(k+1) e ( |x0|2 ) + ( θ23 + θ 2 1 ∆t ) ∆t k+1∑ i=0 (1 −θ2∆t)2i = (1 −θ2∆t) 2k+2 e ( |x0|2 ) + ( θ23 + θ 2 1 ∆t ) ∆t [ 1 −|1 −θ2∆t| 2k+2 1 −|1 −θ2∆t| 2 ] we get: e [∣∣xemk+1∣∣2] = (1 −θ2∆t)2k+2 e (|x0|2) + (θ23 + θ21 ∆t) ∆t [ 1 1 −|1 −θ2∆t|2 ] (3.9) the expression (3.9) as the geometric sequence, we have that it converges when |1 −θ2∆t| < 1 passing to the limit of the equation (3.9), for ∆t → 0 and k → +∞, we find the desired result i.e: lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = θ23 2θ2 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 8 3.3. milshtein’s scheme stabilities. the milshtein scheme associated to the system (3.1) is: xmk+1 = θ1∆t + (1 −θ2∆t)xk + θ3 √ ∆tzk (3.10) with σ = θ3 σ ′ = 0 then mean and mean-square stabilities gives the same results as in the euler-maruyama schemei.e theorem 3.3. (mean stability of milshtein scheme) the milshtein scheme (3.10) of vasicek model(3.1) is mean asymptotically stable if: e [ xmk+1 ] = θ1∆t [ k+1∑ i=0 (1 −θ2∆t)i ] + (1 −θ2∆t)k+1e[x0] and that the following two conditions are satisfied simultaneously: (1) |1 −θ2∆t| < 1 (2) lim ∆t→0 ( lim k→+∞ e [ xmk+1 ]) = θ1 θ2 theorem 3.4. (mean-square stability of milshtein scheme) the milshtein scheme (3.10) of vasicek model (3.1) is mean-square asymptotically stable if: e [∣∣xmk+1∣∣2] = (θ23 + θ21 ∆t) ∆t k+1∑ i=0 (1 −θ2∆t)2i + (1 −θ2∆t)2(k+1) e ( |x0|2 ) and that the following two conditions are satisfied simultaneously: (1) |1 −θ2∆t| < 1 (2) lim ∆t→0 ( lim k→+∞ e [ xmk+1 ]2) = θ23 2θ2 proof. the proofs of these theorems above is done in the same way as the result theorems of theeuler-maruyama scheme for vasicek model. � 3.4. implicit euler-maruyama scheme stabilities. the implicit euler-maruyama scheme associ-ated to the system (3.1) is : xiemk+1 = xk + (θ1 −θ2xk+1)∆t + θ3 √ ∆tzk xk+1 + θ2∆txk+1 = xk + θ1∆t + θ3 √ ∆tzk xk+1 (1 + θ2∆t) = θ1∆t + xk + θ3 √ ∆tzk xk+1 = θ1∆t 1 + θ2∆t + 1 1 + θ2∆t xk + θ3 √ ∆t 1 + θ2∆t zk https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 9 we get: xiemk+1 = θ1∆t 1 + θ2∆t + 1 1 + θ2∆t xk + θ3 √ ∆t 1 + θ2∆t zk (3.11) 3.4.1. mean stability of implicit euler-maruyama scheme. theorem 3.5. (mean stability of implicit euler-maruyama scheme) the implicit euler-maruyama scheme (3.11) of vasicek model (3.1) is mean asymptotically stable if: e ( xiemk+1 ) = ( 1 1 + θ2∆t )k+1 e (x0) + θ1∆t k+1∑ i=0 ( 1 1 + θ2∆t )i and that the following two conditions are satisfied simultaneously: (1) |1 + θ2∆t| > 1 (2) lim ∆t→0 ( lim k→∞ e [ xiemk+1 ]) = θ1 θ2 proof. we start by evaluating the mean of the implicit euler-maruyama scheme of the expressiondefined in (3.11), in effect: e ( xiemk+1 ) = e ( θ1∆t 1 + θ2∆t ) + e ( 1 1 + θ2∆t xk ) + e ( θ3 √ ∆t 1 + θ2∆t zk ) = θ1∆t 1 + θ2∆t + 1 1 + θ2∆t e (xk) = θ1∆t 1 + θ2∆t + 1 1 + θ2∆t {( 1 1 + θ2∆t ) e (xk−1) + θ1∆t 1 + θ2∆t } = θ1∆t 1 + θ2∆t + θ1∆t (1 + θ2∆t) 2 + 1 (1 + θ2∆t) 2 e (xk−1) = ( 1 1 + θ2∆t )2 e (xk−1) + θ1∆t ( 1 (1 + θ2∆t) 2 + 1 (1 + θ2∆t) ) = ( 1 1 + θ2∆t )3 e (xk−2) + θ1∆t (( 1 1 + θ2∆t )3 + ( 1 1 + θ2∆t )2 + ( 1 1 + θ2∆t )) = ( 1 1 + θ2∆t )4 e (xk−3) + θ1∆t (( 1 1 + θ2∆t )4 + ( 1 1 + θ2∆t )3 + · · · + 1 ) by continuing the iterations until k + 1, we obtain: e ( xiemk+1 ) = ( 1 1 + θ2∆t )k+1 e (x0) + θ1∆t k+1∑ i=0 ( 1 1 + θ2∆t )i (3.12) the equation (3.12) is the geometric sum of geometric sequence and geometric series, the expres-sion: ( 1 1 + θ2∆t ) < 1 https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 10 or |1 + θ2∆t| > 1by using limit of (3.12), for ∆t → 0 and k → +∞, we get: lim ∆t→0 ( lim k→+∞ e [ xiemk+1 ]) = θ1 θ2 � 3.4.2. mean-square stability of implicit euler-maruyama scheme. theorem 3.6. (mean-square stability of implicit euler-maruyama scheme) the implicit eulermaruyama of vasicek model (3.1) is mean-square asymptotically stable if: e (∣∣xiemk+1 ∣∣2) = ( 11 + θ2∆t )2(k+1) e (|x0|)2 + ( θ23 + θ 2 1 ∆t ) ∆t k+1∑ i=0 ( 1 1 + θ2∆t )2i , with |1 + θ2∆t| > 1 and lim ∆t→0 ( lim k→∞ e ( |xk+1| 2 )) = θ23 2θ2 proof. let us evaluate the mean-square of the implicit euler-maruyama scheme (3.11), in effect: e (∣∣xiemk+1 ∣∣2) = e (∣∣∣∣ θ1∆t1 + θ2∆t ∣∣∣∣2 ) + e (∣∣∣∣ 11 + θ2∆txk ∣∣∣∣) + ∣∣∣∣ θ23 ∆t1 + θ2∆t ∣∣∣∣2 = θ21 (∆t) 2 (1 + θ2∆t) 2 + 1 (1 + θ2∆t) 2 e ( |xk| 2 ) + θ23 ∆t (1 + θ2∆t) 2 = θ21 (∆t) 2 + θ23 ∆t (1 + θ2∆t) 2 + 1 (1 + θ2∆t) 2 e ( |xk| 2 ) = ( θ23 + θ 2 1 ∆t ) ∆t (1 + θ2∆t) 2 + ( 1 1 + θ2∆t )2 {( 1 1 + θ2∆t )2 e (|xk−1|) 2 + ( θ23 + θ 2 1 ∆t ) ∆t (1 + θ2∆t) 2 } = ( 1 1 + θ2∆t )4 e (|xk−1|) 2 + ( θ23 + θ 2 1 ∆t ) ∆t [( 1 1 + θ2∆t )4 + ( 1 1 + θ2∆t )2] ... = ( 1 1 + θ2∆t )2(k+1) e (|x0|)2 + ( θ23 + θ 2 1 ∆t ) ∆t k+1∑ i=1 ( 1 1 + θ2∆t )2i by using the geometrical sequence and geometrical series, we get: e (∣∣xiemk+1 ∣∣2) = ( 11 + θ2∆t )2(k+1) e (|x0|)2 + ( θ23 + θ 2 1 ∆t ) ∆t k+1∑ i=1 ( 1 1 + θ2∆t )2i we have the geometric sequence and series, converging when |1 + θ2∆t| > 1, by calculating limitof the equation below , for ∆t → 0 and k → +∞, we get: lim ∆t→0 ( lim k→+∞ e [ xiemk+1 ]) = θ23 2θ2 https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 11 � 4. numerical stabilities of geometric brownian motion 4.1. explicit solution of the model. geometric brownian motion known as exponential brownianmotion is a continuous stochastic process whose logarithm follows a brownian motion. it is appliedin the mathematical modeling of certain courses in the financial markets [26]. it represents areasonable approximation of the evolution of stock market prices, because a quantity which followsa geometric brownian motion takes all strictly positive values and only the elementary changesundergone by the random variable are significant. the geometric brownian motion xt is a process which is written in the form [13]:{ dxt = θ1xtdt + θ2xtdbt x(0) = x0 ∀θ1,θ2 ∈r (4.1) this process admits as an explicit solution: xt = x0e {(θ1−12 θ22 )t+θ2bt} (4.2) the variable on the right hand follows a normal distribution, it can also be written in the form: xt = xse {(θ1−12 θ22 )t+θ2(bt−bs )} (4.3) the conditionnal mean is: e (xt|xs) = xseθ1(t−s) (4.4)the (4.3) process is often widely used to model the price of a financial asset the return on theasset between two dates is measured by the difference in the logarithms of the prices and is givenby the gaussian variable below:{ θ1 − 1 2 θ22 } (t − s) + θ2 (bt −bs) the mean and the mean-square give respectively: e(xt) = x0e θ1t e(x2t ) = x 2 0e (2θ1+θ22 )t (4.5) remark 4.1. it should be noted that:(1) for mean if t →∞ and θ1 < 0 we have: lim t→∞ e(xt) = lim k→∞ x0e θ1t = 0 (2) for mean-square if (2θ1 + θ22) < 0 and t →∞ i.e lim t→∞ e(x2t ) = 0 with ( 2θ1 + θ 2 2 ) < 0. now, let’s analyze the stabilities of some numerical schemes (euler-maruyama, milshtein andimplicit euler-maruyama) in mean and mean-square. https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 12 4.2. euler-maruyama scheme stabilities. the euler-maruyama scheme associated to (4.1) is: xemk+1 = xk + θ1xk∆t + θ2xk∆bk xemk+1 = xk (1 + θ1∆t) + θ2xk √ ∆tzk (4.6) 4.2.1. mean stability of euler-maruyama scheme. theorem 4.1. (mean stability of euler-maruyama scheme) the euler-maruyama scheme (4.6) associated to (4.1) model is mean asymptotically stable if e [ xemk+1 ] = (1 + θ2∆t) k+1e (x0) with |1 + θ1∆t| < 1 and lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = 0 proof. by calculating the mean of the expression(4.6), we obtain: e [ xemk+1 ] = e [ xt(1 + θ1∆t) + θ2xt √ ∆tzt ] = e [xt(1 + θ1∆t)] + e [ θ2 √ ∆txtzt ] = e [(1 + θ1∆t)xt] + e [ θ2 √ ∆t ] e [xt] e [zt] as zk 'n(0, 1) e(zk) = 0 e [xk+1] = (1 + θ1∆t)e(xt) = (1 + θ1∆t) ((1 + θ2∆t)e (xk−1)) = (1 + θ2∆t) 2e (xk−1) = (1 + θ1∆t) 2 ((1 + θ2∆t)e (xk−2))... = (1 + θ2∆t) k+1e (x0) we get the following geometric sequence: e [ xemk+1 ] = (1 + θ2∆t) k+1e (x0) which converges if |1 + θ2∆t| < 1 et nd passing to the limit for a ∆t → 0 and k → +∞, we obtain: lim ∆t→0 ( lim k→+∞ e [ xemk+1 ]) = 0 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 13 4.2.2. mean-square stability of euler-maruyama scheme. theorem 4.2. (mean-square stability of euler-maruyama scheme) the euler-maruyama scheme(4.6) associated to (4.1) model is mean-square asymptotically stable if: e (∣∣xemk+1∣∣2) = (|(1 + θ1∆t)|2 + ∣∣∣θ2√∆t∣∣∣2)2k+2 e (|x0|2) , with ∣∣∣|1 + θ1∆t|2 + ∣∣θ2√∆t∣∣2∣∣∣ < 1 and lim ∆t→0 ( lim k→∞ e ( |xk+1| 2 )) = 0 proof. the mean-square of the expression (4.6) gave: e [∣∣xemk+1∣∣2] = e [∣∣∣xk(1 + θ1∆t) + θ2xk√∆tzk∣∣∣2] = e [ |xk(1 + θ1∆t)| 2 + ∣∣∣θ2√∆txkzk∣∣∣2 + 2 ∣∣∣xk(1 + θ1∆t)θ2√∆txkzk∣∣∣] = e [ |xk(1 + θ1∆t)| 2 ] + e [∣∣∣θ2√∆txkzk∣∣∣2] + 2e [|xk(1 + θ1∆t)| ∣∣∣θ2√∆txkzk∣∣∣] = |(1 + θ1∆t)|2 e [ |xk| 2 ] + ∣∣∣θ2√∆t∣∣∣2 e [|xk|2] = ( |(1 + θ1∆t)|2 + ∣∣∣θ2√∆t∣∣∣2)e [|xt|2] = ( |(1 + θ1∆t)|2 + ∣∣∣θ2√∆t∣∣∣2)(|(1 + θ1∆t)|2 + ∣∣∣θ2√∆t∣∣∣2)e [|xk−1|2] ... = ( |(1 + θ1∆t)|2 + ∣∣∣θ2√∆t∣∣∣2)2k+2 e [|x0|2] we get a geometric sequence: e (∣∣xemk+1∣∣2) = (|(1 + θ1∆t)|2 + ∣∣∣θ2√∆t∣∣∣2)2(k+1) e (|x0|2) for ∣∣∣|1 + θ1∆t|2 + ∣∣θ2√∆t∣∣2∣∣∣ < 1 the sequence converges, and passing to the limit, we obtain fora ∀∆t → 0 and k → +∞, lim ∆t→0 ( lim k→∞ e ( |xk+1| 2 )) = 0 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 14 4.3. milshtein’s scheme stabilities. the milshtein schema associated to the expression(4.1) isgiven by : xmk+1 = xk + b(xk)∆t + σ(xk)∆bk + 1 2 σσ′(xk) { (∆bk) 2 − ∆t } = xk + θ1xk∆t + θ2xk∆bk + 1 2 θ2xtθ2 { (∆bk) 2 − ∆t } = xk + θ1xk∆t + θ2xk √ ∆tzk + 1 2 θ22xk ( ∆tz2k − ∆t ) = ( 1 + θ1∆t − 1 2 θ22 ∆t ) xk + θ2xk √ ∆tzk + 1 2 θ22xk∆tz 2 k = ( 1 + ( θ1 − 1 2 θ22 ) ∆t ) xk + θ2xk √ ∆tzk + 1 2 θ22xk∆tz 2 kwe have after calculation: xmk+1 = xk ( 1 + ( θ1 − 1 2 θ22 ) ∆t ) + θ2xk √ ∆tzk + 1 2 θ22xk∆tz 2 k (4.7) we now consider the same model of geometric brownian motion, we state some results on thestabilities following milshtein’s scheme and we prove these results. 4.3.1. mean stability of milshtein’s scheme. theorem 4.3. (mean stability of milshtein’s scheme) the milshtein’s scheme (4.7) assocated to(4.1) model is mean asymptotically stable if e ( xmk+1 ) = [1 + θ1∆t] k+1 e (x0) with |1 + θ1∆t| < 1 and lim ∆t→0 ( lim k→∞ e ( xmk+1 )) = 0 proof. applying the usual approach, let us evaluate the mean of gives: e ( xmk+1 ) = e ( xk ( 1 + ( θ1 − 1 2 θ22 ) ∆t ) + θ2xk √ ∆tzk + 1 2 θ22xk∆tz 2 k ) = e ( xk ( 1 + ( θ1 − 1 2 θ22 ) ∆t )) + e ( θ2xk √ ∆tzk ) + e ( 1 2 θ22xk∆tz 2 k ) = ( 1 + ( θ1 − 1 2 θ22 ) ∆t + 1 2 θ22 ∆t ) e (xk) = (1 + θ1∆t) e (xk)... = (1 + θ1∆t) k+1 e (x0) https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 15 ultimately we get that: e ( xmk+1 ) = [1 + θ1∆t] k+1 e (x0)as the previous expression has the form of a geometric sequence, we know that it converges if |1 + θ1∆t| < 1, passing to the limit, for all ∆t → 0 and k → +∞ we find the results searched i.e: lim ∆t→0 ( lim k→∞ e ( xmk+1 )) = 0 � 4.3.2. mean-square stability of milshtein’s scheme. theorem 4.4. (mean-square stability of milshtein’s scheme) the milshtein scheme (4.7) associated to (4.1) model is mean-square asymptotically stable if e (∣∣xmk+1∣∣) = [∣∣∣∣1 + (θ1 − 12θ22 ) ∆t ∣∣∣∣2 + ∣∣∣θ2√∆t∣∣∣2 + ∣∣∣∣12θ22 ∆t ∣∣∣∣2 ]2(k+1) e ( |x0|2 ) with ∣∣∣∣∣1 + (θ1 − 12θ22) ∆t∣∣2 + ∣∣θ2√∆t∣∣2 + ∣∣12θ22 ∆t∣∣2∣∣∣ < 1 and lim ∆t→0 ( lim k→∞ e (∣∣xmk+1∣∣2)) = 0 proof. let’s start by calculating the mean-sqaure of the model expression, ie: e (∣∣xmk+1∣∣2) = e (∣∣∣∣xk (1 + (θ1 − 12θ22 ) ∆t ) + θ2xk √ ∆tzk + 1 2 θ22xk∆tz 2 k ∣∣∣∣2 ) = e (∣∣∣∣xk (1 + (θ1 − 12θ22 ) ∆t )∣∣∣∣2 ) + e (∣∣∣θ2xk√∆tzk∣∣∣2) + e (∣∣∣∣12θ22xk∆tz2k ∣∣∣∣2 ) = ∣∣∣∣1 + (θ1 − 12θ22 ) ∆t ∣∣∣∣2 e (|xk|2) + ∣∣∣θ2√∆t∣∣∣2 e (|xk|2) + ∣∣∣∣12θ22 ∆t ∣∣∣∣2 e (|xk|2) = [∣∣∣∣1 + (θ1 − 12θ22 ) ∆t ∣∣∣∣2 + ∣∣∣θ2√∆t∣∣∣2 + ∣∣∣∣12θ22 ∆t ∣∣∣∣2 ] e ( |xk| 2 ) ... = [∣∣∣∣1 + (θ1 − 12θ22 ) ∆t ∣∣∣∣2 + ∣∣∣θ2√∆t∣∣∣2 + ∣∣∣∣12θ22 ∆t ∣∣∣∣2 ]2(k+1) e ( |x0|2 ) continuing with the iterations, we get: e (∣∣xmk+1∣∣) = [∣∣∣∣1 + (θ1 − 12θ22 ) ∆t ∣∣∣∣2 + ∣∣∣θ2√∆t∣∣∣2 + ∣∣∣∣12θ22 ∆t ∣∣∣∣2 ]2(k+1) e ( |x0|2 ) passing to the limit with ∆t → 0 and k → +∞, we obtain the stated results, ie: lim ∆t→0 ( lim k→∞ e (∣∣xmk+1∣∣2)) = 0 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 16 4.4. implicit euler-maruyama scheme stabilities. the implicit euler-maruyama scheme (iem)gives: xiemk+1 = xk + b(xk+1)∆t + δ(xt)∆bk xk+1 = xk + θ1xk+1∆t + θ2xt∆bk xk+1 −θ1xk+1∆t = xk + θ2xk∆bk xk+1 (1 −θ1∆t) = xk + θ2xk∆bkwe obtain: xiemk+1 = 1 1 −θ1∆t xk + θ2 √ ∆t 1 −θ1∆t xkzk zk 'n(0, 1) (4.8) 4.4.1. mean stability of implicit euler-maruyama scheme. theorem 4.5. (mean stability of implicit euler-maruyama scheme) the implicit euler-maruyama scheme (iem) (4.8) associated to (4.1) model is mean asymptotically stable if e ( xiemk+1 ) = ( 1 1 −θ1∆t )k+1 e (x0) (4.9) with |1 −θ1∆t| > 1 then, lim ∆t→0 ( lim k→∞ e (∣∣xiemk+1 ∣∣2)) = 0 proof. let’s evaluate the mean associated to the implicit euler-maruyama scheme e ( xiemk+1 ) = e ( 1 1 −θ1∆t xk + θ2 1 −θ1∆t xk √ ∆tzk ) = e ( 1 1 −θ1∆t xk ) + e ( θ2 1 −θ1∆t √ ∆t ) (xk) (zk) = e ( 1 1 −θ1∆t xk ) = 1 1 −θ1∆t e (xk) = ( 1 1 −θ1∆t )2 e (xk−1) = ( 1 1 −θ1∆t )3 e (xk−2) ... = ( 1 1 −θ1∆t )k+1 e (x0) continuing with the iterations we get: e ( xiemk+1 ) = ( 1 1 −θ1∆t )k+1 e (x0) https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 17 passing to the limit with ∆t → 0 and k → +∞, we obtain the stated results, ie lim ∆t→0 ( lim k→∞ e (∣∣xiemk+1 ∣∣2)) = 0 � 4.4.2. mean-square stability of implicit euler-maruyama scheme. theorem 4.6. (mean-square stability of implicit euler-maruyama scheme) the implicit eulermaruyama scheme associated to the model (4.1) is asymptotically mean-square stable if e (∣∣xiemk+1 ∣∣2) = [ 1 + ∣∣θ2√∆t∣∣ 1 −θ1∆t ]2(k+1) e ( |x0|2 ) with ∣∣∣∣1+|θ2√∆t|1−θ1∆t ∣∣∣∣ < 1 and lim ∆t→0 ( lim k→∞ e (∣∣xemik+1 ∣∣2)) = 0 proof. : let us calculate the quadratic mean, in effect, e (∣∣xmk+1∣∣2) = e ∣∣∣∣∣ 11 −θ1∆txt + θ2 √ ∆t 1 −θ1∆t xtzt ∣∣∣∣∣ 2  = ( 1 1 −θ1∆t )2 e (∣∣∣xk + θ2√∆txkzk∣∣∣2) = ( 1 1 −θ1∆t )2 [ e ( |xk| 2 ) + e (∣∣∣+θ2√∆txkzk∣∣∣2)] = ∣∣∣∣ 11 −θ1∆t ∣∣∣∣2 [e (|xk|2) + ∣∣∣+θ2√∆t∣∣∣2 e (|xk|2)] = ( 1 + ∣∣θ2√∆t∣∣2) (1 −θ1∆t)2 e ( |xk| 2 ) = ( 1 + ∣∣θ2√∆t∣∣2) (1 −θ1∆t)2 ( 1 + ∣∣θ2√∆t∣∣2) (1 −θ1∆t)2 e ( |xk−1| 2 ) continuing with the iterations, we get: e (∣∣xemik+1 ∣∣2) = [ 1 + ∣∣θ2√∆t∣∣ 1 −θ1∆t ]2(k+1) e ( |x0|2 ) passing to the limit with ∆t → 0 and k → +∞, we obtain the stated results, i.e : lim ∆t→0 ( lim k→∞ e (∣∣xmk+1∣∣2)) = 0 � https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 18 5. numerical simulations and residual calculations in this section, we present some numerical simulations for vasicek and geometric brownianmotion models using matlab and we calculate the errors between the exact solution and thatobtained by applying the numerical schemes of euler-maruyama, milshtein and implicit euler-maruyama. 5.1. numerical simulation of vasicek and geometric brownian motion models. we present somesimulations of vasicek and brownian geometric motion models in the increasing and decreasingcases. figure 1. increasing vasicek model https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 19 figure 2. decreasing vasicek model figure 3. increasing geometric brownian motion model https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 20 figure 4. decreasing geometric brownian motion model 5.2. interpretation of results. 5.2.1. vasicek model. the figures 1 and 2 show the stability of the vasicek model in the increasingand decreasing cases, in these figures we see that the euler-maruyama scheme coincides with thatof milshtein. we have in the first two figures of figure 1 the following errors: emerr = 0.2280,milerr = 0.2280 and iemerr = 0.2258 in both figures of figure 1 emerr = 0.3007, milerr = 0.3007and iemerr = 0.2851in both figures of figure 2 emerr = 0.2268, milerr = 0.2268 and iemerr = 0.2237. 5.2.2. geometric brownian motion model. the figures figure 3 et figure 4 present the stability ofgeometric brownian motion in the increasing and decreasing cases. indeed, the first three figuresin figure 3 present the increasing stability of geometric motion and the last figure in figure 3 andthe two figures in figure 4 show the decreasing stability of the model. we have in the first twofigures and figure 3 the following errors: emerr = 0.0027, milerr = 0.0011 and iemerr = 0.0013and for the third figure in figure 3: emerr = 0.0177, milerr = 0.0111 and iemerr = 0.0128. in theboth figures of figure 4: emerr = 0.0054, milerr = 0.0022 and iemerr = 0.0026. remark 5.1. from the results bellow, in the cases of increasing and decreasing stabilities of vasicek et geometric brownian motion, we have that, the milshtein scheme is the best scheme because it’s the best approximates the exact solution. 6. conclusion we have presented in this article the analysis of the stability in mean and mean-square forvasicek and geometric brownian motion models. in these models, we established the conditionsof the numerical stabilities of euler-maruyama, implicit euler-maruyama and milshtein schemes.these conditions have been proved by using classical manner and y. saito’s approach. it should benoted that each case is different from the other depending on whether the models examined have https://doi.org/10.28924/ada/ma.3.8 eur. j. math. anal. 10.28924/ada/ma.3.8 21 additive (vasicek model) or multiplicative (geometric brownian motion) white noise type. finally, for these models, we found that the stability conditions of the vasicek model coincideswith the stability of the odes, on the other hand, for the stability conditions of the second modelto coincide with the stability of the odes, it is necessary that θ2 < 0. to support these results,numerical simulations were made and the calculations of the residuals (errors) comes in support ofthe results found. in the next work we will analyze the numerical stabilities of these two modelsby using non-standard euler-maruyama scheme. references [1] a.m. lyapunov, the general problem of the stability of motion, int. j. control. 55 (1992) 531?534. https://doi. org/10.1080/00207179208934253.[2] i. kats, on the stability in first approximation of systems with random lag, j. appl. math. mech. 31 (1967) 478?482. https://doi.org/10.1016/0021-8928(67)90030-5.[3] i.i. gihman, a.v. skorohod, stochastic differential equations, springer berlin heidelberg, 1972. https://doi.org/ 10.1007/978-3-642-88264-7.[4] a. friedman, stochastic differential 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269?278. https://doi.org/10.1016/0898-1221(94)00197-9. https://doi.org/10.28924/ada/ma.3.8 https://doi.org/10.1080/00207179208934253 https://doi.org/10.1080/00207179208934253 https://doi.org/10.1016/0021-8928(67)90030-5 https://doi.org/10.1007/978-3-642-88264-7 https://doi.org/10.1007/978-3-642-88264-7 https://doi.org/10.1016/b978-0-12-268201-8.50010-4 https://openlibrary.org/books/ol2391851m https://openlibrary.org/books/ol2391851m https://doi.org/10.1007/978-3-540-69826-5 https://doi.org/10.1007/978-3-540-69826-5 https://doi.org/10.1007/978-0-387-75839-8 https://doi.org/10.1007/978-0-387-75839-8 https://doi.org/10.1007/3-540-34837-9 https://doi.org/10.1007/3-540-34837-9 https://doi.org/10.1515/mcma.1995.1.4.279 https://doi.org/10.1016/0898-1221(94)00197-9 eur. j. math. anal. 10.28924/ada/ma.3.8 22 [19] y. saito, t. mitsui, stability analysis of numerical schemes for stochastic differential equations, siam j. numer.anal. 33 (1996) 2254-2267. https://doi.org/10.1137/s0036142992228409.[20] k. burrage, p. burrage, t. mitsui, numerical solutions of stochastic differential equations ? implementation and sta-bility issues, j. comput. appl. math. 125 (2000) 171?182. https://doi.org/10.1016/s0377-0427(00)00467-2.[21] y. saito, t. mitsui, t-stability of numerial scheme for stochastic differential equations, contribut. numer. math.(1993) 333?344. https://doi.org/10.1142/9789812798886_0026.[22] y. saito, t. mitsui, mean-square stability of numerical schemes for stochastic differential systems, vietnam j. math.30 (2002) 551-560.[23] r. sakthivel, p. revathi, n.i. mahmudov, asymptotic stability of fractional stochastic neutral differential equationswith infinite delays, abstr. appl. anal. 2013 (2013) 769257. https://doi.org/10.1155/2013/769257.[24] x. mao, exponential stability of large-scale stochastic differential equations, syst. control lett. 19 (1992) 71?81. https://doi.org/10.1016/0167-6911(92)90042-q.[25] r. khasminskii, stochastic stability of differential equations, springer berlin heidelberg, (2012). https://doi. org/10.1007/978-3-642-23280-0.[26] r.m. sheldon, variations sur le mouvement brownien:introduction aux modeles de probabilite (11e edition), elsevier,amsterdam, (2014). https://doi.org/10.28924/ada/ma.3.8 https://doi.org/10.1137/s0036142992228409 https://doi.org/10.1016/s0377-0427(00)00467-2 https://doi.org/10.1142/9789812798886_0026 https://doi.org/10.1155/2013/769257 https://doi.org/10.1016/0167-6911(92)90042-q https://doi.org/10.1007/978-3-642-23280-0 https://doi.org/10.1007/978-3-642-23280-0 1. introduction 2. preliminary notions 2.1. stochastic differential equation and stabilities 2.2. stochastic numerical schemes 3. numerical stabilities of vasicek model 3.1. explicit solution 3.2. euler-maruyama scheme stabilities 3.3. milshtein's scheme stabilities 3.4. implicit euler-maruyama scheme stabilities 4. numerical stabilities of geometric brownian motion 4.1. explicit solution of the model 4.2. euler-maruyama scheme stabilities 4.3. milshtein's scheme stabilities 4.4. implicit euler-maruyama scheme stabilities 5. numerical simulations and residual calculations 5.1. numerical simulation of vasicek and geometric brownian motion models 5.2. interpretation of results 6. conclusion references bibliographie ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 1-18doi: 10.28924/ada/ma.1.1 existence and stability results for second-order neutral stochastic differential equations with random impulses and poisson jumps k. ravikumar1, k. ramkumar1, dimplekumar chalishajar2,∗ 1department of mathematics, psg college of arts and science, coimbatore, 641 046, india; ravikumarkpsg@gmail.com, ramkumarkpsg@gmail.com2department of applied mathematics, mallory hall, virginia military institute, lexington, va 24450, usa ∗correspondence: chalishajardn@vmi.edu abstract. the objective of this paper is to investigate the existence and stability results of second-order neutral stochastic functional differential equations (nsfdes) in hilbert space. initially, weestablish the existence results of mild solutions of the aforementioned system using the banachcontraction principle. the results are formulated using stochastic analysis techniques. in the laterpart, we investigate the stability results through the continuous dependence of solutions on initialconditions. 1. introduction stochastic differential equations (sdes) captures disturbances from random factors. mathe-matical models obtained by integrating stochastic process provide a better understanding of thereal-world system [12]. for elementary study of stochastic differential equations, the reader mayrefer to [7, 12, 14, 24].impulsive differential equations also attracted the attention of researchers (see [4, 11, 13, 21, 22]etc.). impulse in general occurs as deterministic or random models. nevertheless by naturalphenomena, the impulses often occur at random time points. many researches have been undergonesolving various differential equations with fixed time impulses [1, 9, 16, 23]. random impulsivedifferential equations involving fractional derivative are also studied see [20, 25].it is known that impulsive stochastic differential equations play a vital role in modelling practicalprocesses. not only from guassian white noise there are certain other factors that results in therise of random effects. random impulsive stochastic differential equations (isdes) are widely used received: 20 aug 2021. key words and phrases. existence; stability; banach contraction principle; second-order neutral stochastic functionaldifferential equations; random impulse; stochastic differential system.1 https://adac.ee https://doi.org/10.28924/ada/ma.1.1 https://orcid.org/0000-0002-6146-5544 eur. j. math. anal. 1 (2021) 2 in the fields of medicine, biology, economy, finance and so on. for example, the classical stockprice model [28]. d[s(t)] = fs(t)dt + σs(t)dw(t), t ≥ 0, t 6= τk, s(τk) = aks(τ−k ), k = 1, 2, ..., s(0) = s0, is described using an isdes. here wt is a brownian motion or wiener process, s(t) representsthe price of the stock at time t, and {τk} represents the release time of the important informationrelating to the stock. s(τ−k ) = limt→τk−0 s(t) and s0 ∈ r. in reality, {τk} is a sequence ofrandom variables, which satisfies 0 < τ2 < τ3 < · · · . recently, in [10] the authors have contributedthe existence and hyers-ulam stability of mild solutions for random impulsive stochastic functionalordinary differential equations which are studied using krasnoselskii’s fixed point theorem.solving second-order differential equations has been observed by many scholars. many authorssolved second-order stochastic differential equations see [5, 6, 8, 19]. however, there are not manypapers considering the existence and stability results on stochastic differential equations withrandom impulse. anguraj et.al [3], considered the sdes with random impulses and poisson jumpsof the form d[x(t)] = f(t,xt) + g(t,xt)dw(t) + ∫ u h(t,xt,u)ñ(dt,du), t ≥ t0, t 6= τk, x(ζk) = bk(τk)x(ζ−k ), k = 1, 2, ..., xt0 = ζ = {ζ(θ) : −τ ≤ θ ≤ 0} . the authors studied the existence, uniqueness, and stability through continuous dependence oninitial conditions for sdes with random impulses and poisson jumps by using banach fixed pointtheorem. very recently, anguraj et.al [2] investigated the existence and hyers ulam stability ofrandom impulsive stochastic functional integrodifferential equations with finite delays.motivated by the above discussion, here we consider the following second-order nsfdes withrandom impulses and poisson jumps. d [ x′(t) −h(t,xt)] = [ax(t) + f(t,xt)] dt + g(t,xt)dω(t) + ∫ u σ(t,xt,u)ñ(dt,du), , t ≥ t0, t 6= ξk, x(ξk) = bk(δk)x(ξ−k ), x′(ξk) = bk(δk)x′(ξ−k ), k = 1, 2, ..., (1.1) xt0 = φ, x′(t0) = φ, where a : d(a) ⊂ h → h is the infinitesimal generator of a strongly continuous cosine family {c(t), t ≥ 0}. w(t) is a given q-wiener process with a finite trace nuclear covariance operator q > 0. δk is a random variable defined from ω to d ≡ (0,dk) for k = 1, 2 · · · . suppose that δi and δj are independent of each other as i 6= j, (i, j = 1, 2, · · · ). the impulsive moments ξk are randomvariables and satisfy ξk = ξk−1 + δk, k = 1, 2, · · · . obviously, {ξk} is a process with independentincrements. 0 < t0 = ξ0 < ξ2 < ξ3 < · · · < limk→∞ξk = ∞, and x(ξ−k ) = limt→ξk−0 x(t). bk : dk → h, eur. j. math. anal. 1 (2021) 3 for each k = 1, 2, · · · . the time history xt(θ) = {x(t + θ) : −δ ≤ θ ≤ 0} with some given δ > 0.moreover, h, f,g,σ, and φ,φ will be specified later.to the best of authors knowledge, up to now, no work has been reported to derive the second-order nsfdes with random impulses and poisson jumps. the main contributions are summarizedas follows:(1) second-order nsfdes with random impulses and poisson jumps is formulated.(2) initially, we establish the existence results of mild solutions of the aforementioned system usingbanach contraction principle.(3) next, we investigate the stability results through continuous dependence of solutions on initialconditions.(4) an example is provided to illustrate the obtained theoretical results.the rest of the paper is organised as follows. section 2 is devoted to basic definitions, notions andlemma. in section 3, existence of mild solutions of the aforementioned system (1.1) is investigatedusing banach contraction principle. eventually in section 4, the stability of mild solution is obtainedthrough continuous dependence of solutions on initial conditions. 2. preliminaries let (ω,=,p) be a complete probability space equipped with the normal filtration {=}t≥t0 . =t0containing all p-null sets. h and k be two real hilbert spaces. l(h, k) denotes the space of allbounded linear operators from k to h.we may assume that, {n(t), t ≥ t0} be a counting process generated by {ξk, k ≥ 0}. =(1)t denotethe minimal σ algebra denoted by {n(r), r ≤ t} and denote =(2)t the σ-algebra generated by {ω(s),s ≤ t}. we assume that =(1)∞ ,=(2)∞ and ξ are mutually independent and =t = =(1)t ∨=(2)t .we assume that there exist a complete orthonormal system {en}∞n=1 in k, a bounded sequenceof non-negative real numbers λn such that, qen = λnen, n = 1, 2, · · · . let {βn(t)}(n = 1, 2, 3...) bea sequence of real valued one dimensional standard brownian motion mutually independent over (ω,=,p). a q-wiener process can be defined by ω(t) = ∞∑ n=1 √ λnβn(t)en, (t ≥ 0). set φ ∈l(k, h) we define, ∥∥φ∥∥2q = tr(φqφ∗) = ∞∑ n=1 ∥∥∥√λnφen∥∥∥2 if ∥∥φ∥∥2q < ∞, then φ is called a q-hilbert-schmidt operator. let lq(k, h) denote the space ofall q-hilbert-schmidt operator φ : k → h. the completion lq(k, h) of l(k, h) with respect tothe topology induced by the norm ∥.∥q, where ∥∥φ∥∥2q = 〈φ, φ〉 is a hilbert space.let t ∈ (t0, +∞), j := [t0,t ], jk = [ξk,ξk+1) , k = 0, 1, · · · , j̃ = {t : t ∈ j, t 6= ξk, k = 1, 2, · · ·}. l2(ω, h) be the collection of square integrable =t-measurable, h-valued random variables definedby the norm ∥x∥l2 = (e∥x∥2)12 , the expectation being expressed by the form e∥x∥2 = ∫ω ∥x∥2 dp.let pc (j,l2(ω, h)) = {x : j → l2(ω, h)}, x is continuous on every jk, and the left limits x(ξ−k ),x′(ξ−k ) exist k = 1, 2, · · · be a piecewise continuous space. eur. j. math. anal. 1 (2021) 4 we may define the space c = c ([−δ, 0], h) which contains all piecewise continuous functionsmapping from [−δ, 0] to h with the norm ∥x∥t = sup t−δ≤s≤t ∥∥x(s)∥∥ for each t ≥ t0. b be the banachspace, b([t0−δ,t ],l2(ω, h)) consists of continuous, =t-measurable, c-valued processes. the normis defined by ∥x∥b = (sup t∈j e ∥x∥2t)12 . in (1.1), ñ(dt,du) = n(dt,du)−dtv(du) denotes the compensated poisson measure independentof ω(t) and n(dt,du) represents the poisson counting measure associated with a characteristicmeasure v. for a basic study on the poisson jumps we refer to the book by [27].subsequently, we introduce certain definitions of sine and cosine operators.a bounded linear operators family {c(t), t ∈ r} is called a strongly continuous cosine family ifand only if(i) c(0) = i (i is the identity operator in h);(ii) c(t)x is continuous in t, for all x ∈ h;(iii) c(t + s) + c(t −s) = 2c(t)c(s) for all t,s ∈ r.the corresponding strongly continuous sine family {s(t), t ∈ r} is defined by s(t)x = ∫ t0 c(s)xds, x ∈ h, t ∈ rthen the following property holds: a ∫ t t0 s(s)xds = [c(t) −c(t0)] x lemma 2.1. [18] let {c(t), t ∈ r} be a strongly continuous cosine family in h, then for all s,t ∈ r, the following results are true: (i) c(t) = c(−t); (ii) s(s + t) + s(s− t) = 2s(s)c(t); (iii) s(s + t) = s(s)c(t) + s(t)c(s); (iv) s(t) = −s(−t); (v) c(t + s) + c(s− t) = 2c(s)c(t); (vi) c(t + s) −c(t −s) = 2as(t)s(s). before investigating mild solution (1.1), we consider the second-order neutral functional differ-ential equation, which is given byd[u′(t) −g(t,u(t))] = autdt + f(t,ut)dt, t ≥ 0,u0 = φ ∈ c,u′(0) = φ ∈ h, t ∈ (−r, 0], (2.1) where a is the infinitesimal generator of a strongly continuous cosine family {c(t), t ∈ r+} andthe functions g, f ∈l1(0,t ; h). eur. j. math. anal. 1 (2021) 5 lemma 2.2. [15] a continuously differentiable function u(t) : [0,t ] → h is called the mild solution for the cauchy problem (2.1), if it satisfies, u(t) = c(t)φ(0) + s(t)[φ −g(0,φ)] + ∫ t0 c(t −s)g(s,us)ds + ∫ t 0 s(t −s)f(s,xs)ds, t ≥ 0, where s(t) = 12πi ∫ γ eλtr(λ2; a)dλ; c(t) = 12πi ∫ γ eλtλr(λ2; a)dλ, and γ is a suitable path. consider the linear second-order linear differential equation with impulse conditions, u′′(t) = au(t) + f(t), t ≥ 0, t 6= tk, u(0) = u0,u′(0) = v0, u(tk) = bku(t−k ),u′(tk) = bku′(t−k ), k = 1, 2, · · · , (2.2) where 0 = t0 < t1 < t2 < · · · < tk < · · · ,{tk, k ≥ 1} is a sequence of fixed impulsive points, f(t) : [0,t) → h is an integrable function. lemma 2.3. the piecewise continuous differentiable function u(t) : [0,t ] → h is a mild solution of (2.2), if and only if x(t) satisfies the integral equation u(t) = k∏ i=1 bic(t)u0 + k∏ i=1 bis(t)v0 + k∑ i=1 k∏ j=i bj ∫ ti ti−1 s(t −s)f(s)ds × ∫ t tk s(t −s)f(s)ds, t ∈ [tk, tk+1), k = 0, 1, · · · . (2.3) proof. (i)for t ∈ [0, t1), the mild solution is studied in [17], u(t) = c(t)u0 + s(t)v0 + ∫ t0 s(t −s)f(s)ds, t ∈ [0, t1).(ii) for t ∈ [t1, t2), we set u(t) = c(t − t1)u(t1) + s(t − t1)u′(t1) + ∫ t t1 s(t −s)f(s)ds, t ∈ [t1, t2). (2.4) since, u(t1) = b1u(t−1 ), u′(t1) = b1u′(t−1 ),and from (i) we know u(t−1 ) = c(t1)u0 + s(t1)v0 + ∫ t10 s(t1 −s)f(s)ds; (2.5) u′(t−1 ) = as(t1)u0 + c(t1)v0 + ∫ t10 c(t1 −s)f(s)ds. (2.6) eur. j. math. anal. 1 (2021) 6 thus, u(t) = b1c(t − t1)c(t1)u0 + b1s(t − t1)as(t1)u0 + b1c(t − t1)s(t1)v0 + b1s(t − t1)c(t1)v0 + b1c(t − t1)∫ t10 s(t1 −s)f(s)ds + b1s(t − t1) ∫ t1 0 s(t1 −s)f(s)ds + ∫ t t1 s(t −s)f(s)ds, t ∈ [t1, t2). applying lemma 2.1, we get u(t) = b1c(t)u0 + b1s(t)v0 + b1 ∫ t10 s(t1 −s)f(s)ds + ∫ t t1 s(t −s)f(s)ds, t ∈ [t1, t2). (iii) for t ∈ [t2, t3), u(t) = c(t − t2)u(t2) + s(t − t2)u′(t2) + ∫ t t2 s(t −s)f(s)ds = c(t − t2)b2u(t−2 ) + s(t − t2)b2u′(t−2 ) + ∫ t t2 s(t −s)f(s)ds. (2.7) from the conclusions of (ii), it is known that, u(t−2 ) = b1c(t2)u0 + b1s(t2)v0 + b1 ∫ t20 s(t2 −s)f(s)ds + ∫ t2 t1 s(t2 −s)f(s)ds; (2.8) u′(t−2 ) = b1as(t2)u0 + b1c(t2)v0 + b1 ∫ t20 c(t2 −s)f(s)ds + ∫ t2 t1 c(t2 −s)f(s)ds (2.9) along with (2.7) and using lemma 2.1, we have u(t) = b2b1c(t)u0 + b2b1s(t)v0 + b2b1 ∫ t10 s(t −s)f(s)ds + b2 ∫ t2 t1 s(t −s)f(s)ds + ∫ t2 t1 s(t −s)f(s)ds, t ∈ [t2, t3) similarly, for all t ∈ [tk, tk−1). x(t) = k∏ i=1 bic(t)u0 + k∏ i=1 bis(t)v0 + k∑ i=1 k∏ j=i bj ∫ ti ti−1 s(t −s)f(s)ds + ∫ t ξk s(t −s)f(s)ds. � by lemma 2.2, lemma 2.3 the mild solution of the system (1.1) applying index function for t ∈ j. definition 2.1. for a given t ∈ (t0, +∞), a =-adapted process function {x ∈b, t0 −δ ≤ t ≤ t} is called a mild solution of system (1.1), if (i) xt0 (s) = φ(s) ∈l02(ω,b) for δ ≤ s ≤ 0; (ii) x′(t0) = φ(t) ∈l02(ω, h) for t ∈ j; (iii) the functions f(s,xt),g(s,xt),h(s,xt) and σ(s,xs,u) are integrable, and for a.e. t ∈ j, the eur. j. math. anal. 1 (2021) 7 following integral equation is satisfied. x(t) = +∞∑k=0 [ k∏ i=1 bi(δi)c(t − t0)φ(0) + k∏ i=1 bi(δi)s(t − t0)[φ −h(0,φ)] + k∑ i=1 k∏ j=i bj(δj) × ∫ ξi ξi−1 c(t −s)h(s,xs)ds + ∫ t ξk c(t −s)h(s,xs)ds + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s) × f(s,xs)ds + ∫ t ξk s(t −s)f(s,xs)ds + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s)g(s,xs)dω(s) + ∫ t ξk s(t −s)g(s,xs)dω(s) + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 ∫ u s(t −s)σ(s,xs,u)n(ds,du) + ∫ t ξk ∫ u s(t −s)σ(s,xs,u)n(ds,du)]i[ξk,ξk+1)(t), t ∈ [t0,t ]. (2.10) where, k∏ j=i bj(δj) = bk(δk)bk−1(δk−1) · · ·bi(δi), and i(a)(.) is the index function, i.e., ia(t) = 1, if t ∈ a,0, if t /∈ a. lemma 2.4. for any p ≥ 1, and for lq(k, h)-valued predictable process u(.) such that, sup s∈[0,t ] e ∥∥∥∥∫ s0 u(η)dω(η) ∥∥∥∥2p ≤ (p(2p− 1))p (∫ t0 ( e ∥∥u(s)∥∥2pq )1/p ds)p , t ∈ j. 3. existence results of mild solution to prove the existence of mild solutions of random impulsive stochastic differential equations,the following assumptions are to be made.(h1) c(t), s(t)(t ∈ j) are equicontinuous and there exist positive constants m, m̃ such that sup t∈j ∥∥c(t)∥∥ ≤m, sup t∈j ∥∥s(t)∥∥ ≤m̃. (3.1) (h2) the functions f : j × c → h; h : j × c → h; g : j × c →lq(k, h) and σ : j × c ×u → h e ∥∥f(t,xt) − f(t,yt)∥∥2 ≤lf ∥∥x −y∥∥2t , e ∥∥g(t,xt) −g(t,yt)∥∥2 ≤lg ∥∥x −y∥∥2t , e ∥∥h(t,xt) −h(t,yt)∥∥2 ≤lh ∥∥x −y∥∥2t ,∫ u e ∥∥σ(t,xt,u) −σ(t,yt,u)∥∥2 v(du)ds∨∫ u ( e ∥∥σ(t,xt,u) −σ(t,yt,u)∥∥4 v(du)ds)12 ≤lσ ∥∥x −y∥∥2t ,∫ u ( e ∥∥σ(t,xt,u) −σ(t,yt,u)∥∥4 v(du)ds)12 ≤lσ ∥x∥2t . eur. j. math. anal. 1 (2021) 8 (h3) for all t ∈ j, there exist constants κf,κg,κh,κσ ∈l′(j,r+) such that, e ∥∥f(t, 0)∥∥2 ≤ κf, e∥∥g(t, 0)∥∥2 ≤ κg, e ∥∥h(t, 0)∥∥2 ≤ κh, e∥∥σ(t, 0,u)∥∥2 ≤ κσ. (h4) e maxi,k { k∏ j=i ∥∥bj(δj)∥∥}  is uniformly bounded then there exist constant n for all δj ∈ djsuch that e maxi,k { k∏ j=i ∥∥bj(δj)∥∥}  ≤n . theorem 3.1. if assumptions (h1)-(h4) gets satisfied then there exist a unique continuous mild solution of the system (1.1). proof. we define an operator φ : b →b by φx such that, (φx)(t) =  φ(t), t ∈ [t0 −δ,t0],+∞∑ k=0 [ k∏ i=1 bi(δi)c(t − t0)φ(0) + k∏ i=1 bi(δi)s(t − t0)[φ −h(0,φ)] + k∑ i=1 k∏ j=i bj(δj) × ∫ ξi ξi−1 c(t −s)h(s,xs)ds + ∫ t ξk c(t −s)h(s,xs)ds + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s) × f(s,xs)ds + ∫ tξk s(t −s)f(s,xs)ds + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s)g(s,xs)dω(s) +∫ t ξk s(t −s)g(s,xs)dω(s) + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 ∫ u s(t −s)σ(s,xs,u)n(ds,du) +∫ t ξk ∫ u s(t −s)σ(s,xs,u)n(ds,du)]i[ξk,ξk+1)(t), t ∈ [t0,t ]. we need to prove that φ maps b into itself. e ∥∥(φx)(t)∥∥2 ≤ e∥∥∥∥ +∞∑k=0 [ k∏ i=1 bi(δi)c(t − t0)φ(0) + k∏ i=1 bi(δi)s(t − t0)[φ −h(0,φ)] + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 c(t −s)h(s,xs)ds + ∫ t ξk c(t −s)h(s,xs)ds eur. j. math. anal. 1 (2021) 9 + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s)f(s,xs)ds + ∫ t ξk s(t −s)f(s,xs)ds + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s)g(s,xs)dω(s) + ∫ t ξk s(t −s)g(s,xs)dω(s) + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 ∫ u s(t −s)σ(s,xs,u)n(ds,du) + ∫ t ξk ∫ u s(t −s)σ(s,xs,u)n(ds,du)]i[ξk,ξk+1)(t)∥∥∥∥2 ≤ 6e[+∞∑k=0 [ k∏ i=1 ∥∥bi(δi)∥∥∥∥c(t − t0)∥∥∥∥φ(0)∥∥ ] i[ξk,ξk+1)(t) ]2 + 6e[ +∞∑k=0 [ k∏ i=1 ∥∥bi(δi)∥∥∥∥s(t − t0)∥∥ × ∥∥φ −h(0,φ)∥∥]i[ξk,ξk+1)(t)]2 + 6e[ +∞∑k=0 [ k∏ j=i ∥∥bj(δj)∥∥∫ ξi ξi−1 ∥∥c(t −s)∥∥∥∥h(s,xs)∥∥ds + ∫ t ξk ∥∥c(t −s)∥∥∥∥h(s,xs)∥∥ds]i[ξk,ξk+1)(t)]2 + 6e[ +∞∑k=0 [ k∏ j=i ∥∥bj(δj)∥∥∫ ξi ξi−1 ∥∥s(t −s)∥∥ × ∥∥f(s,xs)∥∥ds + ∫ t ξk ∥∥s(t −s)∥∥∥∥f(s,xs)∥∥ds]i[ξk,ξk+1)(t)]2 + 6e[ +∞∑k=0 [ k∏ j=i ∥∥bj(δj)∥∥ × ∫ ξi ξi−1 ∥∥s(t −s)∥∥∥∥g(s,xs)∥∥dω(s) + ∫ t ξk ∥∥s(t −s)∥∥∥∥g(s,xs)∥∥dω(s)]i[ξk,ξk+1)(t) + 6e[ +∞∑k=0 [ k∏ j=i ∥∥bj(δj)∥∥×∫ ξi ξi−1 ∫ u ∥∥s(t −s)∥∥∥∥σ(s,xs,u)∥∥ñ(ds,du) + ∫ t ξk ∫ u ∥∥s(t −s)∥∥∥∥σ(s,xs,u)∥∥ñ(ds,du)]i[ξk,ξk+1)(t)]2, = 6 6∑ i=1 gi. where, g1 ≤ e [+∞∑ k=0 [ k∏ i=1 ∥∥bi(δi)∥∥∥∥c(t − t0)∥∥∥∥φ(0)∥∥ ] i[ξk,ξk+1)(t) ]2 ≤ m2e maxi,k { k∏ j=i ∥∥bj(δj)∥∥}  2 e ∥∥φ(0)∥∥2 ≤ m2n2e∥∥φ(0)∥∥2 , g2 ≤ e [ +∞∑ k=0 [ k∏ i=1 ∥∥bi(δi)∥∥∥∥s(t − t0)∥∥∥∥φ −h(0,φ)∥∥]i[ξk,ξk+1)(t)]2 eur. j. math. anal. 1 (2021) 10 ≤ m̃2e maxi,k { k∏ j=i ∥∥bj(δj)∥∥}  2 e ∥∥φ −h(0,φ)∥∥2 ≤ m̃2n2e∥∥φ −h(0,φ)∥∥2 , g3 ≤ e [ +∞∑ k=0 [ k∏ j=i ∥∥bj(δj)∥∥∫ ξi ξi−1 ∥∥c(t −s)∥∥∥∥h(s,xs)∥∥ds + ∫ t ξk ∥∥c(t −s)∥∥∥∥h(s,xs)∥∥ds]i[ξk,ξk+1)(t)]2 ≤ m2e maxi,k {1, k∏ j=i ∥∥bj(δj)∥∥}  2 (t − t0)∫ t t0 e ∥∥h(s,xs)∥∥2 ds ≤ 2m2 max{1,n2}(t − t0)[∫ t t0 e ∥∥h(s,xs) −h(s, 0)∥∥2 ds + ∫ t t0 ∥∥h(s, 0)∥∥2 ds] ≤ 2m2 max{1,n2}(t − t0)∫ t t0 [ lhe ∥x∥2s + κh]ds ≤ 2m2 max{1,n2}(t − t0)∫ t t0 lhe ∥x∥2s ds + 2m2 max{1,n2}(t − t0)2κh, g4 ≤ e [ +∞∑ k=0 [ k∏ j=i ∥∥bj(δj)∥∥∫ ξi ξi−1 ∥∥s(t −s)∥∥∥∥f(s,xs)∥∥ds + ∫ t ξk ∥∥s(t −s)∥∥∥∥f(s,xs)∥∥ds]i[ξk,ξk+1)(t)]2 ≤ m̃2e maxi,k {1, k∏ j=i ∥∥bj(δj)∥∥}  2 (t − t0)∫ t t0 e ∥∥f(s,xs)∥∥2 ds ≤ 2m̃2 max{1,n2}(t − t0)[∫ t t0 e ∥∥f(s,xs) − f(s, 0)∥∥2 ds + ∫ t t0 ∥∥f(s, 0)∥∥2 ds] ≤ 2m̃2 max{1,n2}(t − t0)∫ t t0 [ lfe ∥x∥2s + κf]ds ≤ 2m̃2 max{1,n2}(t − t0)∫ t t0 lfe ∥x∥2s ds + 2m̃2 max{1,n2}(t − t0)2κf, g5 ≤ e [ +∞∑ k=0 [ k∏ j=i ∥∥bj(δj)∥∥∫ ξi ξi−1 ∥∥s(t −s)∥∥∥∥g(s,xs)∥∥dω(s) + ∫ t ξk ∥∥s(t −s)∥∥∥∥g(s,xs)∥∥dω(s)]i[ξk,ξk+1)(t)]2 ≤ m̃2e maxi,k {1, k∏ j=i ∥∥bj(δj)∥∥}  2 ∫ t t0 e ∥∥g(s,xs)∥∥2 ds ≤ 2m̃2 max{1,n2}tr(q)[∫ t t0 e ∥∥g(s,xs) −g(s, 0)∥∥2 ds + ∫ t t0 ∥∥g(s, 0)∥∥2 ds] ≤ 2m̃2 max{1,n2}tr(q)∫ t t0 [ lge ∥x∥2s + κg]ds ≤ 2m̃2 max{1,n2}tr(q)∫ t t0 lge ∥x∥2s ds + 2m̃2 max{1,n2}(t − t0)tr(q)κg. eur. j. math. anal. 1 (2021) 11 g6 ≤ e [ +∞∑ k=0 [ k∏ j=i ∥∥bj(δj)∥∥×∫ ξi ξi−1 ∫ u ∥∥s(t −s)∥∥∥∥σ(s,xs,u)∥∥ñ(ds,du) + ∫ t ξk ∫ u ∥∥s(t −s)∥∥∥∥σ(s,xs,u)∥∥ñ(ds,du)]i[ξk,ξk+1)(t)]2 ≤ 2m̃2 max{1,n2}∫ t t0 ∫ u [ e ∥∥σ(s,xs,u) −σ(s, 0,u)∥∥2 + ∥∥σ(s, 0,u)∥∥2]ds + 2m̃2 max{1,n2}(∫ t t0 ∫ u e ∥∥σ(s,xs,u)∥∥4 v(du)ds)12 ≤ 4m̃2 max{1,n2}∫ t t0 lσe ∥x∥2s ds + 2m̃2 max{1,n2}(t − t0)κσ, thus we would obtain, e ∥∥(φx)(t)∥∥2t ≤ 6m2n2e∥∥φ(0)∥∥2 + 6m̃2n2e∥∥φ −h(0,φ)∥∥2 + 12m2 max{1,n2}(t − t0) × ∫ t t0 lhe ∥x∥2s ds + 12m2 max{1,n2}(t − t0)2κh + 12m̃2 max{1,n2}(t − t0) × ∫ t t0 lfe ∥x∥2s ds + 12m̃2 max{1,n2}(t − t0)2κf + 12m̃2 max{1,n2}tr(q) × ∫ t t0 lge ∥x∥2s ds + 12m̃2 max{1,n2}(t − t0)tr(q)κg + 24m̃2 max{1,n2}∫ t t0 lσe ∥x∥2s ds + 12m̃2 max{1,n2}(t − t0)κσ. taking supremum over t, sup t0≤t≤t e ∥∥(φx)(t)∥∥2t ≤ 6m2n2e∥∥φ(0)∥∥2 + 6m̃2n2e∥∥φ −h(0,φ)∥∥2 + 12m2 max{1,n2}(t − t0) × ∫ t t0 lh sup t0≤t≤t e∥x∥2s ds + 12m2 max{1,n2}(t − t0)2κh + 12m̃2max{1,n2} × (t − t0)∫ t t0 lf sup t0≤t≤t e∥x∥2s ds + 12m̃2 max{1,n2}(t − t0)2κf + 12m̃2 × max{1,n2}tr(q)∫ t t0 lg sup t0≤t≤t e∥x∥2s ds + 12m̃2max{1,n2}(t − t0)tr(q)κg + 24m̃2 max{1,n2}∫ t t0 lσ sup t0≤t≤t e∥x∥2s ds + 12m̃2 max{1,n2}(t − t0)κσ eur. j. math. anal. 1 (2021) 12 ≤ 6m2n2e∥∥φ(0)∥∥2 + 6m̃2n2e∥∥φ −h(0,φ)∥∥2 + 12m2 max{1,n2}(t − t0)2 × lh sup t0≤t≤t e∥x∥2t + 12m2 max{1,n2}(t − t0)2κh + 12m̃2 max{1,n2} × (t − t0)2lf sup t0≤t≤t e∥x∥2t + 12m̃2 max{1,n2}(t − t0)2κf + 12m̃2 max{1,n2}tr(q)(t − t0)lg sup t0≤t≤t e∥x∥2t + 12m̃2 max{1,n2}(t − t0)tr(q)κg+ 24m̃2 max{1,n2}(t − t0)lσ sup t0≤t≤t e∥x∥2s ds + 12m̃2 max{1,n2}(t − t0)κσ ≤ 6[n2 [m2e∥∥φ(0)∥∥2 + m̃2e∥∥φ −h(0,φ)∥∥2]] + 12 max{1,n2}(t − t0) × [ m2(t − t0)κh + m̃2(t − t0)κf + m̃2tr(q)κg + m̃2κσ] + 12 max{1,n2} × (t − t0)[m2(t − t0)lh + m̃2(t − t0)lf + m̃2tr(q)lg + 2m̃2lσ]∥x∥2t∥∥φx∥∥2b ≤ c1 + c2 ∥x∥2b . where, c1 = 6[n2 [m2e∥∥φ(0)∥∥2 + m̃2e∥∥φ −h(0,φ)∥∥2]] + 12 max{1,n2}(t − t0) × [ m2(t − t0)κh + m̃2(t − t0)κf + m̃2tr(q)κg + m̃2κσ], c2 = 12 max{1,n2}(t − t0)[m2(t − t0)lh + m̃2(t − t0)lf + m̃2tr(q)lg + 2m̃2lσ] . where c1 and c2 are constants.hence φ is bounded.now we need to prove that φ is a contraction mapping. for any x,y ∈b we have,∥∥(φx)(t) − (φy)(t)∥∥2 ≤ ∥∥∥∥ +∞∑k=0 [ k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 c(t −s)h(s,xs)ds + ∫ t ξk c(t −s)h(s,xs)ds + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s)f(s,xs)ds + ∫ t ξk s(t −s)f(s,xs)ds + k∑ i=1 k∏ j=i bj(δj) × ∫ ξi ξi−1 s(t −s)g(s,xs)dω(s) + ∫ t ξk s(t −s)g(s,xs)dω(s) + k∑ i=1 k∏ j=i bj(δj) × ∫ ξi ξi−1 ∫ u s(t −s)σ(s,xs,u)ñ(ds,dt) + ∫ t ξk ∫ u s(t −s)σ(s,xs,u)ñ(ds,dt)]i[ξk,ξk+1)(t)∥∥∥∥2 − ∥∥∥∥ +∞∑k=0 [ k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 c(t −s)h(s,ys)ds + ∫ t ξk c(t −s)h(s,ys)ds eur. j. math. anal. 1 (2021) 13 + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s)f(s,ys)ds + ∫ t ξk s(t −s)f(s,ys)ds + k∑ i=1 k∏ j=i bj(δj) × ∫ ξi ξi−1 s(t −s)g(s,ys)dω(s) + ∫ t ξk s(t −s)g(s,ys)dω(s) + k∑ i=1 k∏ j=i bj(δj) × ∫ ξi ξi−1 ∫ u s(t −s)σ(s,ys,u)ñ(ds,dt) + ∫ t ξk ∫ u s(t −s)σ(s,ys,u)ñ(ds,dt)]i[ξk,ξk+1)(t)∥∥∥∥2 ≤ 4 max{1,n2}m2(t − t0)∫ t t0 ∥∥h(t,xs) −h(t,ys)∥∥2 ds + 4 max{1,n2} × m̃2(t − t0)∫ t t0 ∥∥f(t,xs) − f(t,ys)∥∥2 ds + 4 max{1,n2}m̃2 × ∫ t t0 ∥∥g(t,xs) −g(t,ys)∥∥2 ds + 4 max{1,n2}m̃2 × [∫ t t0 ∫ u ∥∥σ(t,xs,u) −σ(t,ys,u)∥∥2 v(du)ds + (∫ t t0 ∫ u ∥∥σ(t,xs,u) −σ(t,ys,u)∥∥4 v(du)ds)12 ] moreover, sup t0≤t≤t e ∥∥(φx)(t) − (φy)(t)∥∥2 ≤ 4 max{1,n2}m2(t − t0)2lh sup t0≤t≤t e ∥∥x −y∥∥2s ds + 4 max{1,n2} × m̃2(t − t0)2lf sup t0≤t≤t e ∥∥x −y∥∥2s ds + 4 max{1,n2}m̃2tr(q) × (t − t0)lg sup t0≤t≤t e ∥∥x −y∥∥2s ds + 4 max{1,n2}m̃2 × (t − t0)lσ sup t0≤t≤t e ∥∥x −y∥∥2s ds ≤ [4 max{1,n2}m2(t − t0)2lh + 4 max{1,n2}m̃2(t − t0) × [(t − t0)lf + tr(q)lg + lσ ]] sup t0≤t≤t e ∥∥x −y∥∥2t hence, ∥∥(φx) − (φy)∥∥2b ≤ γ(t)∥∥x −y∥∥2b .where, γ(t) = 3 max{1,n2}m2(t − t0)2lh + 3 max{1,n2}m̃2(t − t0) [(t − t0)lf + tr(q)lg + lσ ] . eur. j. math. anal. 1 (2021) 14 by taking suitable 0 < t1 < t sufficiently small such that, γ(t1) < 1.hence φ is a contraction on b . by banach contraction principle, a unique fixed point x ∈ b isobtained for the operator φ and therefore φx = x is a mild solution of the system.the solution can be extended to the entire interval (−δ,t ] in finitely many steps. thus the existenceand uniqueness of the mild solution on (−δ,t ] is proved. � 4. stability the stability through continuous dependence of solutions on initial conditions are established. definition 4.1. a mild solution xξ,x(t) of the system (1.1) with the initial value (ξ,x) is said to be stable in mean square if for all ε > 0 such that e ( sup0≤s≤t ∥∥∥xξ,x(s) −yξ,x(t)∥∥∥2) ≤ ε, when e∥∥ξ −ζ∥∥2 + e∥∥x −y∥∥2 < δ, where xζ,y(t) is another solution of the system (1.1) with initial value (ζ,y). theorem 4.1. let x(t) and x(t) be mild solution of the system (1.1) with the initial condition φ1 and φ2 respectively. if the assumptions of theorem 3.1 gets satisfied, the mean solution of the system (1.1) is stable in the mean square. proof. we may assume that x(t) and x(t) be the mild solutions of the system (1.1) with initialvalues φ1 and φ2 respectively. x(t) −x(t) = +∞∑k=0 [ k∏ i=1 bi(δi)c(t − t0)[φ1 −φ2] + k∏ i=1 bi(δi)s(t − t0)[(φ1 −φ2) − [(h(0,φ1) − (h(0,φ2))]] + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 c(t −s) [h(s,xs) −h(s,x(s))] ds + ∫ t ξk c(t −s) [h(s,xs) −h(s,xs)] ds + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s) [f(s,xs) − f(s,xs)] ds + ∫ t ξk s(t −s) [f(s,xs) − f(s,xs)] ds + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 s(t −s) [g(s,xs) −g(s,xs)] dω(s) + ∫ t ξk s(t −s) [g(s,xs) −g(s,xs)] dω(s) + k∑ i=1 k∏ j=i bj(δj) ∫ ξi ξi−1 ∫ u s(t −s) [σ(s,xs,u) −σ(s,xs,u)] ñ(ds,du) + ∫ t ξk ∫ u s(t −s) [σ(s,xs,u) −σ(s,xs,u)] ñ(ds,du)]i[ξk,ξk+1)(t) eur. j. math. anal. 1 (2021) 15 e ∥∥x(t) −x(t)∥∥2 ≤ 6n2m2e∥∥φ1 −φ2∥∥2 + 12n2m̃2e∥∥φ1 −φ2∥∥2 + 12n2m̃2e∥∥h(0,φ1) −h(0,φ2)∥∥2 + 6m2 max{1,n2}∫ t t0 e ∥∥h(s,xs) −h(s,xs)∥∥2 ds + 6m̃2 max{1,n2} × ∫ t t0 e ∥∥f(s,xs) − f(s,xs)∥∥2 ds + 6m̃2 max{1,n2}∫ t t0 e ∥∥g(s,xs) −g(s,xs)∥∥2 ds + 6 max{1,n2}m̃2 ×[∫ t t0 ∫ u ∥∥σ(t,xs,u) −σ(t,xs,u)∥∥2 v(du)ds + (∫ t t0 ∫ u ∥∥σ(t,xs,u) −σ(t,xs,u)∥∥4 v(du)ds)12 ] ≤ 6n2m2e∥∥φ1 −φ2∥∥2 + 10n2m̃2e∥∥φ1 −φ2∥∥2 + 12n2m̃2lhe∥∥φ1 −φ2∥∥2 + 6m2 max{1,n2}∫ t t0 lhe ∥∥x −x∥∥2s ds + 6m̃2 max{1,n2}∫ t t0 lfe ∥∥x −x∥∥2s ds + 6m̃2 max{1,n2}tr(q)∫ t t0 lge ∥∥x −x∥∥2s ds + 6m̃2 max{1,n2}∫ t t0 lσe ∥∥x −x∥∥2s ds furthermore, sup t0≤t≤t e ∥∥x −x∥∥2t ≤ 6n2m2e∥∥φ1 −φ2∥∥2 + 12n2m̃2e∥∥φ1 −φ2∥∥2 + 12n2m̃2lhe∥∥φ1 −φ2∥∥2 + 6m2 max{1,n2}(t − t0)lh sup t0≤t≤t e ∥∥x −x∥∥2t + 6m̃2 max{1,n2}(t − t0) × lf sup t0≤t≤t e ∥∥x −x∥∥2t + 6m̃2 max{1,n2}(t − t0)tr(q)lg sup t0≤t≤t e ∥∥x −x∥∥2t + 6m̃2 max{1,n2}(t − t0)lσ sup t0≤t≤t e ∥∥x −x∥∥2t sup t0≤t≤t e ∥∥x −x∥∥2t ≤ 6n2 [ m2 + m̃2lh] 1 − 6 max{1,n2}(t − t0)[m2lh + m̃2 [lf + tr(q)lg + lσ ]]e ∥∥φ1 −φ2∥∥2 + 12n2m̃21 − 6 max{1,n2}(t − t0)[m2lh + m̃2 [lf + tr(q)lg + lσ ]]e ∥∥φ1 −φ2∥∥2 ≤ ρe ∥∥φ1 −φ2∥∥2 + υe∥∥φ1 −φ2∥∥2 where, ρ = 5n2 [ m2 + m̃2lh] 1 − 5 max{1,n2}(t − t0)[m2lh + m̃2 [lf + tr(q)lg + lσ ]] υ = 10n2m̃21 − 5 max{1,n2}(t − t0)[m2lh + m̃2 [lf + tr(q)lg + lσ ]] eur. j. math. anal. 1 (2021) 16 given ε > 0,µ > 0 choose, λ = ερ,µ = συ such that, e ∥∥φ1 −φ2∥∥2 ≤ λ and e∥∥φ1 −φ2∥∥2 ≤ µ therefore, ∥∥x −y∥∥2b ≤ ε.thus the proof is complete. 5. illustration in this section, the results obtained are applied to a stochastic partial differential equations withrandom impulses. let us consider a space h = l2([0,π]). the infinitesimal generator a is definedto be a : d(a) ⊂ h → h by a = ∂2∂x2 , with the domain, d(a) = {z ∈ h | z and ∂z ∂x are absolutely continuous, ∂2z ∂x2 ∈ h,z(0) = z(π) = 0} . for z ∈ d(a),az = − ∞∑ n=1 n 2 < z,zn > zn, where {zn : n ∈ z} is an orthonormal basis of h, zn(x) := 1√2πeinx,n ∈ z+,x ∈ [0,π]. it is known that a generates strongly continuous operators c(t) and s(t) in a hilbert space h, such that c(t)z = ∞∑ n=1 cos(nt) < z,zn > zn, and s(t)z = ∞∑ n=1 sin(nt)/n < z,zn > zn, for t ∈ r. and we assume that s(t) is not a compact semigroupand θ(s(t)d) ≤ θ(d), where d ∈ h denotes a bounded set, θ is the hausdroff measure of non-compactness.in the sequel, we may consider second-order neutral stochastic functional differential equation ofthe form, ∂ ∂t [ ∂ ∂t z(t,x) − m15 ∫ 0 −r ε1(s)z(t + s,x)ds ] (5.1) = [ ∂2 ∂x2 z(t,x) + m25 ∫ 0 −r ε1(s)z(t + s)ds ] dt + m35 ∫ 0 −r ε3(s)z(t + s)dω(t) + m45 ∫ u ∫ 0 −r ε4(s)z(t + s)ñ(dt,du), t ≥ t0, t 6= ξk, x ∈ [0,π], z(ξk,x) = ρ(k)δkz(ξ−k ,x), k = 1, 2, 3..., (5.2) ∂ ∂t z(ξk,x) = ρ(k)δk ∂∂tz(ξ−k ,x), z(t0,x) = φ(θ,x),θ ∈ [−r, 0], x ∈ [0,π], r > 0, ∂ ∂t z(t0,x) = φ(x), x ∈ [0,π], z(t, 0) = z(t,π) = 0. eur. j. math. anal. 1 (2021) 17 let δk be a random variable defined on dk ≡ (0,dk) where, 0 < dk < +∞, for k = 1, 2, · · · . ξ0 = t0 > 0 and ξk = ξk−1 + δk for k = 1, 2, · · · . ω(t) denotes a standard cylindrical weiner process inh. furthermore, let ρ be a function of k. εi : [−r, 0] → r are positive functions and mi > 0 for i = 1, 2, 3, 4. ∥∥c(t)∥∥ ,∥∥s(t)∥∥ are bounded on r. ∥∥c(t)∥∥ ≤ e−π2t and ∥∥s(t)∥∥ ≤ e−π2t(t ≥ 0).we may assume that, (i)the function ε(θ) ≥ 0 is continuous on [−r, 0],∫ 0 −r ε2i (θ)dθ < ∞(i = 1, 2, 3, 4.) (ii)max i,k = { k∏ j=i e[ ∥∥ρ(j)δj∥∥2]} < n . using above assumptions and functions ε1,ε2,ε3,ρ we can show that lg = rm125 ∫ 0−r ε21(θ)dθ,lf = rm225 ∫ 0−r ε21(θ)dθ, lh = rm325 ∫ 0−r ε21(θ)dθ and lσ = rm425 ∫ 0−r ε21(θ)dθ. hence stability in mean squareof mild solution (5.1) is obtained. � 6. conclusion in this paper, the existence and stability results of second-order neutral stochastic functionalsystems with random impulse is presented. the existence results of aforementioned system is estab-lished using banach contraction principle. then the stability of mild solutions through continuousdependence of solutions on initial conditions are calculated. references [1] a. anguraj, m. mallika arjunan, e. hernández m, existence results for an impulsive neutral functional dif-ferential equation with state-dependent delay, appl. anal. 86 (2007) 861–872. https://doi.org/10.1080/ 00036810701354995.[2] a. anguraj, k. ramkumar, k. ravikumar, existence and hyers-ulam stability of random impulsive stochastic func-tional integrodifferential equations with finite delays, comput. methods differ. equ. 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lipschitz conditions, chinese. j. appl. probab. statist. 26 (2010), 347-356.[27] d. applebaum, levy process and stochastic calculus, cambridge university press, cambridge, 2009.[28] t. wang, s wu, random impulsive model for stock prices and its application for insurers, master thesis (in chinese),shanghai, east china normal university, 2008. https://doi.org/10.1080/17442508.2018.1551400 https://doi.org/10.1080/17442508.2018.1551400 https://doi.org/10.11948/20190089 https://doi.org/10.11948/20190089 https://doi.org/10.1016/j.na.2010.11.007 https://doi.org/10.1016/j.camwa.2006.04.026 https://doi.org/10.1007/s10255-004-0157-z https://doi.org/10.3934/mbe.2018069 https://doi.org/10.1002/asjc.918 https://doi.org/10.1186/s13662-018-1779-4 1. introduction 2. preliminaries 3. existence results of mild solution 4. stability 5. illustration 6. conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 6doi: 10.28924/ada/ma.3.6 fractionalization of hankel type integral transforms and their relevance b. b. waphare∗, r. z. shaikh mit arts, commerce and science college, alandi(d), pune, maharashtra, india balasahebwaphare@gmail.com, shaikhrahilanaz@gmail.com ∗correspondence: balasahebwaphare@gmail.com abstract. in this paper, the fractionalization of certain types of hankel transforms is suggested.barut-girardello type transforms are then introduced along with the relevant fractional order forms.finally some further generalizations are suggested. 1. introduction the theory of hankel transforms is very vast and it is studied by many researchers in recent aswell as in past. the forward and inverse transforms are completely symmetric and resemble thefourier transform, with the complex exponent as kernal being replaced by the bessel function offirst kind jα−β of order α−β ≥−12 .the formal equivalence between the zeroth-order hankel transform and the abel transform fol-lowed by the fourier transform is used as basis for developing fast algorithms for the computationof the zeroth-order hankel transform [5]. algorithms for the computation of the hankel transformof integer order n > 0 have been proposed. on the basis of the general relation involving thehankel transform of integer order n > 0 and the abel transform, whose kernel is modulated by thechebyshev polynomial of the first kind of order n, followed by the fourier sine or cosine transformaccording to whether n is odd or even [5, 13].it has been evidenced in [18] the formal equivalence between the hankel transform of order α−βand the erdelyi-kober fractional integral of order (α − β + 1 2 ) followed by the fourier cosinetransform, with both acted on function and the resulting transform being modulated by properly α−β dependent power functions of the inherent variables.notably such as equivalence suggests a tool for the optical computation of erdelyi-kober typefractional integrals of order (n + 1 2 ),n integer, through the optical implementation of the hankeland id fourier transforms. in a sense, the erdelyi-kober type fractional integrals of order n + 1 2 received: 4 jun 2022. key words and phrases. hankel type transform; barut-girardello transform; fractional transform; erdelyi-kobertransform. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 2 can be regarded as the (2n + 1)-plane abe1 transform on rm, with 0 < 2n + 1 < m.various forms of hankel like integral transforms have been considered in detail in a series of pa-pers [6,7,10–12,22]. we will be concerned here with certain hankel type transform having relevanceamong others, in connection with the solution to evolution problems involving the bessel type differ-ential operators xα+3β−1 ( ∂ ∂x ) x2(α−β)+1 ( ∂ ∂x ) x−3α−β and x−3α−β ( ∂ ∂x ) x2(α−β)+1 ( ∂ ∂x ) xα+3β−1with α−β ≥−1 2 and −2(α + β) real parameter [6, 11]. we introduced the fractional order formsof such hankel-type transforms by following the lines of the fractionalization of the conventionalhankel transform [9, 16].work of torre [17] motivated us to prepare this paper. 2. hankel type transforms: the first and second hankel-type transforms of bessel order α−β, depending on an arbitraryreal parameter −2(α + β), respectively defined by the operational relations [6, 11]. f̃1,α,β(y) = [ h1,α−β,−2(α+β)f ] (y) = y1−4(α+β) ∫ ∞ 0 (xy)2(α+β)jα−β(xy)f (x)dx (1) f̃2,α,β(y) = [h2,α−β,−2(α+β)f ](y) = ∫ ∞ 0 x1−4(α+β)(xy)2(α+β)jα−β(xy)f (x)dx (2) where jα−β is the bessel type function of the first kind and order (α−β) ≥ −12 . here f ∈ l2(r+)the space of the complex-valued functions which are lebesgue integrable on r+ = (0, +∞).thetransform [h2,α−β,−2(α+β)f ](y) for α = −13 β was originally considered in [15], where the relevantcondition for its inversion were established. also (1) and (2) relate to the hankel type cliffordtransforms [10].for suitable values of α,β the able transforms can be framed within the formalism, developedin [20, 21] concerning the integral transforms associated with complex linear transformations inquantum mechanics, which maps the position and momentum operators to canonically conjugate,but not necessarily hermitian operator. thus according to that formalism, the able transformscan be seen as the radial parts of n-dimentional linear cannonical transformations, specificallyrepresenting a π/2-rotation for each pair of the cannonically conjugate operators in the respec-tive n-component position and momentum operator vectors. precisely, n = 4(α + β) for (1) and n = 2[1 − 2(α + β)] = 2 − 4(α + β) for (2).the order α−β of the bessel type function relates to the eigen value λ = −l(l+n−2), l = 0, 1, 2, ...of the angular momentum; specifically, it turns out that l = α−β− n 2 + 1, and so l = −(α+ 3β−1)for (1) and l = 3α + β for (2), thus respectively yeilding λ = 3(α2 + β2) + 10αβ − 4(α + β) + 1and λ = 3(α2 + β2) + 10αβ. when α + β = 1 4 ,n = 1 in both cases, and accordingly both trans-forms yield the conventional hankel type transform. the symmetry of (1) and (2) reflects into therelation between the respective integral kernels k1,α−β,−2(α+β)(x,y) and k2,α−β,−2(α+β)(y,x);i.e. k1,α−β,−2(α+β)(x,y) = y1−2(α+β) x2(α+β) jα−β(xy) = k2,α−β,−2(α+β)(y,x). https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 3 as a consequence of well known orthogonality relation of the bessel functions, both transforms (1)and (2) are self reciprocal. h−1 1,α−β,−2(α+β) = h1,α−β,−2(α+β), h −1 2,α−β,−2(α+β) = h2,α−β,−2(α+β). (3) interestingly, the adjoint operator of h1,α−β,−2(α+β) is h2,α−β,−2(α+β) and so h∗1,α−β,−2(α+β) = h2,α−β,−2(α+β), h ∗ 2,α−β,−2(α+β) = h1,α−β,−2(α+β). (4) also one can prove for the operator h1,α−β,−2(α+β) and h2,α−β,−2(α+β) the parsevel equalities [12]∫ ∞ 0 x−1+4(α+β) f ∗(x) g(x)dx = ∫ ∞ 0 x−1+4(α+β) f̃ ∗1,α−β,−2(α+β) g̃1,α−β,−2(α+β)(x)dx,∫ ∞ 0 x1−4(α+β) f ∗(x) g(x)dx = ∫ ∞ 0 x1−4(α+β) f̃ ∗2,α−β,−2(α+β) g̃2,α−β,−2(α+β)(x)dx (5) both containing a weight function i.e. x−1+4(α+β) and x1−4(α+β) respectively. a mixed parsevelrelation holds as well, which writes as∫ ∞ 0 f ∗(x) g(x)dx = ∫ ∞ 0 f̃ ∗2,α−β,−2(α+β) g̃2,α−β,−2(α+β)(x)dx. (6) note that it does not contain any weight function and involves both transforms [12].relations (4) and (6) express the complementary of (1) and (2).for α + β = 1 4 , we recover the conventional hankel type transform of bessel order α−β; ĥ 1,α−β,−1 2 = ĥ 2,α−β,−1 2 ≡ ĥα−β with [ ĥα−β,−2(α+β)f ] (y) ≡ f̃α−β(y) = ∫ ∞ 0 (xy)2(α+β)jα−β(xy)f (x)dx = ∫ ∞ 0 kα,β(x,y)f (x)dx (7) the latter expressing the transform in terms of the kernel kα,β(x,y) = (xy)2(α+β) jα−β(xy) = kα,β(y,x).now we evidence the similarity transformations like structure of the transforms (1) and (2), beingindeed [ ĥ1,α−β,−2(α+β)f ] (y) = (y)−2(α+β)+1/2 ∫ ∞ 0 (x)2(α+β)−1/2 kα,β(x,y) f (x)dx = (y)−2(α+β)+1/2 [ ĥα,β(x) 2(α+β)−1/2f ] (y)[ ĥ2,α−β,−2(α+β)f ] (y) = (y)2(α+β)−1/2 ∫ ∞ 0 (x)−2(α+β)+1/2 kα,β(x,y) f (x)dx = (y)2(α+β)−1/2 [ ĥα,β(x) −2(α+β)+1/2f ] (y) (8) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 4 and so for the kernels k1,α−β,−2(α+β)(x,y) = (y)−2(α+β)+1/2 kα,β(x,y)x2(α+β)−1/2 = k2,α−β,−2(α+β)(y,x). it can be easily verified that [6, 11][ ĥ1,α−β,−2(α+β) b̂ ∗ α−β,−2(α+β)f ] (y) = −y2 [ ĥ1,α−β,−2(α+β)f ] (y)[ ĥ2,α−β,−2(α+β) b̂α−β,−2(α+β)f ] (y) = −y2 [ ĥ2,α−β,−2(α+β)f ] (y) (9) where b̂α−β,−2(α+β) is the bessel type differential operator b̂α−β,−2(α+β) = x α+3β−1 dxx 2(α−β)+1 dxx −3α−β = d2x + [1 − 4(α + β)] 1 x dx + (3α + β)(α + 3β) x2 (10) whose adjoint is then b̂∗α−β,−2(α+β) = x −3α−β dxx 2(α−β)+1 dxx α+3β−1 = d2x + [4(α + β) − 1] 1 x dx + (α + 3β − 1)(3α + β − 1) x2 (11) notice that for α + β = 1 4 both operators turn into the self-adjoint operator b̂α,β, being b̂ α−β,−1 2 = b̂∗ α−β,−1 2 = d2x + 64 αβ + 3 16 x2 ≡ b̂α−β. (12) from (9) it follows, for instance, that the solution of the differential equation [6, 11]. k ∂ ∂τ h(x,τ) = b̂∗α−β,−2(α+β) (13) satisfying the initial condition h(x, 0) = f (x) can be written into the transform conjugate y-space as ĥ1,α−β,−2(α+β)(y,τ) = e −y 2 kτ f̂1,α,β(y) for any value of the arbitrary constant k. then transformingback to the x-space, one obtains h(x,τ) = k 2τ x1−4(α+β) ∫ ∞ 0 (xy)2(α+β)−( k 4τ )(x 2+y2) iα−β ( k 2τ xy ) f (y)dy (14) under the condition that |arg(τ k )| ≤ π 4 , which for both τ and k real turns into τ k > 0. 3. hankel-type transforms of fractional order: it is well known that equation (13) has the formal solution h(x,τ) = e τk b̂∗α,βf (x). equation(14) yields an explicit functional representation of the exponential operator eb b̂∗α−β,−2(α+β) :[ e b b̂∗ α−β,−2(α+β) ] f (y) = 1 2b y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e−( 1 4b )(x 2+y2) iα−β (xy 2b ) f (x)dx (15) where iα−β denotes the modified bessel function of the first kind of order α − β : jα−β(ix) = iα−β iα−β(x).in particular, setting b = i 2 we obtain a representation of ĥ1,α−β,−2(α+β) in the form of a symmetricfractional product of the exponential of the generators of the su(1, 1) algebra: ĥ1,α−β,−2(α+β) = i α−β+1e−( i 2 )x2e ( i 2 )b̂∗ α−β,−2(α+β)e−( i 2 )x2 (16) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 5 in this connection, we may note that the operators k̂ (1) + = 1 2 x2, k̂ (1) − = − 1 2 b̂∗α−β,−2(α+β), k̂ (1) 3 = − i 2 ( x d dx + 2(α + β) ) (17) conjugate a non self-adjoint one variable realization of the su(1, 1) algebra generators accordingto the inherent commutation relations[ k̂ (1) + ,k̂ (1) − ] = 2ik̂ (1) 3 , [ k̂ (1) ± ,k̂ (1) 3 ] = ±ik̂(1)± . (18) thus (16) can be formally be rewritten in terms of the operators k̂(1)+ and k̂(1)− , and further recastin the single exponential form ĥα−β,−2(α+β) = i α−β+1 e−i π 2 [ k̂ (1) + + k̂ (1) − ] . (19) this is on account of disentanglement relation for the su(1, 1) algebra generators e −iφ [ k̂ (1) + +k̂ (1) − ] = e−i tan(φ/2)k̂ (1) + e−i sinφk̂ (1) − e−i tan(φ/2)k̂ (1) + (20) holding for −π < φ < π. expressing (19) corresponds to the value φ = (π/2).exploiting the integral transform representation (15) of the centred operator in equation (20), weobtain an expression for the operator e−iφ[k̂(1)+ +k̂(1)− ] in the form of a hankel -type integral transform.then writing φ = a(π/2) and multiplying both sides by ei(aπ/2)(α−β+1), one ends up on the l.h.s.with the ath power of the operator iα−β+1e−i π2 [k̂(1)+ +k̂(1)− ] and corresponding on the r.h.s. with ath power of the first hankel-type transform, ĥa 1,α−β,−2(α+β), or the first hankel-type transform offractional order a. accordingly, we can write[ ĥa1,α−β,−2(α+β)f ] (y) = ei(α−β+1)(φ−π/2) sin φ x1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e i 2 cotφ[x2+y2] jα−β ( xy sin φ ) f (x)dx (21) = [eiφ(α−β+1) e −iφ [ k̂ (1) + +k̂ (1) − ] f ](y) = f̃ (a) 1,α,β (y) where φ = a(π/2).we can interpret it as the functional representation of the operator associated with the equation i ∂ ∂τ h(x,τ) = − 1 2 { ∂2 ∂x2 − (1 − 4(α + β)) 1 x ∂ ∂x + [1 − 2(α + β)2 − (α−β)2] 1 x2 −x2 + 2(α−β + 1) } h(x,τ) (22) with the relevant initial condition h(x, 0) = f (x).the evolution variable is here measured in unitsof (π/2) : τ = a(π/2), and conventionally denoted by φ.we can also develop similar results in relation with the second hankel-type transform ĥ2,α−β,−2(α+β),the inherent su(1, 1) algebra generators being the adjoint of equation (17), namely, k̂ (2) + = 1 2 x2, k̂ (2) − = − 1 2 b̂α−β,−2(α+β), k̂ (2) 3 = − i 2 ( x d dx − 2(α + β) + 1 ) . (23) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 6 now we introduce the second hankel-type transform of fractional order a, ĥa 2,α−β,−2(α+β) as[ ĥa2,α−β,−2(α+β)f ] (y) = ei(α−β+1)(φ−π/2) sin φ ∫ ∞ 0 x1−4(α+β) (xy)2(α+β) e i 2 cotφ[x2+y2] jα−β ( xy sin φ ) f (x)dx (24) = [ eiφ(α−β+1) e −iφ [ k̂ (2) + +k̂ (2) − ] f ] (y) = f̃ (a) 2,α−β,−2(α+β)(y), φ = a(π/2). this yields the functional representation of the evolution operator for the equation i ∂ ∂τ h(x,τ) = − 1 2 { ∂2 ∂x2 + (1 − 2(α + β)) 1 x ∂ ∂x + [−2(α + β)2 − (α−β)2] 1 x2 −x2 + 2(α−β + 1) } h(x,τ) (25) with the initial condition h(x, 0) = f (x).the evolution variable parameterized as τ = a(π/2). for α+β = 1 4 , both equations (21) and (24) yields the expression of the conventional hankel transform offractional order, originally introduced by namia [9] and further investigated in [16, 21].in particular,for α + β = 1 4 , equations (17) and (23) turn into the same set of self-adjoint operators k̂+ = 1 2 x2, k̂− = − 1 2 b̂α−β, k̂3 = − i 2 ( x d dx + 1 2 ) (26) which pertain to the conventional hankel transform (7).thus we have ĥα−β = i α−β+1e i( π2 ) [ d2 dx2 −(α−β)2−(1/4) x2 −x 2 2 ] (27) and corresponding for the transform of fractional order a ĥaα−β = e i( aπ2 )(α−β+1)e i( aπ2 ) [ d2 dx2 −(α−β)2−(1/4) x2 −x 2 2 ] (28) whose fractional equation is [9, 16, 21] [ ĥaα−βf ] (y) ≡ f̃ (a) α−β(y) = ei(α−β+1)(φ−π/2) sin φ ∫ ∞ 0 (xy)( 1 2 )e i 2 cotφ[x2+y2] jα−β ( xy sin φ ) f (x)dx (29) since (1) and (2), also ĥa 1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) can be framed for suitable values of α,β,within the formalism of [20, 21], the relevant canonical transformation being now the rotation bythe angle φ for each pair of corresponding canonically conjugate position and momentum operatorsin the relevant n-component operator vectors. 4. properties of ĥa 1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β): the fractionalization of order a of an integral transform t̂ is intended to produce an integraltransform t̂a which satisfy specific properties; more precisely we need that (1) t̂a is continuous with respect to the order,i.e. t̂b −→ t̂a as b −→ a https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 7 (2) t̂a obeys the semigroup property, so that composing two fractional transform of order a1and a2 yeilds the fractional transform of order a1 + a2 t̂a1t̂a2 = t̂a2t̂a1 = t̂a1+a2, (3) t̂a reduces to the identity operator with a = 0 and to the ordinary transform for a = 1; insymbols : t̂0 = 1̂ and t̂1 = t̂. from the procedure we followed to introduced the fractional order transforms ĥa 1,α−β,−2(α+β) and ĥa 2,α−β,−2(α+β), it is clear that both transforms satisfy the above properties.thus the additivityproperties follows from ĥa1,α−β,−2(α+β) = [ ĥ1,α−β,−2(α+β) ]a , ĥa2,α−β,−2(α+β) = [ ĥ2,α−β,−2(α+β) ]a (30) which in turn implies that ĥ01,α−β,−2(α+β) = 1̂, ĥ 0 2,α−β,−2(α+β) = 1̂. (31) in our case, the ordinary transform ĥa 1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) are recovered for a = ±1: ĥ11,α−β,−2(α+β) = ĥ1,α−β,−2(α+β), ĥ −1 1,α−β,−2(α+β) = ĥ1,α−β,−2(α+β) ĥ12,α−β,−2(α+β) = ĥ2,α−β,−2(α+β), ĥ −1 2,α−β,−2(α+β) = ĥ2,α−β,−2(α+β). (32) this confirms to the self-reciprocal property (3) of the ordinary transform. we have ĥ−a 1,α−β,−2(α+β) = [ ĥa1,α−β,−2(α+β) ]−1 , ĥ−a 2,α−β,−2(α+β) = [ ĥa2,α−β,−2(α+β) ]−1 . (33) in fact, both transforms are periodic with respect to order parameter a i.e. ĥ a+2j 1,α−β,−2(α+β) = ĥ a 1,α−β,−2(α+β), ĥ a+2j 2,α−β,−2(α+β) = ĥ a 2,α−β,−2(α+β) (34) so that a can be taken in [-1,1].intrestingly, for the adjoint operator, we find the cross-relations:[ ĥa1,α−β,−2(α+β) ]∗ = ĥ−a 2,α−β,−2(α+β), [ ĥa2,α−β,−2(α+β) ]∗ = ĥ−a 1,α−β,−2(α+β) (35) which reproduce (4) for a = ±1.parsevel’s equalities separately pertaining to ĥa 1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) are easily proceedi.e.∫ ∞ 0 x−1+4(α+β)f (x)∗g(x)dx = ∫ ∞ 0 x1+2(α+β) [ f̃ (a) 1,α−β,−2(α+β)(x) ]∗ g̃ (a) 1,α−β,−2(α+β)(x)dx∫ ∞ 0 x1−2(α+β)f (x)∗g(x)dx = ∫ ∞ 0 x1−4(α+β) [ f̃ (a) 2,α−β,−2(α+β)(x) ]∗ g̃ (a) 2,α−β,−2(α+β)(x)dx (36) as well as the mixed parsevel’s relation:∫ ∞ 0 f (x)∗g(x)dx = ∫ ∞ 0 [ f̃ (a) 1,α−β,−2(α+β)(x) ]∗ g̃ (a) 2,α−β,−2(α+β)(x)dx. (37) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 8 for α + β = 1 4 , the above equalities turn into the energy preserving relation of the conventionaltransform: ∫ ∞ 0 f (x)∗g(x)dx = ∫ ∞ 0 [ f̃ (a) α−β(x) ]∗ g̃ (a) α−β(x)dx. operational rules similar to (9) can be stated for the fractional transforms, involving of courseappropriate bessel-type differential operators. indeed we see that[ ĥa1,α−β,−2(α+β) b̂ ∗ α−β,−2(α+β),af ] (y) = − y2 sin2φ [ ĥa1,α−β,−2(α+β)f ] (y),[ ĥa2,α−β,−2(α+β) b̂ ∗ α−β,−2(α+β),af ] (y) = − y2 sin2φ [ ĥa2,α−β,−2(α+β)f ] (y), (38) with b̂α−β,−2(α+β),a = −2k̂ (2) − − 2 cot 2φk̂ (2) + − 4 cot φk̂ (2) 3 , (39)and in particular b̂α−β,a ≡ b̂α−β,−(1 2 ),a = b̂∗ α−β,−(1 2 ),a = −2k̂− − 2 cot2φk̂+ − 4 cot φk̂3. (40) relation (38) gives the relevance of the fractional transforms for the solution of differential equationsinvolving the operators b̂∗ α−β,−2(α+β),a and b̂α−β,−2(α+β),a like, for instance, the evolution equation k ∂ ∂τ h(x,τ) = b̂∗α−β,−2(α+β),a h(x,τ), (41) or, more in general, the following k ∂ ∂τ h(x,τ) = p (b̂∗α−β,−2(α+β),a) h(x,τ) (42) involving polynomial function of b̂∗ α−β,−2(α+β),a (or the adjoint involving a polynomial of b̂α−β,−2(α+β),a).then considering equation (41), we note that the solution turns out to be h(x,τ) = [ ĥ−a 1,α−β,−2(α+β) ĥ a α−β,−2(α+β),a(y,τ) ] (x,τ) (43) before giving details of the expression of h(x,τ) from the above scheme, let us note that theoperators b̂α−β,−2(α+β),a and b̂∗α−β,−2(α+β),a arise from the adjoint transformation respectively of b̂α−β,−2(α+β),a and b̂∗α−β,−2(α+β),a through the operator k̂(2)+ = k̂(1)+ = (x22 ). in other words: b̂α−β,−2(α+β),a = e −i cot(φ) ( x2 2 ) b̂α−β,−2(α+β)e i cot(φ) ( x2 2 ) b̂∗α−β,−2(α+β),a = e −i cot(φ) ( x2 2 ) b̂∗α−β,−2(α+β)e i cot(φ) ( x2 2 ) (44) which can be recast as b̂α−β,−2(α+β),a = x α+3β−1d̂ax 2(α−β)+1d̂ax −3α−β b̂∗α−β,−2(α+β),a = x −3α−βd̂ax 2(α+β)+1d̂ax α+3β−1 (45) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 9 where d̂a being a linear combination of the heisenberg operators x and −i ∂∂x as d̂a = e −i cot(φ) ( x2 2 ) ∂ ∂x e i cot(φ) ( x2 2 ) = i sin φ [ cos(φ)x − i sin(φ) ∂ ∂x ] (46) the exponential operators ebb̂α−β,−2(α+β),a and ebb̂∗α−β,−2(α+β),a arise from the same adjoint transfor-mations of ebb̂α−β,−2(α+β) and ebb̂∗α−β,−2(α+β) respectively; viz ebb̂α−β,−2(α+β),a = e −i cot(φ) ( x2 2 ) ebb̂α−β,−2(α+β)e i cot(φ) ( x2 2 ) e bb̂∗ α−β,−2(α+β),a = e −i cot(φ) ( x2 2 ) e bb̂∗ α−β,−2(α+β)e i cot(φ) ( x2 2 ) (47) which on account of (15) yields an explicit functional expression for both operators.accordingly, thesolution of equation (41) can be written as h(x,τ) = k 2τ x1−4(α+β) ∫ ∞ 0 (xy)2(α+β)e−( k 4τ )(x2+y2)e−( i 2 )cot(φ)(x2−y2)iα−β ( k 2τ xy ) f (y)dy (48) under the same condition specified in connection with equation (14). one can also note thatthe similarity transformation like link between the operators b̂α−β,−2(α+β),a and b̂α−β,−2(α+β)suggests to recover equation(48) from(41) transforming h(x,τ) to h(x,τ) = h(x,τ)ei cot(φ)(x22 ).in fact the fractional transforms ĥa 1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) are linked to ĥaα−β through thesame similarity transformation holding between the ordinary transforms; viz. ĥa1,α−β,−2(α+β) = x −2(α+β)+1 2 ĥaα−β x 2(α+β)−1 2 ĥa2,α−β,−2(α+β) = x 2(α+β)−1 2 ĥaα−β x −2(α+β)+1 2 as a straightforward consequence of the relations b̂α−β,−2(α+β),a = x 2(α+β)−1 2 b̂α−β,a x −2(α+β)+1 2 b̂∗α−β,−2(α+β),a = x −2(α+β)+1 2 b̂α−β,a x 2(α+β)−1 2 . 5. barut-girardello-type transformations: the functional expression (15) of ebb̂∗α−β,−2(α+β) resembles the barut-girardello-type transform.the barut-girardello-type transform of bessel order α−β is defined by [3]. [ ĝα−βf ] (y) = √ 2 ∫ ∞ 0 (xy)1/2e−(1/2)(x 2+y2)iα−β( √ 2xy)f (x)dx. (49) as a straightforward generalization of the notion of coherent stakes associated with the heisenbergalgebra, such generalized coherent stakes were introduced as eigenstates of the lowering operator ofthe aforementioned algebra in the relative discrete representations d±(k),k = −1/2,−1,−3/2, ... https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 10 then taking 2b = 1√ 2 in equation (15) and multiplying the integrand function by e−(1/2)(x 2+y2)e(1/2)(x 2+y2), we end up expression[ e−(1/ √ 2)k̂ (1) − f ] (y) = e−(y 2/2)( √ 2−1) √ 2 y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e−(1/2)(x 2+y2) iα−β( √ 2xy) e−(x 2/2)( √ 2−1)f (x)dx. this allows us to define the first barut-girardello-type transform of bessel order α−β, dependingon a real parameter −2(α + β), through the expression[ ĝ1,α−β,−2(α+β)f ] (y) = √ 2 y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e−(1/2)(x 2+y2) iα−β( √ 2xy)f (x)dx. (50) thus we may write ĝ1,α−β,−2(α+β) = e ( √ 2−1)k̂(1)+ ek̂ (1) − / √ 2 e( √ 2−1)k̂(1)+ which can eventually be imposed into the single exponential form: ĝ1,α−β,−2(α+β) = e (π/4) [ k̂ (1) + −k̂ (1) − ] . (51) it clearly states that ĝ1,α−β,−2(α+β) can be regarded as the evolution operator e−iτĥ, associatedwith the dynamical problem ruled by the hamiltonian operator ĥ = k̂ (1) + − k̂ (1) − = 1 2 [ x2 + b̂∗α−β,−2(α+β) ] and evaluated at the purely imaginary value τ = i(π/4) of the evolution variable.now we can define the second barut-girardello-type transform of bessel order α−β as[ ĝ2,α−β,−2(α+β)f ] (y) = √ 2 ∫ ∞ 0 x1−4(α+β) (xy)2(α+β) e−(1/2)(x 2+y2)iα−β( √ 2xy)f (x)dx (52) for which the following operational relatoins can be stated as: ĝ2,α−β,−2(α+β) = e ( √ 2−1)k̂(2)+ ek̂ (2) − / √ 2 e( √ 2−1)k̂(2)+ = e (π/4) [ k̂ (2) + −k̂ (2) − ] , (53) involving of course the operators (23). accordingly, ĝ2,α−β,−2(α+β) can be interpreted as theevolution operator operator e−iτĥ, associated with the dynamical problem ruled by the hamiltonianoperator ĥ = k̂ (2) + − k̂ (2) − = 1 2 [ x2 + b̂α−β,−2(α+β) ] and evaluated at the same complex value τ = i(π/4) of the evolution variable as ĝ1,α−β,−2(α+β) .both definitions (50) and(52) can be recast into the comprehensive expression[ ĝj,α−β,−2(α+β)f ] (y) = ∫ ∞ 0 k (bg) j,α−β,−2(α+β)(x,y)f (x)dx, j = 1, 2. (54) in terms of the kernels k (bg) 1,α−β,−2(α+β)(x,y) = √ 2y1−2(α+β)(x)2(α+β)iα−β( √ 2xy)e−(1/2)(x 2+y2) = k (bg) 2,α−β,−2(α+β)(y,x) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 11 which relate to the kernel k(bg) α−β (x,y) = √ 2iα−β( √ 2xy)e−(1/2)(x 2+y2) of the conven-tional transform (49) through the same similarity transformation like relation holding between k1,α−β,−2(α+β)(x,y), k2,α−β,−2(α+β)(y,x) and kα−β(x,y) .as the hankel-type transforms, the transforms ĝ1,α−β,−2(α+β) and ĝ2,α−β,−2(α+β) are adjoint toeach other: ĝ∗1,α−β,−2(α+β) = ĝ2,α−β,−2(α+β), ĝ ∗ 2,α−β,−2(α+β) = ĝ1,α−β,−2(α+β). (55) however, they are not self reciprocal; the respective inverse transforms can be easily obtained fromthe corresponding factored representation in equations (51) and (53) which yield[ ĝ−1 1,α−β,−2(α+β)f ] (y) = (−1)α−β+1 √ 2 y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e(1/2)(x 2+y2) iα−β( √ 2xy)f (x)dx = {[ ĝ−1 2,α−β,−2(α+β) ]∗ f } (y). operational relations similar to (9) can be deduced for ĝ1,α−β,−2(α+β) and ĝ2,α−β,−2(α+β). in fact:[ ĝ1,α−β,−2(α+β) î ∗ α−β,−2(α+β)f ] (y) = 2y2 [ ĝ1,α−β,−2(α+β)f ] (y),[ ĝ2,α−β,−2(α+β) îα−β,−2(α+β)f ] (y) = 2y2 [ ĝ2,α−β,−2(α+β)f ] (y), (56) with the differential operator îα−β,−2(α+β) being îα−β,−2(α+β) = 2 [ k̂ (2) + − k̂ (2) − + 2ik̂ (2) 3 ] . (57) it can be easily seen that îα−β,−2(α+β) = e −(x2/2) b̂α−β,−2(α+β) e −(x2/2) = x2(α+β)−(α−β)−1 × ( x + ∂ ∂x ) x2(α−β)+1 ( x + ∂ ∂x ) x−(3α+β) (58) even though equation (56) correspond to equation (9) , pertaining to the hankel transform, in-volve operators îα−β,−2(α+β) and î∗α−β,−2(α+β) comprise also the operators k̂+ and k̂3 of thecorresponding algebras. 6. barut-girardello-type transforms of fractional order: we may introduce fractional order versions of the transforms ĝ1,α−β,−2(α+β) and ĝ2,α−β,−2(α+β). let us consider, the disentanglement relation for the su(1, 1) algebra generators eζ [k̂+−k̂−] = etan(ζ/2)k̂+ e−sin(ζ)k̂− etan(ζ/2)k̂+, (59) holding for −π < ζ < π. the expressions above obtained for ĝ1,α−β,−2(α+β) and ĝ2,α−β,−2(α+β)correspond to value ζ = (π/4) with appropriate set of operations (21) and (23) being respectively https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 12 involved.let us refer in particular to the operators (17) so that on account of (15) one ends up with[ eζ [k̂ (1) + −k̂ (1) − ]f ] = 1 sin ζ y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e−(1/2)cot(ζ)(x 2+y2) iα−β ( xy sin ζ ) f (x)dx then,writing ζ = (aπ/4), one can obtains the ath power of (51), with the first barut-girardello-typetransforms of fractional order a being accordingly defined by the functional expression:[ ĝa1,α−β,−2(α+β)f ] (y) = 1 sin (φ/2) y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) e−cot(φ/2)(x 2+y2) iα−β ( xy sin (φ/2) ) f (x)dx(60)with φ = (aπ/2), as beforethe second barut-girardello-type transforms of fractional order a is similarly introduced through e (aπ/4) [ k̂ (2) + −k̂ (2) − ] = ĝa2,α−β,−2(α+β), (61) the relevant functional expression being then:[ ĝa2,α−β,−2(α+β)f ] (y) = 1 sin (φ/2) ∫ ∞ 0 x1−2(α+β)(xy)2(α+β)e−(1/2)cot(φ/2)(x 2+y2) iα−β ( xy sin (φ/2) ) f (x)dx.(62)the ordinary transforms are recovered, of course with a = 1, while for α + β = 1 4 , we obtainthe conventional barut-girardello-type transforms of fractional order a, ĝaα−β, introduced in [16]. ĝa 1,α−β,1/4 = ĝ a 2,α−β,1/4 ≡ ĝ a α−β, with [ ĝaα−βf ] (y) = 1 sin (φ/2) ∫ ∞ 0 √ xy e−(1/2)cot(φ/2)(x 2+y2) iα−β ( xy sin (φ/2) ) f (x)dx. (63) the fractional transforms ĝa 1,α−β,−2(α+β) and ĝa2,α−β,−2(α+β) are cyclic with respect to order a,being ĝa+8j 1,α−β,−2(α+β) = ĝ a 1,α−β,−2(α+β), ĝ a+8j 2,α−β,−2(α+β) = ĝ a 2,α−β,−2(α+β), (64) which allows us to limit the values of a to the interval a ∈ [−4, 4].the operational relations (55)can be generalized to ĝa 1,α−β,−2(α+β) and ĝa2,α−β,−2(α+β) for which we obtain[ ĝa1,α−β,−2(α+β) î ∗ α−β,−2(α+β),af ] (y) = y2 sin2 (φ/2) [ ĝ1,α−β,−2(α+β)f ] (y), [ ĝa2,α−β,−2(α+β) îα−β,−2(α+β),af ] (y) = y2 sin2 (φ/2) [ ĝ2,α−β,−2(α+β)f ] (y). (65) the differential operator îα−β,−2(α+β),a is given by îα−β,−2(α+β),a = 2 cot 2(φ/2) k̂ (2) + − k̂ (2) − + 4i cot(φ/2) k̂ (2) 3 = e[1−cot(φ/2)](x 2/2) îα−β,−2(α+β) e −[1−cot(φ/2)](x2/2). (66) https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 13 by using equation (57) we have îα−β,−2(α+β),a = e −cot(φ/2)(x2/2) b̂α−β,−2(α+β) e cot(φ/2)(x2/2) = e2(α+β)−(α−β)−1îax 2(α−β)+1îax −2(α+β)−(α−β) (67) with îa = e −cot(φ/2)(x2/2) ∂ ∂x ecot(φ/2)(x 2/2) = 1 sin2 (φ/2) [ cos(φ/2)x + sin(φ/2) ∂ ∂x ] . (68) therefore, ĝa 1,α−β,−2(α+β) and ĝa2,α−β,−2(α+β) are of relevance in connection with evolution equa-tions like k ∂ ∂τ h(x,τ) = p (î∗α−β,−2(α+β),a) h(x,τ),or k ∂ ∂τ h(x,τ) = p (îα−β,−2(α+β),a) h(x,τ) involving polynomial function of îα−β,−2(α+β),a and î∗α−β,−2(α+β),a respectively. 7. generalized hankel transforms: the h and g transform discussed above are associated with hamiltonian operators involvinga linear combination of the generators k̂+ and k̂− of the relevant su(1, 1) algebra realizationsin a form that naturally suggests an arbitrary respectively with the attractive and repulsive radialquantum mechanics oscillator.the dynamical symmetry of the linear quantum mechanical oscillator is that of the su(1, 1) algebra,whose generator are defined in terms of the position and momentum operators are defined in terms ofthe position and momentum operators x̂ and p̂ = i ( d dx ) (h = 1) through the self-adjoint quadranticexpressions k̂+ = 1 2 x̂2 = 1 2 x2, k̂− = − i 2 p̂2 = − 1 2 d2 dx2 , k̂3 = 1 4 (x̂p̂ + p̂x̂) = − i 2 ( x d dx + 1 2 ) . thus the conventional hankel transform of any order a, being associated with the sum operator k̂1 = k̂+ + k̂−, turns out to be linked to the dynamics of the alternative radial oscillator, therelevant k̂− generator (12) being the radical part of the 2d laplacian operator. in fact, as notedearlier, the hankel transform of integer bessel order can be regarded as the radial part of the 2dfourier transform of rotationally symmetric function, when polar co-ordinates are adopted.in otherwords we have[ f̂a f (ζ,η) ] (x,y) = e−imφ e−imθ [ ĥam ρ 1/2 g(ρ) ] (r), m = 0, 1, 2, ... (69) where (ρ,φ) and (r,θ) are polar co-ordinates respectively in the function and transform domainand f is a rotationally symmetric function: f (ζ,η) = g(ρ) eimφ. https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 14 evidently (69) can be generalised to the transform (21) and (24) as[ f̂a f (ζ,η) ] (x,y) = e−imφ e−imθ r−1+2(α+β) [ ĥa1,m ρ 1−2(α+β) g(ρ) ] (r),[ f̂a f (ζ,η) ] (x,y) = e−imφ e−imθ r−2(α+β) [ ĥa2,m,−2(α+β) ρ 2(α+β) g(ρ) ] (r) for non-negative integers m and rotationally symmetric functions.likewise, the barut-girardello transform resorting to the difference operator k̂2 = k̂+ + k̂−,can be associated with dynamics of the repulsive radical oscillator.therefore barut-girardellotransform of integer bessel order can be regarded as the radial part of the 2d-bargman transform b̂a [2, 16, 20], the inherent relation being similar to (69) i.e. [ b̂a f (ζ,η) ] (x,y) = e−imθ r−1/2 [ ĝam ρ 1/2 g(ρ) ] (r), m = 0, 1, 2, ... (70) with the same meaning of the symbols as in equation (69).the generalization of (70) to the transforms of the first and second type is obvious.following the correspondence of the hankel to the fourier transform, we may introduce a gener-alized fractional hankel transforms as the operator associated with evolution equation driven by ageneric operator belonging to the su(1, 1) algebra namely ĥ(1,2) = ak̂ (1,2) + + bk̂ (1,2) − + ck̂ (1,2) 3 + d(v + 1)1̂ being its pertinent to the algebra realization (17) or (23).we exploit the disentanglement scheme e−iτ[ak̂ (1,2) + +bk̂ (1,2) − +ck̂ (1,2) 3 +d(v+1)1̂] = e−idτ(α−β+1) eak̂ (1,2) + eck̂ (1,2) 3 e−iφk̂ (1,2) 1 , giving the operator e−iτĥ(1,2) in three-term factored form, apart from the phase factor e−idτ(v+1).we may introduce the generalized hankel-type transforms of first and second type ,depending onthe parameter p,m and γ, [ ĥa,p,m,γ 1,α−β,−2(α+β)f ] (y) = ei(α−β+1)(γ−π/2) m sin(φ) e−ip(y 2/2) y1−4(α+β) ∫ ∞ 0 (xy)2(α+β) ×e(i/2)cot(φ)(x 2+(y2/m2)) jα−β ( xy m sin (φ/2) ) f (x)dx (71) [ ĥa,p,m,γ 2,α−β,−2(α+β)f ] (y) = ei(α−β+1)(γ−π/2) m sin(φ) e−ip(y 2/2) ∫ ∞ 0 x1−4(α+β) (xy)2(α+β) ×e(i/2)cot(φ)(x 2+(y2/m2)) jα−β ( xy m sin (φ/2) ) f (x)dx. the above relations reproduce (21) and (24) respectively, for b = a,d = −a,c = 0 and aτ = φ;also relations like (9) can be deduced for the generalized transform.in addition , generalized borut-girardello type transforms can be introduced on the basis of a https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 15 disentanglement scheme involving the operators k̂(1,2)2 instead of k̂(1,2)1 .as a conclusion, we note that from equation (71) we may recover with p = ib/m2,m2 = 1 − b2, tan(φ) = −ib and γ = 0, the integral transformations corresponding to the exponential forms e bb̂∗ α−β,−2(α+β) and ebb̂α−β,−2(α+β) (see (15) and the relevant adjoint expression, easily deducible)which can respectively be regarded as the first and second weiestrass-gauss integral transformsof bessel order α−β depending on the real parameter −2(α+β). they generalize to α+β = 1/4the expression of the radical weiestrass-gauss integral transform, for which we have [16][ ŵα−β,bf ] (y) = [ e−(b/2)b̂α−βf ] (y) = 1 b ∫ ∞ 0 (xy)(1/2) e−(1/2b)(x 2+y2) iα−β (xy b ) f (x)dx for any real parameter β > 0. ŵα−β,b arises as the transfer operator associated with the radial part of heat conduction likeequations.it can be in fact considered as the radial part of 2d fresnel transform for real parametersis the optical operator for free propagation. a linear weiestrass-gauss integral transform has alsobeen introduced [19] the relation of ŵm,b, m = 0, 1, 2, ... to it is evidently similar to (69), whenrationally symmetric function are involved. remark. [(i)](1) if we take α = ν 2 − µ 4 , β = −µ 4 − ν 2 throught this paper then all the results studied in this paper reduce to the results studied in torre [17].(2) authors claim that results of this paper are stronger than that of torre [17]. 8. conclusions: following the scheme already applied to other type of transforms, like for instance, the fouriertransform , we have introduced the fractional forms of two adjoint self-reciprocal variants of thehankel type transform, which, as noted are of interest in connection with evolution problems ruledby the bessel-type differential operators, b̂α−β,−2(α+β) = x α+3β−1dxx 2(α−β)+1 dxx −3α−β and b̂α−β,−2(α+β) = x −3α−βdxx 2(α−β)+1dxx α+3β−1. the fractional order transform relate to evolution problems ruled the operators b̂α−β,−2(α+β),a = x α+3β−1 la ( x, ( ∂ ∂x )) x2(α−β)+1 la ( x, ( ∂ ∂x )) x−3α−β and b̂∗α−β,−2(α+β),a = x −3α−β la ( x, ( ∂ ∂x )) x2(α−β)+1 la ( x, ( ∂ ∂x )) xα+3β−1 https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 16 where la (x,( ∂∂x)) is linear a-depending combination of x and ( ∂∂x). since a ranges from -1 to 1,it is evident that the set of evolution problems inherent in the transforms has been greatly enlargeby the fractionalization.in general we have shown that introduced transforms can be regarded as the evolution operatorsassociated with evolution problems having an underlying su(1, 1) symmetry, the specific realizationof the algebra resorting to b̂α−β,−2(α+β) and b̂∗α−β,−2(α+β) as the relative ladder operators.evidently, transforms of complex fractional order can be considered although of course the space offunctions on which they can meaningfully be applied must be carefully investigated. disregardinghere this aspect of the question, we simply note that due to the additivity with respect to the order,we may write ĥar+ial j,α−β,−2(α+β) = ĥ ar j,α−β,−2(α+β) + ĥ ial j,α−β,−2(α+β), j = 1, 2. where ar and al respectively denote the real and imaginary part of the order a = ar + ial .thus ĥarj,α−β,−2(α+β), j = 1, 2 have just the expression considered in this paper, while ĥial j,α−β,−2(α+β), j = 1, 2 are from then easily deducible replacing φ with iφ.as earlier mentioned, the hankel transform is of interest within the context of the fractional calculus[17]. following the arguments in [18], for the transforms of our concern we find that[ ĥ1,α−β,−2(α+β)f ] (y) = 21−(α−β) x−α−3β+1 √ π ∫ ∞ 0 cos(yτ) [ k̂α−β+1/2g1 ] (τ)dτ, [ ĥ2,α−β,−2(α+β)f ] (y) = 21−(α−β) x−α−3β+1 √ π ∫ ∞ 0 cos(yτ) [ k̂α−β+1/2g2 ] (τ)dτ (72) where k̂α−β+1/2 is the left hand sided erdelyi-kober fractional integral operator, represented by[ k̂bg ] (y) = 1 γ(b) ∫ ∞ y (y2 −x2)b−1 x g(x) dx, r(b) > 0, y ∈r, and the functions g1(x) and g2(x) on which it acts in (72) involve f as g1(x) = xα+3β−1f (x), g2(x) = x −3α−βf (x).relations (72) holds under the assumption that both xg1(x) and xg2(x) are inegrable. note thataccording to sonine’s first integral for bessel functions, we may say that the right hand sidederdelyi-kober operator of order b acts on the function xα−β jα−β(x) as a rising operator turningit into xα−β+b jα−β+b(x).expressions similar to (72) can be deduced for the fractional order transforms, of course.in addition (72) as paralleled by similar expressions involving the barut-girardello transform ofthe first and second type. as an example, we deduce here the relation involving the conventionaltransform (49). on account of the integral representation of the modified bessel function of thefirst kind, iα−β, i.e. iα−β(xy) = 21−(α−β) yα−β x−(α−β) √ π γ(α−β + 1/2) ∫ x 0 (x2 −τ2)α−β−1/2 cosh(yτ)dτ, r(α−β + 1/2) > 0, https://doi.org/10.28924/ada/ma.3.6 eur. j. math. anal. 10.28924/ada/ma.3.6 17 it is easy to rewrite (49) in the form [ ĝα−βf ] (y) = √ 2 π 21+( α−β 2 ) e−(1/2)y 2 ∫ ∞ 0 cosh( √ 2yτ) [k̂α−β+1/2h](τ)dτ, with the function h(x) being h(x) = x−1−( α−β 2 ) e−(1/2)y 2 f (x). the above relation holds under the assumption that xh(x) is integrable.we finally note 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hankel transform and its application for studying the propagation ofcylindrical electromagnetic fields, opt. express. 10 (2002), 521-525. https://doi.org/10.1364/oe.10.000521. https://doi.org/10.28924/ada/ma.3.6 https://doi.org/10.1016/j.sigpro.2005.10.001 https://doi.org/10.1090/s0002-9939-1969-0243294-0 https://doi.org/10.1090/s0002-9939-1969-0243294-0 https://doi.org/10.1016/s0377-0427(02)00637-4 https://doi.org/10.1080/10652460701827848 https://doi.org/10.1006/jmaa.1997.5351 https://doi.org/10.1111/j.1365-2478.1986.tb00481.x https://doi.org/10.1063/1.1666811 https://doi.org/10.1063/1.1666811 https://doi.org/10.1063/1.1666590 https://doi.org/10.1063/1.1666590 https://doi.org/10.1364/oe.10.000521 1. introduction 2. hankel type transforms: 3. hankel-type transforms of fractional order: 4. properties of a1,-,-2(+) and a2,-,-2(+): 5. barut-girardello-type transformations: 6. barut-girardello-type transforms of fractional order: 7. generalized hankel transforms: 8. conclusions: references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 9doi: 10.28924/ada/ma.3.9 on norm estimates for derivations in norm-attainable classes j. z. nyabonyi1,∗, n. b. okelo2, r. k. obogi1 1department of mathematics and actuarial science, kisii university, kenya nyabonyijanes@yahoo.com, krbertobogi@yahoo.com 2department of pure and applied mathematics, jaramogi oginga odinga university of science and technology, kenya bnyaare@yahoo.com ∗correspondence: nyabonyijanes@yahoo.com abstract. in this note, we provide detailed characterization of operators in terms of norm-attainabilityand norm estimates in banach algebras. in particular, we establish the necessary and sufficientconditions for norm-attainability of the derivations and also give their norm bounds in the norm-attainable classes. 1. introduction the norm of a derivation was first introduced by stampfli [49], who determined the inner derivation δt0 : a0 → t0a0 − a0t0 which acts on b(h), the algebra of all bounded linear operators on acomplex hilbert space h. further, ‖δt0‖ = inf 2‖t0 −λi0‖, for every complex λ was shown. fora normal operator t , ‖δt0‖ can be expressed as the geometry of the spectrum of t0. johnson [21]established methods which apply to a uniformly convex spaces with a large class, i.e the formula ‖δt‖ is false in lp and lp(0, 1) 1 < p < ∞, p 6= 2. for l1 space the formula is true for areal case and not for a complex case whose space dimension is 3 or more. johnson [20] foundthat a derivation on b(h) is a mapping ∆ : b(h) → b(h) with ∆(as) = a∆(s) + ∆(a)s, where a,s ∈ b(h). such derivations are necessarily continuous and if s ∈ b(h) then ∆s(a) = as−sais a derivation on b(h). gajendragadka [18] was concerned with the von neumann algebra andcomputed the norm of a derivation. specifically, it was proved that the von neumann algebra actson a separable hilbert space h, whereby if t is in u and δt is the derivation induced by t, then ‖δt |u‖ = 2 inf ‖t −z‖, where z is the centre of u. therefore, anderson [3] in his investigation onnormal derivations with the operators a,c ∈ b(h) proved if a is normal and ac commute, for every x ∈ b(h), ‖δa(x) + c‖≥‖c‖. therefore, the inequality showed that the kernel and the range of δa are orthogonal to δa which is the commutation of {a}′ of a. kyle [24] examined the relationship received: 22 jun 2022. key words and phrases. derivation; norm; norm-attainability; banach algebra.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 2 of the numerical range of inner derivation and that of the implementing element. kyle [25] studiednorms of inner derivations and used their properties and concluded that a closed subset of allderivations on a c∗-algebra, forms the set of inner derivations and obtained the result which is aconverse of stampfli [49]. charles and steve [11] answered the question when x = t by structurecharacterization of compact derivations of c∗-algebras. moreover, the structure of weak compactderivations of c∗-algebras was determined and as immediate corollaries of these results, conditionsthat were necessary and sufficient were obtained so that c∗-algebras can admit a non-zero compactor weakly compact derivation. stampfli [50] studied operators on hilbert spaces and their propertiesinducing a derivation whose closure is self-adjoint after the range of such operators are termed d-symmetric and then characterized compact d-symmetric operators. erik [16] established thatany operator t on a hilbert space h with a cyclic vector has a property with a finite spectrum.mecheri [31] established that t (x) is linear for any m-linear derivation and hence, the topologyof von neumann algebra x of type i is automatically continuous in measure with center m and thesemi-finite trace τ which is normal is faithful. therefore, t (x) is the algebra of all τ-measurableoperators affiliated with x. mathieu [29] proved that for non-zero derivations, the product of twoprime c∗-algebras are bounded if both of them are bounded. in [51], two automatic continuityproblems for derivations on commutating banach algebras were discussed, that is, derivation on acommutative algebra is mapped onto the radical, and banach algebras are continuous on semiprimederivations. bresar, zalar [9] showed that a jordan ∗-derivation is the map δa(x) = ax − x∗a forfixed a ∈ u; hence, the derivation is inner. douglas [15] continued the study of ws(y ) which wasconsiderably more amenable where archbold [1] defined the smallest numbers to be [0,∞] andintroduced two constants w (y ) and wt(y ) such that d(y,z(y )) ≤ w (y )‖d(y,y )‖, for all y ∈ yand d(y,z(y )) ≤ ws(y )‖d(y,y )‖, for all y = y∗ ∈ y. the author in [26] showed that for the nthorder commutator [[[k(b),y ],y ], ...,y ], a formula was obtained in terms of the frechet derivatives smk(b) in which the formula illustrated was used to obtain bounds for norms of a generalizedcommutator k(b)y −y k(b) and their higher order analogues. in [17], numerical ranges of 2 x 2matrices were determined and the convex of the numerical range for any hilbert space operator wasestablished in toeplitz-hausdorff theorem and the relation of the numerical range to that of spectrumwas discussed. further, the closure of the numerical range is contained in the spectrum and theintersection of closures of the numerical range of all operators were asserted by hildebrandt’stheorem. considering results on special cases [10], established that ‖pxq + qxp‖ ≥ ‖p‖‖q‖.chi-kwong [13] established that for an n x n matrix x, the numerical range w (x) has manyproperties which can be used to locate eigenvalues to obtain norm bounds. algebraic and analyticproperties were deduced which help in finding the dilations of simple structures. let the linearoperators xi and yi , 1 ≤ i ≤ n act on separate hilbert space h, therefore, hong-ke, yue-qing [19]proved that sup{‖∑ni=1pixqi‖ : x ∈ b(h),‖x‖ ≤ 1} = sup{‖∑ni=1pitqi‖ : uu∗ = t∗u = https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 3 i,u ∈ b(h)}. in addition, okelo, agure and ambogo [35] established the norm of jordan elementaryoperator ua,b : b(h) → b(h) which is given by ua,b = ay b + by a, ∀y ∈ b(h) and a,b fixedin b(h) and showed that ‖ua,b ‖≥‖a‖‖b‖ and then characterized the norm-attainable operatorsusing this norm. inner derivations implemented by norm-attainable elements of a c∗-algebra hasrelation to those of ideals and primitive ideals. since there is a relationship between the constants a(ξ) and asξ of c∗-algebras to the ideals and primitive ideals then related results have beengiven in general banch settins. okelo, agure and oleche [38] gave results on necessary andsufficient conditions for norm-attainable operators and also studied norm-attainable operators andgeneralized derivations. okelo [37] extended the work by presenting new results on conditionsthat are necessary and sufficient for norm-attainability for operators in hilbert space, elementaryoperators and generalized derivations. further, okelo [37] established that a unit vector exists λ ∈ h, ‖λ‖ = 1 such that ‖sλ‖ = ‖s‖ with 〈sλ,λ〉 = η. results from [23] showed that every jordanderivation of the trivial extension of a by m, under certain conditions, is the sum of a derivationand antiderivation. in [10], the author studied norm-attainable operators that are convergent andestablished norm-attainability of operators via projective tensor norm. wickstead [52] showed thatif an atomic banach lattice z with a continuous norm order, x,y ∈ t r and mx,y is the operatoron t r (z) defined by mx,y (a) = xay, then ‖mx,y‖r = ‖x‖r‖y‖r but there is no real β > 0such that ‖mx,y‖r = β‖x‖r‖y‖r. okelo [36] outlined the theory of normal, self-adjoint and norm-attainable operators then presented norms of operators in hilbert spaces. in [8] the author provedthat for a linear map ∆ : u → u, ∆(xy ) = ∆(x)y + ∆x(y ) for each x,y ∈ u is a derivation,and for any two derivations ∆ and ∆′ on a c∗-algebra u there exists a derivation δ ∈ u suchthat ∆∆′ = δ2 if and only if either ∆′ = 0 or ∆ = f ∆′ for any f ∈ c. clifford [12] studiedhypercyclic generalized derivations acting on separable ideals of operators and also identifiedconcrete examples and established some conditions that are necessary and sufficient for theirhypercyclicity. okelo [36] considered orthogonal and norm-attainable operators in banach spaces,gave in details the characterization and generalizations of norm-attainability and orthogonality.the conditions that are sufficient and necessary for norm-attainability of operators on a hilbertspace, the result on orthogonal range and the kernel of elementary operators implemented by norm-attainable operators in banach spaces were also given. okelo [34] characterized norm-attainableclasses in terms of orthogonality by giving norm-attainability conditions that were necessary andsufficient for hilbert space operators first and the orthogonality result on the range and kernel ofelementary operators when implemented by norm-attainable operators in norm-attainable classeswere also given. okelo [38] gave conditions for norm-attainability for linear functionals in banachspaces, non-power operators on h and elementary operators and also gave a new notion of norm-attainability for power operators then characterized norm-attainable operators in normed spaces.in [51] determined the norm of the inner jordan ∗-derivation δs : x → sx − x∗s acting on the https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 4 banach algebra b(h). it was shown that ‖δs‖≥ 2 supλ∈w0(s) |=λ| in which w0(s) is the maximalnumerical range of operator s. the work of [1] obtained precisely when zero belongs to maximalnumerical range of composition operators on h and then characterized the norm-attainability ofderivations on b(h). in okelo [41] norm-attainability for hyponormal operators that are compactwere characterized, sufficient conditions for a compact hyponormal operator that is linear andbounded on an infinite dimension for a complex hilbert space to be norm attainable were given.further, the structure and other properties of compact hyponormal operators when they are self-adjoint, normal and norm attainable with their commutators were discussed in general. lumer [27]obtained a sharp estimate not only from |sp(r)| equal to spectral radius of r but indeed for |sp(r)|in terms of sup(|x(r)|, |x(rn)|1/n),n being any positive even integer. in [18] the author studiedthe algebra of functions that are continuous on [0, 1] and are ‖.‖w -approximate polynomial; i.epoint-wise functions of limits of ‖.‖w -cauchy sequence of polynomial. archbold [1] investigatedwhether the simple triangle inequality ‖t (a,a)‖ ≤ 2t(a,z) if applied holds. d(a) was definedto be a minimum value d in [0,∞] such that t(a,z) ≤ d‖t (a,a)‖. the behaviour of d in idealsand quotients were discussed which proved that ds(a) ≤ 1 for a weakly central c∗-algebra a andconsidered a class of n-homogeneous c∗-algebras that are special. d and ds were investigatedand approximated finite-dimension (af )c∗-algebra in that context and an example was given toshow certain estimates. the results of [44] showed that for a certain von neumann algebra u,a constant f existed such that dist(t,u) ≤ f supp∈latu ‖p⊥tp‖∀t ∈ b(h). the work wasextended to a von neumann algebra u and showed that there exists a constant g ∈ b(h),dist(t,u) ≤ g‖∆t |u′‖ where δt is the derivation δt (s) = st − ts thus proving that theinequality holds for large classes of von neumann algebras. in [14] the researcher considered λ(m) defined as the smallest number ‖z‖2 of z that satisfy [z∗,z] = m and showed that 1 ≤ λ(m) ≤ 2. matej [28] estimated the distance of d1 and d2 to the generalized derivations andthe normed algebra of p and considered the cases when p is an ultraprime, when d1 = d2 and p are ultrasemiprime and when p is a von neumann algebra we have the equation ‖p + q‖ = ‖p‖ + ‖q‖, p,q ∈ b(h). further, a constructive proof was provided that a minimum bound isnot valid and a relevant method to analyze the problem on estimation of eigenvalues such aninterpolation matrix was commented on. the norm property was done by cabrera, rodriguez [10]for basic elementary operators and obtained ‖ma,b‖ ≤ 2‖a‖‖b‖, for jordan elementary operator ‖u‖ = ‖ma,b‖ + ‖ma,b‖, ‖ma,b‖ + ‖ma,b‖ ≤ 2‖a‖‖b‖ for the upper estimates. in fact, [30] gavean estimate on matrix-valued function that is regular and showed that for normal matrices it isattainable and investigated their stability. kittaneh [26] established the orthogonality, kerneland the range of a normal derivation associated with norm ideals of operators with respect tothe unitarily invariant norms. results related to orthorgonality of some derivation that are notnormal were also obtained. stacho and zalar [48] established the lower estimates for elementary https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 5 operators of jordan type in standard banach algebras. danko [14] established that for all unitarilyinvariant norms and for bounded hilbert space operators there exist {xn}n ⊆ h which is a unitsequence such that limn‖c −ω‖xn = 0. from [11], ‖a‖ ∈ σ(a) if and only if ‖a‖ ∈ σap(a) also σ(a) ⊆ w (a) (spectral inclusion) and if ω(a) = ‖a‖, then γ(a) = ‖a‖. therefore, the resultimplied that ‖a‖ ⊆ w (a) if and only if ‖a‖ ∈ σ(a). in fact, megginson [32] established that forall y ∈ k, then δb(y ) ∈ j and ‖by − y b‖k = ‖(b − λ)y − y (b − α)‖j ≤ 2‖b − α‖‖y‖kfor all α ∈ c. hence, ‖δb(y )‖k ≤ 2d(b)‖y‖k, implying that ‖δb|k‖ ≤ 2d(b). further, thenotion of r-universal operators was introduced and that r-universal is an operator a ∈ b(h) if ‖δb|k‖ = 2d(b) for every norm ideal k ∈ b(h). landsman [23] proved that for a standard operatoralgebra on h ‖ma,b‖+‖ma,b‖≥ 2(√2−1)‖a‖‖b‖. therefore, both the lower norm and upper normbounds have been established for normally represented elementary operators. the work of [3] had anestimate on transfer functions of stable linear time-invariant systems on stochastic assumptions. theapproach of nonparametric minimax was adopted to measure the estimate accurately, an estimator ofquality was measured over a family of transfer functions by its worst case error. in [32] the authorestablished that for a holomorphic functions f with re{gf ′(g)} > α and re{gf ′′(g)/f ′(g)} > α−1, (0 ≤ α < 1) respectively in {|g| < 1}, estimates of sup|g|<1(1−|g|2)|f ′′(g)/f ′(g)| were givenand functions gelfer-convex of exponential order α,β was also considered. milos, dragoljub [33]considered elementary operators x → ∑nj=1vjxwj that acts on a banach algebra. the ascentestimation and lower bound estimation of an operator was given. barraa and boumazgour [4]showed that the norm of bounded operators more than one on a hilbert space is the same asthe sum of the norms and showed that δs,a,b is convexoid with the convex hull of its spectrumif and only if a and b are convexoid. richard [44] established the cb-norms of elementaryoperators and the lower bounds for norms on b(h). the result was concerned with the operator ua,bx = axb+bxa which showed that ‖ua,b‖≥‖a‖‖b‖ which proved a conjecture of mathieu,other results and formula of ‖ua,b‖cb and ‖ua,b‖ were established. richard [45] provided thehaagerup estimation on the norm of elementary operators that are completely bounded. seddik [46]proved that lower estimate bound ‖tm,n‖ ≥ 2(√2 − 1)‖m‖‖n‖ holds, if it is either a standardoperator algebra or a norm ideal on b(h) and m,n ∈ b(h). florin, alexandra [17] estimatedthe norm of operator hθ,λ = uθ + u∗θ + (λ/2)(vθ + v ∗θ ) which is an element on a c∗-algebra aθ = c ∗(uθ,vθ unitaries : uθvθ = e2πiθvθuθ), and proved that for every λ ∈c and θ ∈ [14, 12] theinequality ‖hθ,λ‖≤ √4 + λ2 − (1 − 1tanθ,λ)(1 − √1+cos24πθ2 )min{4,λ2} holds. this significantlyimproved the inequality ‖hθ,2‖≤ 2√2,θ ∈ [14, 12], conjectured by [18]. the author in [31] consideredcommuting matrices of matrix valued analytic function and established a norm estimate, in particular,two matrices of matrix valued functions on a tensor product in a euclidean space were explored. in [5]the research communicated results on complex symmetric operator theory and showed that two non-trivial examples were of great use in studying schrödinger operators. the work of [43] showed that https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 6 triangle inequality served an upper norm bound for the sum operators that is sup{‖t∗rt +v ∗sv‖ : tandv} are unitaries. the result discussed had relationship to normal dilations, spectral setsand the von neumann inequality. yong, toshiyuki [53] gave a norm estimate on pre-schwarzianderivatives of a specific type of convex functions by introducing a maximal operator of independentinterest of a given kind. the relationship between the convex functions and the hardy spaces wasdiscussed. in [16] the author analyzed the structure of the set d = {y ∈ d(δ) : limn→∞ ∆n(y) = ∆(y)} for convergence of the generators that are pointwise where α is an approximate innerflow on a c∗-algebra t with generator ∆ and ∆n for bounded generators of the approximateflows αn. in fact, the relationship of d and various cores related to spectral subspaces wereexamined. seddik [47] showed that q is a normal operator which is invertible in b(h) if theestimate ‖q⊗q−1 + q−1 ⊗q‖λ ≤ ‖q‖‖q−1‖ + 1‖q‖‖q−1‖ holds, such that ‖.‖λ is the injectivenorm on the tensor product b(h) ⊗ b(h), when q is invertible self-adjoint then the equationbecomes an equality. bonyo and agure [7] characterized the norm of inner derivation on normideal to be equal to the quotient algebra and investigated them when they are implemented bynormal and hyponormal operators on norm ideals. a hyponormal x is a bounded linear operatoron a hilbert space h if x∗x − xx∗ ≥ 0 and is normal if x∗x = xx∗. bonyo and agure [8]investigated the relation of the diameter of the numerical range of an operator b ∈ b(h) and thenorm of inner derivation implemented by b on a norm ideal j and considered the application of s-universality to the relation. bonyo and agure [6] defined inner derivations implemented by a,brespectively on b(h) by δa(y ) = ay − y a, δb(y ) = by − y b and generalized derivation by δa,b (y ) = ay −y b ∀ y ∈ b(h). further, a relationship between the norms of δa,δb and δa,bon b(h) was established, specifically when the operators a,b are s-universal. ber, sukochev [5]showed that for every self-adjoint element b ∈ s(n) a scalar λ0 ∈ r exists such that ∀ ε > 0,then there exists a unital element uε from n satisfy |[b,uε]| ≥ (1 −ε)|b−λ01|. from this result aconsequence is that for any derivation δ on n with the range on an ideal i ⊆ n the derivation δis inner i.e δ(.) = δa(.) = [a,.] and a ∈ i. pablo, jussi, mikael [42] provided theoretic estimate oftwo functions for the essential norm as a composition operator cϕ that acts on the space bmoa;one in terms of the n-th power ϕn denoted by ϕ and the other involved the nevanlinna countingfunction. the research of [20] introduced a new type of norm for semimartangles, the defined norm ofquasimartangales and then characterized the square integrable semimartangales. in [4] the authorgave the result on lower bound of the norms for finite dimensional operators. the work of [14]determined the norm of two-sided symmetric operator in an algebra. more precisely, the lowerbound of the operator using injective tensor norm was investigated. further, the inner derivationnorm on irreducible c∗-algebra was determined and stampfli’s [49] result for these algebras wasconfirmed. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 7 2. preliminaries this section provides the basic concepts which are useful in the sequel. definition 1 ( [1], definition 1.5). a banach ∗-algebra t is called c∗-algebra if ‖tt∗‖ = ‖t‖2, ∀ t ∈t . definition 2 ( [37], definition 2.1). elementary operator t : b(h) → b(h) is defined by tdi,ei (x) = ∑n i=1di x ei ∀ x ∈ b(h) and ∀ di,ei fixed in b(h) where i = 1, ...,n. for b(h), we define the particular elementary operators as below: (i). left multiplication operator ld : b(h) → b(h) by ld(x) = dx, ∀ x ∈ b(h).(ii). right multiplication operator re : b(h) → b(h) by re(x) = xe, ∀ x ∈ b(h).(iii). generalized derivation (implemented by d,e) by δd,e = ld −re.(iv). inner derivation (implemented by d) by δd(x) = dx −xd.(v). basic elementary operator (implemented by d,e) by md,e(x) = dxe, ∀ x ∈ b(h).(vi). jordan elementary operator (implemented by d,e) by ud,e(x) = dxe + exd, ∀ x ∈ b(h). definition 3 ( [49], definition 2.3). a derivation is a map d : u → u satisfying d(f g) = f d(g) + d(f )g for all f ,g ∈ u. definition 4 ( [39], definition 1.2). the maximal numerical range of an operator s is defined by: w0(s) = {β : 〈st,t〉→ β, where ‖t‖ = 1 and ‖st‖→‖s‖}. definition 5 ( [35], definition 2.1). an operator k is norm-attainable if t ∈ h exists which is a unit vector such that ‖kt‖ = ‖k‖. moreover, it is self-adjoint if k = k∗. 3. main results in this section, we give results on norm-attainability conditions an norm estimates for derivations.we begin with the following proposition. proposition 6. let h be a complex hilbert space and b(h) the algebra of all bounded linear operators on h. a ∈ b(h) is norm-attainable if and only if its adjoint a∗ ∈ b(h) is normattainable. proof. given a ∈ b(h) is norm-attainable then we need to show that a∗ ∈ b(h) is norm-attainable. if a ∈ b(h) is norm-attainable then by definition of norm-attainability there exists aunit vector x ∈ h with ‖x‖ = 1 such that ‖ax‖ = ‖a‖. that is, ‖aa∗x‖ = ‖a2x‖. let η = ax‖a‖,then η is a unit vector such that ‖η‖ = 1 this implies that ‖a∗η‖ = ‖a‖ = ‖a∗‖. hence, a∗ isnorm-attainable. � https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 8 the next result gives norm-attainability conditions for operators via the essential numerical range.an analogy of the same can be found in [37]. proposition 7. let a ∈ b(h), λ ∈ wess(a) and η > 0. then there exists a0 ∈ b(h) such that ‖a‖ = ‖a0‖ with ‖a−a0‖ > η. proof. see [37] for the proof. � remark 8. the set of all norm-attainable operators is denoted by na(h), the set of all normattainable self adjoint operators is denoted by na∗(h) and the set of all norm-attainable elementary operators is denoted by ena[b(h)]. at this point, we consider norm-attainability in a general set up. we begin with the followingproposition. proposition 9. let d be the unit disc of a complex hilbert space h and a : h → h be compact and self adjoint. then there exists x ∈ d such that ‖ax‖ = ‖a‖. proof. by the definition of usual norm, we have ‖a‖ = supx∈d ‖ax‖. so, there exists a sequence x1,x2, ...,xn in d such that ‖axn‖ = ‖a‖. but a is compact so let y0 = limn→∞axn exist in h. suppose y = span{x1,x2}, then it is a closed subspace of h. if we pick a subsequence xnkof xn, then it converges weakly to x and we have done 〈x,x〉 = limk→∞〈xnk,x〉 and |〈xnk,x〉| ≤ ‖xnk‖‖x‖ = 1 for all k. therefore, ‖x‖ ≤ 1 but we cannot have ‖x‖ < 1 since then ‖ax‖ = ‖a‖‖x‖ < ‖t‖ which is a contradiction. thus, ‖x‖ = 1 i.e x ∈ d. hence, the existence of x isshown and thus completes the proof. � at this point, we consider q-normality and q-norm-attainability. lemma 10. let a ∈ na(h) then a is q-norm-attainable if it is q-normal. proof. let a ∈ na(h) be q-normal i.e aqa∗ = a∗aq. raising a∗ to power q and using it toreplace a∗ we have aq(a∗)q = (a∗)qaq. this shows that aq is normal. now aqa∗ = a∗aq byfuglede property. therefore, a is q-normal. however, a ∈ na(h) and aq is normal so it followsthat there exists a unit vector x ∈ h such that ‖aqx‖ = ‖aq‖, for any q ∈ n. hence, aq isnorm-attainable. � remark 11. every norm-attainable operator and every self adjoint operator is q-norm-attainable and q-normal for any q ∈n. however, the converse need not be true in general see [66]. lemma 12. let naq(h) be the set of all q-norm-attainable operators on h. then naq(h)is a closed subset of na(h) which is algebraic if and only if for any a ∈ na(h), a is q-normal. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 9 proof. let a be q-normal and pick λ ∈ k. by premultiplying by λ and postmultiplying by q asa power on the normal a we have (λa)q(λa)∗ = (λa)∗(λa)q. this proves the normality of λa.now if a ∈ na(h) then the converse is true if we take limits over a sequence of vectors in h andalso by proposition 9. therefore, a is a q-normal. � theorem 13. let a ∈ naq(h). then the following conditions are true. (i). a∗ is q-norm-attainable.(ii). v av ∗ is q-normal, for a unitary operator v ∈ naq(h).(iii). a−1 is q-norm-attainable if it exists.(iv). a0 = a/g is q-norm-attainable for some g which is a uniformly invariable subspace of hwhich reduces to a.(v). a0 is uniformly equivalent to a implies a0 is norm-attainable. proof. (i). since a ∈ naq(h), then from lemma 10, aq is q-norm-attainable and so (a∗)q isnorm-attainable. consequently, a∗ is q-norm-attainable.(ii). since v is unitary then v v ∗ = v ∗v = i, where i is the identity operator. by definition ofnorm-attainability and lemma 10 we obtain the desired results.(iii). if a−1 exists then since a is q-norm-attainable, aq is q-norm-attainable. now since a is q-norm-attainable then by lemma 10 aq is q-norm-attainable. but (aq)−1 = (a−1)q is q-norm-attainable. so a−1 is q-norm-attainable.(iv). follows from the fact that g invariant under a.(v). follows from (iii) since v is unitary. � corollary 14. let aq,aq0 ∈ naq(h) be commuting operators, then a,a0 ∈ naq(h). proof. since aq,aq0 ∈ naq(h) are commuting then a,a0 are commuting normal operators. bysupraposinormality of operators in dense classes we have a,a0 ∈ naq(h) and hence are norm-attainable. indeed, aqaq0 = (aa0)q = (a0a)q which is normal and norm-attainable. hence, a,a0 ∈ naq(h). � remark 15. not all q-norm-attainable operators are q-normal. thus, the following example shows that the two commuting q-normal operators need not be q-normal. example 16. let a = [ 1 0 0 1 ] and a0 = [ 0 1 0 0 ] . now a + a0 = [ 1 1 0 1 ] and (a + a0)2 =[ 1 2 0 1 ] are not normal. so a + a0 is not 2-normal. we note that a0 is self-adjoint. lemma 17. the sum of norm-attainable operators is norm-attainable. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 10 proof. consider a,b ∈ b(h). we need to show that the sum of a and b is norm-attainable. for a,b to be norm-attainable then there exists a unit vector x ∈ h such that ‖x‖ = 1, ‖(a + b)x‖ = ‖ax +bx‖ = ‖a+b‖ = ‖a‖+‖b‖. since ‖ax +bx‖≤‖ax‖+‖bx‖≤‖a‖+‖bx‖≤‖a‖+‖b‖then for an orthonormal sequence xn ∈ h we have limn→∞(‖axn + bxn‖) = ‖ax + bx‖. but since a and b are norm-attainable we have ‖ax +bx‖ = ‖(a+b)x‖ = ‖a+b‖ is norm-attainable. � theorem 18. a norm-attainable operator perturbed by an identity operators is norm-attainable. proof. let b ∈ b(h) be norm-attainable. since b is norm-attainable then there exists a unitvector x0 ∈ h, an identity i ∈ b(h) and for every ε > 0 we have ‖(bi)x0‖ ≤ ‖bix0‖ + ε ≤ ‖b‖‖i‖‖x0‖ + ε. since ε is arbitrary then it follows that ‖(bi)x0‖ ≤ ‖b‖‖i‖‖x0‖ = ‖b‖. hence, ‖(bi)x0‖ = ‖b‖. � at this point, we consider norm-attainability for elementary operators. we begin with inner deriva-tions. lemma 19. let δa ∈ e[b(h)], then δa is norm-attainable if there exists a unit vector x0 ∈ h, a ∈ na(h) and 〈ax0,x0〉 ∈ wess(a). proof. for an operator a ∈ na(h) we know that an operator is norm-attainable via essentialnumerical range from proposition 4.2. now, we need to show that δa ∈e[b(h)] is norm-attainable.by the definition of inner derivation, δa = ay0−y0a. since a is norm-attainable then there exists aunit vector x0 ∈ h such that ‖x0‖ = 1, ‖ax0‖ = ‖a‖. by orthogonality let y0 satisfy y0⊥{ax0,x0}and a contractive y0 be defined as a linear transformation y0 : x0 → x0 with ax0 → −ax0 as y0 → 0. since y0 is a bounded linear operator on h, then by norm-attainability ‖y0x0‖ = ‖y0‖ = 1and ‖ay0x0 −y0ax0‖ = ‖ax0 − (−ax0)‖ = 2‖a‖. it follows from lemma 3.1 in [49] that ‖δa‖ = 2‖a‖. by the inner product 〈ax0,x0〉 = 0 ∈ wess(a),it follows that ‖δa‖ = 2‖a‖. therefore, ‖ay0 − y0a‖ = 2‖a‖ = ‖δa‖. hence, δa is norm-attainable. � lemma 20. let a,a0 ∈ b(h). if there exists unit vectors y and y0 on h such that a,a0 are norm-attainable then δa,a0 is also norm-attainable. proof. given the operators a,a0 ∈ b(h) are norm-attainable then we need to show that δa,a0 isalso norm-attainable. we define the generalized derivation by δa,a0 = ay −y a0. since a,a0 arenorm-attainable then there exists unit vectors y and y0 on h such that ‖y‖ = ‖y0‖ = 1, ‖ay‖ = ‖a‖and ‖a0y0‖ = ‖a0‖. by linear dependence of vectors, if y and ay are linearly dependent then wehave ‖ay‖ = η‖a‖y where |η| = 1 and |〈ay,y〉| = ‖a‖. it follows that |〈a0y0,y0〉| = ‖a0‖ whichimplies that ‖a0y0‖ = φ‖a0‖y0 and |φ| = 1. therefore, 〈a0y0‖a0‖,y0〉 = φ = −〈 ay‖a‖,y〉 = −η. if y is https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 11 defined as y : y → y0 and y0 → 0, ‖y‖ = 1 then (ay −y a0)y0 = φ(‖a‖+‖a0‖)y0 which implies ‖ay −y a0‖ = ‖(ay −y a0)y0‖ = ‖a‖ + ‖a0‖ = ‖δa,a0‖. hence, δa,a0 is norm-attainable. � lemma 21. every inner derivation is norm-attainable if and only if it is self-adjoint. proof. let δa ∈ b(h) be norm-attainable then we show that δa = δ∗a. now since δa ∈ b(h)is norm-attainable then there exists a contraction y ∈ b(h) such that ‖δay‖ = ‖δa‖. that is, ‖δ∗aδay‖ = ‖δ 2 ay‖. let η ∈ h be defined as η = δa‖δa‖ then η is contractive such that ‖δ∗aη‖ = ‖δa‖ = ‖δ∗a‖. hence, δa is self-adjoint. conversely, let δa be self-adjoint. now since δ∗a is norm-attainable from the first part, then there exists a contractive m ∈ b(h) such that ‖δ∗am‖ = ‖δ∗a‖, i.e ‖δaδ∗am‖ = ‖δ 2 am‖. let ζ be denoted by ζ = δ∗a‖δ∗ a ‖ where ‖ζ‖ = 1 such that ‖δaζ‖ = ‖δ∗a‖ = ‖δa‖.hence, δa is norm-attainable. � lemma 22. every generalized derivation is norm-attainable if and only if it is implemented by orthogonal projections. proof. let a,a0 ∈ b(h) be orthogonal projections. indeed, to show that a generalized derivationis implemented by orthogonal projections a and a0, it is enough to show that it is self-adjoint ifand only if it is normal as proved in [22]. let δa,a0 : b(h) → b(h) be bounded linear operator on b(h). then exists a unique bounded linear operator δ∗a,a0 : b(h) → b(h) such that 〈δa,a0x,y 〉 = 〈x,δ∗a,a0y 〉, for all x,y ∈ b(h). now, ‖δ∗a,a0y‖ = sup ‖x‖=1 〈δa,a0x,y 〉 ≤ sup ‖x‖=‖y‖=1 ‖δa,a0‖‖x‖‖y‖ = ‖δa,a0‖ so, we conclude that δ∗a,a0 is norm-attainable. conversely, let δa,a0 be norm-attainable. we needto show that it is implemented by orthogonal projections. this follows immediately from [22] andthis completes the proof. � at this point, we give results on upper norm estimates for norm-attainable derivations. we con-sider both inner derivations and generalized derivations. we begin with the following proposition. proposition 23. let a,b ∈ na(h) and δa,b be bounded then ‖δa,b‖≤‖a‖ + ‖b‖. proof. since δa,b is bounded then for fixed a,b ∈ na(h) we have ‖δa,b(x)‖ ≤ ‖ax −xb‖ ≤ ‖ax‖ + ‖xb‖ ≤ ‖a‖‖x‖ + ‖x‖‖b‖. let x be of norm 1 and take supremum over x ∈ na(h)then ‖δa,b‖≤‖a‖ + ‖b‖. � remark 24. if a = b then ‖δa‖≤ 2‖a‖. next, we consider upper bounds in the unit ball of na(h) denoted by [na(h)]0. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 12 lemma 25. let [na(h)]0 be the unit ball of na(h) and s be a fixed element of na(h). let x ∈ [na(h)]0 then ‖δs|[na(h)]0‖≤ 2d(s). proof. since x ∈ [na(h)]0 has norm 1 then we have ‖δs|[na(h)]0(x)‖ = ‖sx −xs‖[na(h)]0 = ‖(s−λ)x−x(s−λ)‖[na(h)]0 ≤‖s−λ‖‖x‖[na(h)]0 +‖x‖‖s−λ‖[na(h)]0. taking the supremumover [na(h)]0, we obtain ‖δs|[na(h)]0‖ ≤ 2‖s −λ‖ and considering the infimum over λ ∈ c weobtain ‖δs|[na(h)]0‖≤ 2 infλ∈c‖s −λ‖ = 2d(s). � remark 26. the restriction of δa|[na(h)]0 i.e δa to [na(h)]0 is a bounded linear operator. next we give an extension of lemma 25 to a generalized derivation in the following theorem. theorem 27. let s,s0 be fixed elements of na(h) then ‖δs,s0|[na(h)]0‖≤‖δs,s0‖. proof. since x ∈ [na(h)]0 has norm 1 then we have ‖δs,s0|[na(h)]0(x)‖ = ‖sx−xs0‖. followingproof of lemma 25 anologously we have ‖δs,s0|[na(h)]0(x)‖≤‖s −λ‖‖x‖[na(h)]0 + ‖x‖‖s0 −λ‖[na(h)]0. taking the supremum over x ∈ [na(h)]0 we obtain ‖δs,s0|[na(h)]0‖≤ infλ∈c(‖s −λ‖ + ‖s0 −λ‖) = ‖δs,s0‖. � corollary 28. every generalized derivation δs,s0 is norm-bounded. proof. this follows immediately from [49] and from theorem 27. this completes the proof. � now, we consider lower bounds for norms of derivations. we begin the following proposition ongeneralized derivation. proposition 29. let s,s0 be fixed elements of na(h) then ‖δs,s0|[na(h)]0‖≥‖s‖ + ‖s0‖. proof. let η,ξ and x be unit vectors in h and φ,ϕ be positive linear functionals such that φ⊗η : h → c and ϕ ⊗ ξ : h → c be of rank 1 defined as (φ ⊗ η)x = φ(x)η and (ϕ ⊗ ξ)x = ϕ(x)ξ,∀x ∈ h,‖x‖ = 1. now we have that ‖(φ⊗η)x‖ = sup{‖(φ⊗η)x‖,‖x‖ = 1} = |φ(x)| = |φ|.similarly, we have ‖(ϕ ⊗ ξ)x‖ = ‖ϕ‖. letting s = φ ⊗ η and s0 = ϕ ⊗ ξ then ‖s‖ = ‖φ‖and ‖s0‖ = ‖ϕ‖. now from corollary 28 we have that every generalized derivation is norm-bounded this implies that ‖δs,s0|[na(h)]0(x)‖ ≥ ‖δs,s0(x)‖ where x ∈ [na(h)]0. therefore, ‖δs,s0|[na(h)]0‖ 2 ≥‖sx−xs0‖2 implying that ‖δs,s0|[na(h)]0‖2 ≥ [‖s‖+‖s0‖]2. taking positivesquare root on both sides we obtain ‖δs,s0|[na(h)]0‖ = ‖δs,s0‖≥‖s‖ + ‖s0‖. � remark 30. if s = s0 then ‖δs,s0‖ = ‖δs‖≥ 2‖s‖. remark 31. from theorem 27 and proposition 3 it is easy to see that ‖δs,s0‖ = ‖s‖ + ‖s0‖ and hence ‖δs‖ = 2‖s‖. https://doi.org/10.28924/ada/ma.3.9 eur. j. math. anal. 10.28924/ada/ma.3.9 13 theorem 32. let s,s0 ∈ na(h) and α1 ∈ w0(s) and α2 ∈ w0(s0). then ‖δs,s0‖ ≥ (‖s‖ 2 − |α1|2)1/2 + (‖s0‖2 −|α2|2)1/2. proof. by definition of w0(s) we have xn ∈ h such that ‖sxn‖ = ‖s‖ and 〈sxn,xn〉 → α1for α1 ∈ w0(s). 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https://doi.org/10.1215/ijm/1258138100 https://doi.org/10.1215/ijm/1258138100 https://doi.org/10.1080/0308108031000122515 https://doi.org/10.1016/j.jmaa.2008.10.008 https://doi.org/10.2140/pjm.1970.33.737 https://doi.org/10.2140/pjm.1970.33.737 https://doi.org/10.2140/pjm.1979.82.257 https://doi.org/10.2140/pjm.1979.82.257 https://doi.org/10.2140/pjm.1991.147.365 https://doi.org/10.2140/pjm.1991.147.365 https://doi.org/10.1090/proc/12664 https://doi.org/10.1017/s0013091504000306 1. introduction 2. preliminaries 3. main results 4. conclusion references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 5doi: 10.28924/ada/ma.2.5 unilateral problem for a viscoelastic beam equation type p-laplacian with strong damping and logarithmic source ducival c. pereira1, geraldo m. de araújo2, carlos a. raposo3,∗ 1department of mathematics, state university of pará, belém, pa, 66113-200, brazil ducival@uepa.br 2department of mathematics, federal university of pará, belém, pa, 66075-110, brazil gera@ufpa.br 3department of mathematics, federal university of são joão del-rei, são joão del-rei, 36307-352, brazil ∗correspondence: raposo@ufsj.edu.br abstract. in this manuscript, we investigate the unilateral problem for a viscoelastic beam equationof p-laplacian type. the competition of the strong damping versus the logarithmic source term isconsidered. we use the potential well theory. taking into account the initial data is in the stabilityset created by the nehari surface, we prove the existence and uniqueness of global solutions by usingthe penalization method and faedo-galerkin’s approximation. 1. introduction we denote the p-laplacian operator by ∆pu = div (|∇u|p−2∇u), which can be extended to amonotone, bounded, hemicontinuos and coercive operator between the spaces w 1,p0 (ω) and its dualby −∆p : w 1,p 0 (ω) → w −1,q(ω), 〈−∆pu,v〉p = ∫ ω |∇u|p−2∇u ·∇v dx. in [3] the authors establish existence of global solution to the problem utt + ∆ 2u − ∆pu + ∫ t 0 g(t − s)∆u(s)ds − ∆ut + f (u) = 0 in ω ×r+, (1.1) u = ∆u = 0 on γ ×r+, (1.2) u(x, 0) = u0, ut(x, 0) = u1 in ω, (1.3) where ω is a bounded domain of rn with smooth boundary γ = ∂ω.equations of the type (1.1) are related to models of elastoplastic microstructure flows. asconsidered by an and peirce [1, 2], they are essentially of the form utt + uxxxx −a(u2x )x = 0. received: 13 nov 2021. key words and phrases. unilateral problem; viscoelastic beam equation type p-laplacian; logarithmic source.1 https://adac.ee https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 2 a more general equation, utt + ∆ 2u −div(σ(|∇u|2)∇u) − ∆ut + h1(ut) + h2(u) = h3(x), was considered by yang et al [22–24]. they studied de existence of attractors and their hausdorffdimensions. another related equation is utt + ∆ 2u −div(f0(∇u)) + kut = ∆(f1(u)) − f2(u), which was considered by chueshov and lasiecka [12] the problem (1.1), with its memory term ∫ t 0 g(t−s)∆u(s)ds, can be regarded as a fourth-orderviscoelastic plate equation with a lower order perturbation of the p-laplacian type. this kind ofproblem can be also regarded as an elastoplastic flow equation with some kind of memory effect.we observe that for viscoelastic plate equation, it is usual consider a memory of the form∫ t 0 g(t − s)∆2u(s)ds, see for instance [10]. however, because the main dissipation of the system (1.1) is given by strongdamping −∆ut, here we consider a weaker memory, acting only on ∆u. there is a large literatureabout stability in viscoelasticity. we refer the reader to [11, 13].a nonlinear perturbation of problem (1.1) is given by utt + ∆ 2u − ∆pu + ∫ t 0 g(t − s)∆u(s)ds − ∆ut + f (u) ≥ 0. (1.4) variational inequality theory was introduced by hartman and stampacchia (1966) [14] as a toolfor the study of partial differential equations with applications principally in mechanics.in [7] the authors investigated the unilateral problem associated with this perturbation, thatis, a variational inequality given for (1.4) (see [16]). making use of the penalization method andgalerkin’s approximations, they established existence and the uniqueness of strong solutions.the unilateral problem is very interesting because, in general, dynamic contact problems arecharacterized by nonlinear hyperbolic variational inequalities. variational inequality theory wasintroduced by hartman and stampacchia (1966) [14] as a tool for the study of partial differentialequations with applications principally in mechanics. bensoussan and lions (1982) [9] used vari-ational inequalities initially in the study of stochastic control. in [5] was obtained a variationalinequality for the navier-stokes operator with variable viscosity. in [6] was studied the contactproblem on the oldroyd model of viscoelastic fluids. by using results from the theory of monotoneoperators, was established the existence of weak solutions. in [8] was studied the problem forparabolic variational inequalities with volterra type operators. the authors proved the existenceand the uniqueness of the solution. for contact problems on elasticity and finite element method,see kikuchi-oden [15] and reference therein. in [18] was studied the unilateral problem for the https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 3 klein-gordon operator with the nonlinearity of kirchhoff-carrier type. by using an appropriate pe-nalization was shown the existence and uniqueness of solutions for the perturbed equation. in [19]was considered the unilateral problem for a nonlinear wave equation with p-laplacian operatorand source term. by using an appropriate penalization, authors obtained a variational inequalityfor the equation perturbed and then the existence of solutions was proved.in this work, we propose to investigate the existence and uniqueness of solutions for the vari-ational inequality associated with the problem (1.4) with the source term f (u) = −|u|r−2u ln |u|.more precisely, we investigate the existence and uniqueness of solutions for the unilateral problem utt + ∆ 2u − ∆pu + ∫ t 0 g(t − s)∆u(s)ds − ∆ut ≥ |u|r−2u ln |u| in ω×r+, (1.5) u(x, 0) = u0(x),ut(x, 0) = u1(x) in ω, (1.6) u(x,t) = ∆u(x,t) = 0 on γ ×r+. (1.7) this work is organized as follows: in section 2 we introduce the notation and some well-knownresults. in section 3 we introduce the potential theory suitable for our problem. in section 4 definestrong solution to the boundary value problem (1.5)-(1.7) and present the theorem of existence ofstrong solution. in section 5 we apply the penalization method. the existence of global solutionsis given by using faedo-galerkin approximation. finally, in section 6 we prove the result ofuniqueness. 2. preliminaries let ω be a bounded domain in rn with the boundary γ of class c2. for t > 0, we denote by qthe cylinder ω×(0,t ), with lateral boundary σ = γ×(0,t ). by 〈·, ·〉 we will represent the dualitypairing between a banach space x and x′, x′ being the topological dual of the space x, and by c we denote various positive constants. the inner product in h10 (ω) and l2(ω) , respectively, willbe denoted by (∇·,∇·), (·, ·). the norm in lp(ω) will be denoted by | · |p.the inequality (1.5) must be satisfied in the following sense. let k = {v ∈ h10 (ω); v ≥ 0 a.e. in ω} be a closed and convex subset of h10 (ω), the unilateral problem consists to find a solution u(x,t)satisfying∫ q (utt + ∆ 2u − ∆pu + ∫ t 0 g(t − s)∆u(s)ds − ∆ut −|u|r−2u ln |u|)(v −ut) ≥ 0, (2.1) for all v ∈ k with ut(x,t) ∈ k a.e. on [0,t ] and the initial and boundary data u = ∆u = 0 in γ × (0,t ), (2.2) u(x, 0) = u0, ut(x, 0) = u1 in ω. (2.3) https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 4 to study the existence and uniqueness of the problem (1.5)-(1.7), let us consider the followinghypotheses: h1. suppose that  2 ≤ p , if n = 1, 2 2 ≤ p ≤ 2n− 2 n− 2 , if n ≥ 3. h2. with respect to the power r , let us suppose that 2 < r < +∞ , if n = 1, 2 2 < r < 2n n− 2 , if n ≥ 3. h3. with respect to the function g : [0, +∞) →r, we will assume that g ∈ c1[0,t ] and g(0) > 0, i = 1 −µ ∫ ∞ 0 g(s)ds > 0, where µ > 0 is the embedding constant for |∇u| ≤√µ |∆u|, for all u ∈ h10 (ω) ∩h2(ω). h4. there exists a constant k1 > 0 such that g′(t) ≤−k1g(t), ∀t ≥ 0. by h1 we have h10 (ω) ∩h 2(ω) ↪→ w 1,2(p−1)0 (ω) ↪→ h 1 0 (ω) ↪→ l 2(ω). the lemmas below will be a important role in this manuscript. lemma 2.1. (sobolev poincaré inequality) let p be a number with 2 < p < ∞ if n = 1, 2 or 2 ≤ p ≤ 2n n− 2 if n ≥ 3, then there exists a constant c > 0 such that |u|p ≤ c|∇u|,∀u ∈ h10 (ω) lemma 2.2. (technical lemma) for v ∈ c1(0,t ; h10 (ω)), we have∫ ω ∫ t 0 g(t − s)∇v ·∇vtdsdx = 1 2 (g′ �∇v)(t) − 1 2 g(t)|∇v(t)|2 − 1 2 d dt [ (g �∇v)(t) − (∫ t 0 g(s)ds ) |∇v(t)|2 ] , where (g �∇u)(t) = ∫ t 0 g(t − s)|∇u(s) −∇u(t)|2ds. proof. differentiating the term (g �∇u)(t) we arrive to the above inequality. � https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 5 3. potential well in this section, we use the potential well theory, a power full tool in the study of the globalexistence of solution in partial differential equation. see payne-sattinger [17]. it is well-knownthat the energy of a pde system, in some sense, splits into kinetic and potential energy. the sourceterm induces potential energy in the system that acts in opposition to the effect of the stabilizingmechanism. in this sense, it is possible that the energy from the source term destabilizes all thesystem and produces a blow-up in a finite time. to provide a global solution, we are able toconstruct a stability set corresponding to the source term created from the nehari manifold, see y.ye [20]. in the stability set, there exists a valley or a well of the depth d created in the potentialenergy. if d is strictly positive, then we find that, for solutions with the initial data in the goodpart of the potential well, the potential energy of the solution can never escape the potential well.in general, the energy from the source term causes the blow-up in a finite time. however, thegood part of the potential well is an invariant set where it remains bounded. as a result, the totalenergy of the solution remains finite for any time interval [0,t ], providing the global existence ofthe solution.for the model considered here, the total energy is given by e(t) = 1 2 [ |ut(t)|2 + |∆u(t)|2 + 2 p |∇u(t)|pp + (g �∇u)(t) − (∫ t 0 g(s)ds ) |∇u(t)|2 + 2 r2 |u(t)|rr − 2 r ∫ ω |u(t)|r ln |u(t)|dx ] (3.1) and satisfies d dt e(t) ≤−|∇ut(t)|2. (3.2) from (h3) we get i(t) = 1 2 [ |ut(t)|2 + ( 1 −µ ∫ t 0 g(s)ds ) |∆u(t)|2 +(g �∇u(t)+ 2 p |∇u(t)|pp + 2 r2 |u(t)|rr − 2 r ∫ ω |u(t)|r ln |u(t)|dx ] . (3.3) then, we introduce the functional j : h10 (ω) ∩h2(ω) →r defined by j(u) = 1 2 [( 1 −µ ∫ t 0 g(s)ds ) |∆u|2 + 2 p |∇u(t)|pp + (g �∇u)(t) + 2 r2 |u(t)|rr − 2 r ∫ ω |u(t)|r ln |u(t)|dx ] . (3.4) https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 6 for u ∈ h10 (ω) ∩h2(ω), we have j(λu) = λ2 2 ( 1 −µ ∫ t 0 g(s)ds ) |∆u|2 + λp p |∇u(t)|pp + λ 2 (g �∇u)(t) + λr r2 |u(t)|rr − λr r ∫ ω |u(t)|r ln |u(t)|dx. (3.5) associated with j, we have the well-known nehari manifold given by n def= { u ∈ h10 (ω) ∩h 2(ω) \{0}; [ d dλ j(λu) ] λ=1 = 0 } (3.6) or equivalently, n = { u ∈ h10 (ω) ∩h 2(ω)) \{0}; ( 1 −µ ∫ t 0 g(s)ds ) |∆u|2 + |∇u(t)|pp + 1 2 (g �∇u)(t) = ∫ ω |u(t)|r ln |u(t)|dx } . (3.7) we define as in the mountain pass theorem due to ambrosetti and rabinowitz [4] d def = inf u∈(h10 (ω)∩h2(ω)\{0} sup λ≥0 j(λu). similar to the result in [21] one has 0 < d = inf u∈n j(u).now, we introduce w = {u ∈ h10 (ω) ∩h 2(ω); j(u) < d}∪{0} and partition it into two sets w = w1 ∪w2 as follows w1 = { u ∈ w ; ( 1 −µ ∫ t 0 g(s)ds ) |∆u|2 + 2 p |∇u(t)|pp + (g �∇u)(t) + 2 r2 |u(t)|rr > 2 r ∫ ω |u(t)|r ln |u(t)|dx } ∪{0} (3.8) and w2 = { u ∈ w ; ( 1 −µ ∫ t 0 g(s)ds ) |∆u|2 + 2 p |∇u(t)|pp + (g �∇u)(t) + 2 r2 |u(t)|rr < 2 r ∫ ω |u(t)|r ln |u(t)|dx } . (3.9) so, we define by w1 the set of stability for the problem (1.5)-(1.7), and before starting the sectionof existence and uniqueness of solution, we will prove that w1 is invariant set for sub-critical initialenergy. proposition 1. let u0 ∈ w1 and u1 ∈ h10 (ω). if e(0) < d then u(t) ∈w1. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 7 proof. let t > 0 be the maximum existence time. from (3.2) we get e(t) ≤ e(0) < d, for all t ∈ [0,t ). and then, 1 2 ∫ ω |ut(t)|2 dx + j(u(t)) < d, for all t ∈ [0,t ). (3.10) arguing by contradiction, we suppose that there exists a first t0 ∈ (0,t ) such that i(u(t0)) = 0and i(u(t)) > 0 for all 0 ≤ t < t0, that is,( 1 −µ ∫ t 0 g(s)ds ) |∆u(t0)|2 + 2 p |∇u(t0)|pp + (g �∇u)(t0) + 2 r2 |u(t0)|rr = 2 r ∫ ω |u(t0)|r ln |u(t0)|dx from the definition of n , we have that u(t0) ∈n , which leads to j(u(t0)) ≥ inf u(t)∈n j(u(t)) = d. we deduce 1 2 ∫ ω |ut(t0)|2 dx + j(u(t0)) ≥ d, which contradicts with (3.10). then u(t) ∈w1 for all t ∈ [0,t ). � 4. existence of strong solutions next, we shall state the main results of this paper. theorem 4.1. consider the space h3γ(ω) = {u ∈ h 3(ω)|u = ∆u = 0 on γ}. if u0 ∈ w1 ∩h3γ(ω),j(u0) < d,u1 ∈ h 1 0 (ω) and the hypothesis (h1)-(h4) holds, then there exists a function u : ω × (0,t ) →r such that u ∈ l∞(0,t ; (h10 (ω) ∩h 2(ω))) ∩l∞(0,t ; h3γ(ω)), (4.1) ut ∈ l∞(0,t ; l2(ω)) ∩l2(0,t ; h10 (ω) ∩h 2(ω)), (4.2) utt ∈ l∞(0,t ; h−1(ω)), (4.3) ut(t) ∈ k a.e. in [0,t ], (4.4) ∫ t 0 [ 〈utt,v −ut〉 + (∆2u,v −ut) − (∆pu,v −ut) + (∫ t 0 g(t − s)∆u(s)ds,v −ut ) − (∆ut,v −ut) − (|u|r−2u ln |u|,v −ut) ] ≥ 0, (4.5) https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 8 for all v ∈ l2(0,t ; h10 (ω)), v(t) ∈ k a.e. in t and initial data u(0) = u0, ut(0) = u1. the proof of theorem 4.1 is given in section 5 by the penalization method. it consists in con-sidering a perturbation of the problem (1.5) adding a singular term called penalization, dependingon a parameter � > 0. we solve the mixed problem in q for the penalization operator and theestimates obtained for the local solution of the penalized equation, allow to pass to limits, when �goes to zero, in order to obtain a function u which is the solution of our problem. first of all, letus consider the penalization operator β : h10 (ω) −→ h −1(ω) associated to the closed convex set k, cf. lions [16], p. 370. the operator β is monotonous,hemicontinuous, takes bounded sets of h10 (ω) into bounded sets of h−1(ω), its kernel is k and β : l2(0,t ; h10 (ω)) −→ l 2(0,t ; (h−1(ω)) is monotone and hemicontinous. the penalized problem associated with the variational inequality(1.5)-(1.7), consists in given 0 < � < 1, find u� satisfying u�tt + ∆ 2u� − ∆pu� + ∫ t 0 g(t − s)∆u�(s)ds − ∆u�t + 1 � (β(u�t )) −|u �|r−2u� ln |u| = 0, in q (4.6) and u�(x, 0) = u�0(x),u � t (x, 0) = u�1(x) in ω. u�(x.t) = ∆u�(x,t) = 0 on ∂ω ×r+. (4.7) definition 4.2. suppose that u�0 ∈ w1, j(u�0) < d, u�1 ∈ h10 (ω) and hypothesis (h1) − (h4) holds. a strong solution to the boundary value problem (4.6)-(4.7) is a function u� such that u� ∈ l∞(0,t ; h10 (ω) ∩h 2(ω)), u�t ∈ l ∞(0,t ; l2(ω)) ∩l2(0,t ; h10 (ω)), u�tt ∈ l 2(0,t ; (h10 (ω) ∩h 2(ω))′) satisfying for all w ∈ h10 (ω) ∩h 2(ω) d dt (u�t (t),w) + (∆u �(t), ∆w) + (−∆pu�(t),w) + ∫ t 0 g(t − s)(∆u�(s),w)ds +(∇u�t (t),∇w) + 1 � (β(u�t (t)),w)−(|u �(t)|r−2u�(t) ln |u�(t)|),w) = 0 https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 9 and initial data u�(0) = u�0, u � t (0) = u�1. the solution of problem (4.6)-(4.7) is given by the following theorem: theorem 4.3. assume that hypotheses (h1) − (h4) holds, u�0 ∈ w1, j(u�0) < d and u�1 ∈ h10 (ω), (4.8) then, for each 0 < � < 1, there exists a function u� strong solution of (4.6)-(4.7). 5. penalization method in order to prove theorem 4.1, we first prove the penalized theorem 4.3. the existence ofglobal solutions will be given by using faedo-galerkin method. first we consider the approximateproblem. then we obtain the a priori estimates needed to passage to the limit in the approximatesolutions. 5.1. approximate problem. let {wj} be the galerkin basis given by eigenfunctions of ∆2 withboundary condition u = ∆u = 0 on γ ×r+ and let vm ⊂ n be the subspace spanned by thevectors w1,w2, ...,wm.. consider u�m(t) = m∑ j=1 g�jm(t)wj solution of approximate problem (u�mtt (t),w)+(∆u �m(t), ∆w)+(−∆pu�m(t),w)+ ∫ t 0 g(t − s)(∆u�m(t),w)ds −(|u�m(t)|r−2u�m(t) ln |u�m|,w) + (∇u�m(t),∇w) + 1 � (β(u�mt )(t),w) = 0 (5.1) with initial conditions u�m(0) = u�0m → u�0 strongly in h2(ω) ∩h10 (ω), (5.2) u�mt (0) = u�1m → u�1 strongly in l2(ω). (5.3) the system of ordinary differential equation (5.1) in the variable t has a local solution u�m(t)defined in [0,tm[, 0 < tm ≤ t . in the next step obtain priori estimates for the solution u�m(t) thatpermits us to extend this solution to the whole interval [0,t ]. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 10 5.2. first estimate. we consider w = u�mt in (5.1) to obtain d dt [ 1 2 |u�mt (t)| 2 + 1 2 |∆u�m(t)|2 + 1 p |∇u�m(t)|pp + 1 r2 |u�m(t)|pp − 1 r ∫ ω |u�m(t)|r ln |u�m(t)|dx ] + |∇u�mt (t)|2 + 1 � (β(u�mt (t)),u �m t (t)) = ∫ t 0 g(t − s)(∇u�m(s),∇u�mt (t))ds. (5.4) we have (β(u�mt (t)),u�mt (t)) ≥ 0. then from lemma 2.2 and (h4) 1 2 d dt [ |u�mt (t)| 2 + |∆u�m(t)|2 + 2 p |∇u�m(t)|pp + (g �∇u �m)(t) − (∫ t 0 g(s)ds ) |∇u�m(t)|2 + 2 r2 |u�m(t)|rr − 2 r ∫ ω |u�m(t)|r ln |u�m(t)|dx ] + |∇u�mt (t)|2+ ≤ 1 2 (g′ �∇u�m)(t) − 1 2 g(t)|∇u�m(t)|2 ≤ 0. (5.5) let e�m(t) = 1 2 [ |u�mt (t)| 2 + |∆u�m(t)|2 + 2 p |∇u�m(t)|pp + (g �∇u �m)(t) − (∫ t 0 g(s)ds ) |∇u�m(t)|2 + 2 r2 |u�m(t)|rr − 2 r ∫ ω |u�m(t)|r ln |u�m(t)|dx ] . (5.6) so, by (5.5) and (5.8), we have d dt e�m(t) ≤−|∇u�mt (t)| 2. integrating from 0 to t, t ≤ tm, we obtain e�m(t) + ∫ t 0 |∇u�mt (t)| 2 ≤ e�m(0). (5.7) by (h3), it follows 1 2 [ |u�mt (t)| 2 + ( 1 −µ ∫ t 0 g(s)ds ) |∆u�m(t)|2 +(g �∇u�m)(t)+ 2 p |∇u�m(t)|pp + 2 r2 |u�m(t)|rr − 2 r ∫ ω |u�m(t)|r ln |u�m(t)|dx ] + ∫ t 0 |∇u�mt | 2ds ≤ e�m(t) ≤ e�m(0) = 1 2 |u�1m|2 + c1j(u�0m), (5.8) where c1 > 0 is a positive constant, independent of m and t. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 11 we have j(u0�m) < d and by (5.3), there exists a constant c2 > 0 such that |u�mt (t)| 2 + ( 1 −µ ∫ t 0 g(s)ds ) |∆u�m(t)|2 +(g �∇u�m)(t)+ 2 p |∇u�m(t)|pp + 2 r2 |u�m(t)|rr − 2 r ∫ ω |u�m(t)|r ln |u�m(t)|dx + ∫ t 0 |∇u�mt | 2ds ≤ c2. (5.9) from (3.7) and (5.9) we get ∆u�m ⇀ ∆u� in l∞(0,t ; l2(ω)), (5.10) u�m ⇀ u� in l∞(0,t ; h10 (ω) ∩h2(ω)), (5.11) −∆pu�m ⇀ χ in l2(0,t ; h−1(ω)), (5.12) u�mt ⇀ u � t in l∞(0,t ; l2(ω)) ∩l2(0,t ; h10 (ω)), (5.13) β(u�mt ) ⇀ ψ in l2(0,t ; h−1(ω)). (5.14) follows from(5.11), (5.13) and aubin-lions theorem, for any t > 0, u�m → u� in l2(0,t ; h10 (ω)), strong and a.e. in q. (5.15) now, we prove that χ(t) = −∆pu�(t). we consider x,y ∈ r,p ≥ 2. then the elementaryinequality ∣∣|x|p−2x −|y|p−2y∣∣ ≤ c (|x|p−2 + |y|p−2) |x −y| (5.16) is a consequence of the mean value theorem. using (5.16) and hölder generalized inequality with p− 2 2(p− 1) + 1 2 + 1 2(p− 1) = 1, we deduce for θ ∈d(0,t ) and v ∈ vm,∣∣∣∣∫ t 0 〈(−∆u�mp (t)) − (−∆u � p(t)),v〉pθ(t)dt ∣∣∣∣ = ∣∣∣∣∫ t 0 ∫ ω ( |∇u�m(t)|p−2∇u�m(t) −|∇u�(t)|p−2∇u�(t) ) ∇v dx θ(t) dt ∣∣∣∣ ≤ c|θ|∞ ∫ t 0 ∫ ω ( |∇u�m(t)|p−2 + |∇u�(t)|p−2 ) |∇u�m(t) −∇u�(t)||∇v|dx dt ≤ c1 ∫ t 0 ( |∇u�m(t)|p−2 2(p−1) + |∇u �(t)|p−2 2(p−1) ) |∇u�m(t) −∇u�(t)||∇v|2(p−1) dt ≤ c2 ∫ t 0 |∇u�m(t) −∇u�(t)|dt (5.17) where c1 and c2 are positive constants independent of m and t. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 12 now, from estimate (5.10) and (5.11), we have d dt |∇u�m(t) −∇u�(t)|2 ≤ 2|∆(u�m(t) −u�(t))||∇(u�mt (t) −u � t (t))| ≤ c3, where c3 is a constant independent of m and t. so, |u�m(t) −u�(t)|h10 (ω) ∈ h 1[0,t ] ↪→ c[0,t ], whence ∇u�m(t) →∇u�(t) a. e. in [0,t ]. therefore, χ = −∆pu�. now, we observe that sobolev inequality∫ ω ||u�m(t)|r−2u�m(t) ln |u�m(t)||2dx ≤ |u�m(t)|2r2r ≤ c 2r|∇u�m(t)|2r ≤ µrc2r|∆u�m(t)|r ≤ c4, where c4 is a constant independent of m and t.then (|u�m|r−2u�m ln |u�m|) is bounded in l2(0,t ; l2(ω)) = l2(q). (5.18) using continuity of function s →|s|r−2s ln |s| and (5.15) we have |u�m|r−2u�m ln |u�m|→ |u�|r−2u� ln |u�| a.e. in q. (5.19) by (5.18), (5.19) and applying lions lemma (lemma 1.3, page 12, [16]), we get |u�m|r−2u�m ln |u�m| ⇀ |u�|r−2u� ln |u�| weakly in l2(0,t ; l2(ω)). (5.20) 5.3. second estimate. let us consider the initial data u�0 ∈ h3γ(ω),u�1 ∈ h 1 0 (ω) and u�m0 = ∆u�m0 = 0 on γ. (5.21) we consider w = −∆u�mt in approximate equation (5.1).then we have d dt { 1 2 |∇u�mt (t)| 2 + 1 2 |∇∆u�m(t)|2 } + 〈∆pu�mt (t), ∆u �m t (t)〉 +|∆u�mt (t)|2 + 1 � (β(u�mt (t),−∆u �m t (t)) = (|u�m(t)|r−2u�m ln |u�m(t)|,−∆u�mt (t)) + ∫ t 0 g(t − s)(∆u�m(s), ∆u�mt (t))ds. now, 〈∆pu�m(t), ∆u�mt (t)〉 = d dt 〈∆pu�m(t), ∆u�m(t)〉−j1, https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 13 where j1 = ∫ ω { (p− 2)|∇u�m(t)|p−4(∇u�m(t) ·∇u�mt (t))∇u �m(t) +|∇u�m(t)|p−2∇u�mt (t) } ·∇∆u�m(t)dx. then d dt { 1 2 |∇u�mt (t)| 2 + 1 2 |∇∆u�m(t)|2 + 〈∆pu�m(t), ∆u�m(t)〉 } +|∆u�mt (t)|2 + 1 � (β(u�mt (t)),−∆u �m t (t)) = j1 + j2 + j3. (5.22) where j2 = ∫ ω |u�m(t)|r−2u�m ln |u�m(t)|∆u�mt (t) and j3 = ∫ t 0 g(t − s)(∆u�m(s), ∆u�mt (t))ds. let us the right hand side of (5.22). we denote by c a generic positive constant not dependingon m,t. by estimate (5.9) and p− 2 2(p− 1) + 1 2(p− 1) + 1 2 = 1, |j1| ≤ (p− 1) ∫ ω |∇u�m(t)|p−2|∇u�mt (t)||∇∆u �m(t)|dx ≤ (p− 1)|∇u�m(t)|p−2 2(p−1)|∇u �m t (t)|2(p−1)|∇∆u �m(t)| ≤ c|∇u�mt (t)|2(p−1)|∇∆u �m(t)|. how h10 (ω) ∩h2(ω) ↪→ w 1,20 (ω), we have |∇u�mt (t)| 2 2(p−1) ≤ µ2|∆u �m t (t)| 2, where µ2 > 0 is the corresponding embedding constant. then |j1| ≤ 1 2 |∆u�mt (t)| 2 + c|∇∆u�m(t)|2. (5.23) https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 14 let ω1 = {x ∈ ω : |u�m(t)| < 1} and ω2 = {x ∈ ω : |u�m(t)| ≥ 1}. by (5.9) and sobolevinequality |j2| ≤ ∫ ω1 ||u�m(t)|r−2u�m ln |u�m(t)|∆u�mt (t)|dx + ∫ ω2 ||u�m(t)|r−2u�m ln |u�m(t)|∆u�mt (t)|dx ≤ (e(r − 1))−1 ∫ ω |∆u�mt (t)|dx + (e(r − 1)) −1 ∫ ω |u�m(t)|r−1|∆u�mt (t)|dx ≤ 2(e(r − 1))−2 + 1 8 |∆u�mt (t)| 2 + 2(e(r − 1))−2|u�m(t)|2(r−1) 2(r−1) + 1 8 |∆u�mt (t)| 2 ≤ 2(e(r − 1))−2 + 1 4 |∆u�mt (t)| 2 + 2c(e(r − 1))−2|∇u�m(t)|2(r−1) ≤ c + 1 4 |∆u�mt (t)| 2 (5.24) where we have used |xr−1 ln x| ≤ (e(r − 1))−1 for 0 < x < 1 and ln x ≤ (e(r − 1))−1xr−1, if x ≥ 1. remark 5.1. we note from the cauchy-schwarz inequality and fubini’s theorem follows ‖g �∇u‖l2(q) ≤‖g‖l1(0,∞)‖∇u‖l2(q) again from estimate (5.9) and remark 5.1 |j3| ≤ (∫ t 0 g(t − s)|∆u�m(t)|ds ) |∆u�mt (t)| (5.25) ≤ c‖g‖l1(r+)|∆u �m t (t)| ≤ c + 1 4 |∆u�mt (t)| 2. follows from (5.22)-(5.25) that d dt [ 1 2 |∇u�mt (t)| 2 + 1 2 |∇∆u�m(t)|2 + 〈∆pu�m(t), ∆u�m(t)〉 ] + 1 2 |∆u�mt (t)| 2 + 1 � (β(u�mt (t)),−∆u �m t (t)) ≤ c + c|∇∆u�m(t)|2. (5.26) now, observe that |〈∆pu�m(t), ∆u�m(t)〉| ≤ ∫ ω |∇u�m(t)|p−1|∇∆u�m(t)|dx ≤ |∆u�m(t)|p−1 2(p−1)|∇∆u �m(t)| (5.27) ≤ c + |∇∆u�m(t)|2, https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 15 and then c + |∇∆u�m(t)|2 + 〈∆pu�m(t), ∆u�m(t)〉≥ 0. therefore, there exists c0 > 0 such that d dt [ 1 2 |∇u�mt (t)| 2 + 1 2 |∇∆u�m(t)|2 + 〈∆pu�m(t), ∆u�m(t)〉 ] + 1 2 |∆u�mt (t)| 2 + 1 � (β(u�mt (t)),−∆u �m t (t)) ≤ c0 + c0|∇∆u�m(t)|2 + 〈∆pu�m(t), ∆u�m(t)〉. (5.28) taking into account that (β(u�mt (t),−∆u�mt (t)) ≥ 0, (5.21), integrating from 0 to t and applyinggronwall inequality, we obtain |∇u�mt (t)| 2 + |∇∆u�m(t)|2 + ∫ t 0 |∆u�mt (t)| 2 ≤ c, (5.29) then u�m ⇀ u� in l∞(0,t ; h3γ(ω)), weakly star. (5.30) u�mt ⇀ u � t in l2(0,t ; h10 (ω) ∩h2(ω)), weakly (5.31) ∆u�m ⇀ ∆u� in l∞(0,t ; h10 (ω)), weakly star. (5.32) 5.4. third estimate. let pm be the ortogonal projection pm : l2(ω) → vm, that is pmφ = m∑ n=1 (φ,wj)wj, φ ∈ l2(ω). remark 5.2. by remark 5.1, we observe that if ψ ∈ l2(0,t ; h10 (ω)) then ∫ t 0 g(t − s)ψ(s)ds ∈ l2(0,t ; h−1(ω)) and by (5.12) −∆pu�m ∈ l2(0,t ; (h−1(ω)). we obtain using the notation and ideas of lions [16], pages 75-76, remark 5.2 and estimatesabove that u�mtt ⇀ u � tt in l2(0,t ; (h−1(ω)), weakly. (5.33) (5.31), (5.33) and aubin-lions compactness theorem imply that there exists a subsequence from (u�mt ), still denoted by (u�mt ), such that u�mt → u � t strongly in l2(0,t ; h10 (ω)) and a.e. in q. (5.34) now, we are in position to prove theorem 4.1. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 16 5.5. strong solution. let v ∈ l2(0,t ; h10 (ω)) be v(t) ∈ k a. e. for t ∈ (0,t ). from (4.6)1follows that ∫ t 0 (u�tt,v −u � t )dt + ∫ t 0 (∆2u�,v −u�t )dt + ∫ t 0 (−∆pu�,v −u�t )dt + ∫ t 0 (∫ t 0 g(t − s)∆u�(s)ds,v −u�t ) dt + ∫ t 0 (−∆u�t,v −u � t )dt − ∫ t 0 (|u�|r−2u� ln |u�|,v −u�t )dt = 1 � ∫ t 0 (β(u�t ),u � t −v) dt = 1 � ∫ t 0 (β(u�t ) −βv,u � t −v) dt ≥ 0, (5.35) because v ∈ k (β(v) = 0) and β is monotone.from (5.11), (5.12), (5.15), (5.20), (5.30), (5.31), (5.33), (5.34) and the bannach-steinhauss the-orem, it follows that there exists a subsequence (u�)0<�<1, such that it converge to u as � → 0,that is u� ⇀ u in l∞(r+; h10 (ω) ∩h2(ω)), (5.36) −∆pu� ⇀ −∆pu in l2(0,t ; h−1(ω), (5.37) u� → u in l2(0,t ; h10 (ω))and a.e. in q, (5.38) u� ⇀ u in l∞(0,t ; h3γ(ω)), (5.39) u�t ⇀ ut in l2(0,t ; h10 (ω) ∩h2(ω)), (5.40) u�tt ⇀ utt in l2(0,t ; h−1(ω)), (5.41) |u�|r−2u� ln |u�| ⇀ |u|r−2u ln |u| in l2(0,t ; l2(ω)), (5.42) u�t → ut in l2(0,t ; h10 (ω)) and a.e. in q. (5.43) the convergences above are sufficient to pass to the limit in (5.35) with � > 0 to conclude that(4.5) is valid. to complete the proof of theorem 4.1, it remains to show that ut(t) ∈ k a.e.in the position, we observe that using convergences (5.10)-(5.16) and (5.30)-(5.32), making m → ∞ in (5.1), we can find u� such that u�tt + ∆ 2u� − ∆pu� + ∫ t 0 g(t − s)∆u�(s)ds − ∆u�t −|u �|r−2u� ln |u�| + 1 � β(u�t ) = 0 in l2(0,t ; h−1(ω). (5.44) then, β(u�t ) = �[−u � tt − ∆ 2u� + ∆pu � − ∫ t 0 g(t − s)∆u�(s)ds + ∆u�t + |u �|r−2u� ln |u�|]. (5.45) so, β(u�t ) → 0 in d′(0,t ; h−1ω). https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 17 from (5.45) it follows that β(u�t ) is bounded in l2(0,t ; h−1(ω)), therefore β(u�t ) ⇀ 0 weak in l2(0,t ; h−1ω). (5.46) on the other hand we deduce from (5.45) that 0 ≤ ∫ t 0 (β(u�t ),u � t ) dt ≤ � c. (5.47) thus ∫ t 0 (β(u�t ),u � t )dt −→ 0. (5.48) we have that ∫ t 0 (β(u�t ) −β(ϕ),u � t −ϕ) dt ≥ 0, ∀ϕ in l2(0,t ; h10 (ω)), because β is a monotonous operator. thus,∫ t 0 (β(u�t ),u � t ) dt − ∫ t 0 (β(u�t ),ϕ) dt − ∫ t 0 (β(ϕ),u�t −ϕ) dt ≥ 0. (5.49) from (5.40), (5.46) and (5.48) we obtain∫ t 0 (β(ϕ),ut(t) −ϕ) dt ≤ 0. (5.50) taking ϕ = ut −λv , with v ∈ l2(0,t ; h10 (ω)) and λ > 0, we deduce using the hemicontinuityof β that β(ut(t)) = 0, (5.51) and this implies that ut(t) ∈ k a. e. 6. uniqueness let u1,u2 two solutions of (4.5) , w = u2 −u1 and t ∈ (0,t ). because ut ∈ l2(0,t ; h10 (ω), wecan talking u1t (resp. u2t ) in the inequality (4.5) relative to v2 (resp. v1) and adding up the resultswe obtain − ∫ t 0 (wtt,wt)ds − ∫ t 0 (∆2w,wt)ds + ∫ t 0 (∆pu 1,wt)ds − ∫ t 0 (∆pu 2,wt)ds + ∫ t 0 (∫ t 0 g(t − s)∆w(s)ds,wt ) ds + ∫ t 0 (∆wt,wt)ds − ∫ t 0 (|u1|r−2u1 ln |u1|,wt)ds + ∫ t 0 (|u2|r−2u2 ln |u2|,wt)ds ≥ 0, https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 18 thus, we have 1 2 ∫ t 0 d dt ( |wt(t)|2 + |∆w(t)|2 ) ds + ∫ t 0 |∇wt(t)|2ds ≤ ∫ t 0 〈∆pu1(t) − ∆pu2(t),wt(t)〉ds + ∫ t 0 ∫ t 0 g(t − s)(∇w(s),∇wt(t))dsdσ∫ t 0 ( |u1(t)|r−2u1(t) ln |u1(t)|− |u2(t)|r−2u2(t) ln |u2(t)|,wt(t) ) ds. by lemma 2.2, we derive 1 2 ∫ t 0 d dt { |wt(t)|2 + |∆w(t)|2 − (∫ t 0 g(s)ds ) |∇w(t)|2 + (g �∇w)(t) } ds + ∫ t 0 |∇wt(t)|2ds ≤ ∫ t 0 |〈∆pu1(t) − ∆pu2(t),wt(t)〉|ds + ∫ t 0 ∫ ω ( |u1(t)|r−2u1(t) ln |u1(t)|− |u2(t)|r−2u2(t) ln |u2(t)|,wt(t) ) dxds. (6.1) from mean value theorem, |〈∆pu1(t) − ∆pu2(t),wt(t)〉| ≤ c ( |∇u1(t)|p−2 2(p−1) + |∇u2(t)| p−2 2(p−1) ) |∇w(t)|2(p−1)|∇wt(t)| ≤ c|∆w(t)|2 + 1 4 |∇wt(t)|2, (6.2) for some constant c > 0, and∫ t 0 ∫ ω ( |u1(t)|r−2u1(t) ln |u1(t)|− |u2(t)|r−2u2(t) ln |u2(t)|,wt(t) ) dxds ≤ ∫ t 0 ∫ ω |θu1(t) + (1 −θ)u2(t))|r−2|w(t)||wt(t)|dxds +(r − 1) ∫ t 0 ∫ ω |θu1(t) + (1 −θ)u2|r−2 ln |θu1(t) +(1 −θ)u2(t)||w(t)|wt(t)|dxds = i1 + i2, 0 < θ < 1. (6.3) hence, from the hölder inequality and sobolev inequality, we have∫ ω |θu1(t) + (1 −θ)u2(t))|r−2|w(t)||wt(t)|dx ≤ |θu1(t) + (1 −θ)u2(t)|r−2 n(r−2)|w(t)| 2nn−2 |wt(t)| ≤ cr−21 c2c3|∆w(t)||∇wt(t)| ≤ c|∆w(t)| 2 + 1 4 |∇wt(t)|2, (6.4) where c1 ,c2 and c3 are constants satisfying |θu1(t) + (1 −θ)u2(t)|r−2 n(r−2)| ≤ c1|θu 1(t) + (1 −θ)u2(t)|, |w(t)| 2n n−2 ≤ c|w(t)| ≤ c2|∆w(t)| and |w(t)| ≤ c3|∇w(t)|. https://doi.org/10.28924/ada/ma.2.5 eur. j. math. anal. 10.28924/ada/ma.2.5 19 also we used the condition n(p− 2) < 2n n− 2 . now, using the calculation similar to (5.24), it follows that∫ ω |θu1(t) + (1 −θ)u2|r−2 ln |θu1(t) + (1 −θ)u2(t)|ndx ≤ (e(r − 2)−n)|ω| + (e(r − 2))−n|θu1(t) + (1 −θ)u2(t)|n(r−2) n(r−2) ≤ (e(r − 2)−n)|ω| + (e(r − 2))−1c4|θu1(t) + (1 −θ)u2(t)|n(r−2) ≤ c. (6.5) inserting (6.5) into i2, we have i2 = (r − 1) ∫ t 0 ∫ ω |θu1(t) + (1 −θ)u2|r−2 ln |θu1(t) +(1 −θ)u2(t)||w(t)|wt(t)|dxds ≤ (r − 1) ∫ t 0 (∫ ω ||θu1(t) + (1 −θ)u2|r−2 ln |θu1(t) + θu2(t)||ndx )1 n ×|wt(t)||w(t)| 2n n−2 ds ≤ c|∆w(t)|2 + 1 4 |∇wt(t)|2. (6.6) by (6.1), (6.2), (6.4) and (6.6) we get∫ t 0 d dt { |wt(t)|2 + |∆w(t)|2 − (∫ t 0 g(s)ds ) |∇w(t)|2 + (g �∇w)(t) } ds + ∫ t 0 |∇wt(t)|2ds ≤ c ∫ t 0 (|∆w(t)|2 + |∇wt(t)|2)ds. (6.7) putting, φ(t) = |wt(t)|2 + |∆w(t)|2 − (∫ t 0 g(s)ds ) |∇w(t)|2 + (g �∇w)(t) and using (h3), we have |∆w(t)|2 − (∫ t 0 g(s)ds ) |∇w(t)|2 ≥ i|∆w(t)|2 ≥ 0. as (g �∇w)(t) ≥ 0, we have from (6.7) that ∫ t 0 d dt φ(t) ≤ cφ(t) and because φ(0) = 0, followsfrom the gronwall lemma that |wt(t)|2 + i|∆w(t)|2 ≤ φ(t) ≤ 0, which proves that w = 0 in h10 (ω) ∩h2(ω). references [1] l. an, a. pierce, the effect of microstructure on elastic-plastic models, siam j. appl. math. 54(3) (1994) 708-730. https://doi.org/10.1137/s0036139992238498[2] l. an, a. pierce, a weakly nonlinear analysis of elastoplastic-microstructure models, siam j. appl. math. 55(1)(1995) 136-155. https://doi.org/10.1137/s0036139993255327[3] a. andrade, m. a. jorge silva, t. f. ma, exponential stability for a plate equation with p-laplacian and memoryterms, math. meth. appl. sci. 35(4) (2012) 417-426. https://doi.org/10.1002/mma.1552 https://doi.org/10.28924/ada/ma.2.5 https://doi.org/10.1137/s0036139992238498 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p-laplacianoperator, j. appl. anal. comput. 11(1) (2021) 546-555. https://doi.org/10.11948/20200147[20] y. ye, global existence and asymptotic behavior of solutions for a class of nonlinear degenerate wave equations,differ. equ. nonlinear mech. 2007 (2007) 1-9. https://doi.org/10.1155/2007/19685[21] m. willem, minimax theorems. progress in nonlinear differential equations and their applications 24, birkhouserboston inc. boston, ma, 1996.[22] y. zhijian, longtime behavior for a nonlinear wave equation arising in elastoplastic flow, math. meth. appl. sci.32(9) (2009) 1082-1104. https://doi.org/10.1002/mma.1080[23] y. zhijian, global attractores and their hausdorff dimensions for a class of kirchhoff models, j. math. phys. 51(3)(2010) 032701. https://doi.org/10.1063/1.3303633[24] y. zhijian, j. baoxia, global attractor for a class of kirchhoff models, j. math. phys. 50(3) (2009) 032701. https: //doi.org/10.1063/1.3085951 https://doi.org/10.28924/ada/ma.2.5 https://doi.org/10.1016/0022-1236(73)90051-7 https://doi.org/10.3934/cpaa.2006.5.583 http://ejde.math.txstate.edu https://doi.org/10.1080/00036811.2020.1766028 https://doi.org/10.13189/ms.2019.070504 https://doi.org/10.1137/s0363012902408010 https://doi.org/10.3934/dcds.2006.15.777 https://doi.org/10.3934/dcds.2006.15.777 https://doi.org/10.1007/bf00251609 https://doi.org/10.1007/bf00251609 https://doi.org/10.1007/bf02392210 https://doi.org/10.1007/bf02761595 https://ejde.math.txstate.edu/volumes/2015/137/abstr.html https://ejde.math.txstate.edu/volumes/2015/137/abstr.html https://doi.org/10.11948/20200147 https://doi.org/10.1155/2007/19685 https://doi.org/10.1002/mma.1080 https://doi.org/10.1063/1.3303633 https://doi.org/10.1063/1.3085951 https://doi.org/10.1063/1.3085951 1. introduction 2. preliminaries 3. potential well 4. existence of strong solutions 5. penalization method 5.1. approximate problem 5.2. first estimate 5.3. second estimate 5.4. third estimate 5.5. strong solution 6. uniqueness references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 9doi: 10.28924/ada/ma.2.9 efficient numerical schemes for computations of european options with transaction costs md. shorif hossan1 , md. shafiqul islam1 , md. kamrujjaman2,∗ 1department of applied mathematics, university of dhaka, dhaka, bangladesh shorif@du.ac.bd, mdshafiqul@du.ac.bd 2department of mathematics, university of dhaka, dhaka, bangladesh kamrujjaman@du.ac.bd ∗correspondence: kamrujjaman@du.ac.bd abstract. this paper aims to find numerical solutions of the non-linear black-scholes partial dif-ferential equation (pde), which often appears in financial markets, for european option pricing inthe appearance of the transaction costs. here we exploit the transformations for the computationalpurpose of a non-linear black-scholes pde to modify as a non-linear parabolic type pde with reli-able initial and boundary conditions for call and put options. several schemes are derived rigorouslyusing the finite volume method (fvm) and finite difference method (fdm), which is the novelty of thispaper. stability and consistency analysis assure the convergence of these schemes. we apply theseschemes to various volatility models, such as the leland, boyle and vorst, barles and soner, andrisk-adjusted pricing methodology (rapm). all the schemes are tested numerically. the convergenceof the obtained results is observed, and we find that they are also reliable. finally, we display allthe approximate results together with the exact values through graphical and tabular representations. 1. introduction understanding and accurately evaluating transaction costs in a financial market is vital forsecurity trading, asset pricing, stock market regulation, and many other issues. during the last fewdecades, pricing options more accurately after including realistic assumptions-such as transactioncost, getting more importance from both the traders and the investors.the literature’s [1–6], contains descriptive discussions of options. fischer black and myronscholes [7] worked jointly, and first disclosed the concept of the black-scholes model for options pricing and corporate liabilities, and was published in 1973, while robert merton [8] advanced thismodel in the article "theory of rational option pricing" in the same year. their derived equation isbased on the assumption that there are no fees for buying and selling options and stocks, as wellas no trade barriers (i.e., no commissions and transaction costs). in other words, this model makes received: 20 dec 2021. key words and phrases. nonlinear black-scholes pde; option pricing; volatility model; finite volume method; finitedifference method. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.9 https://orcid.org/0000-0003-4115-9002 https://orcid.org/0000-0001-7121-3386 https://orcid.org/0000-0002-4892-745x eur. j. math. anal. 10.28924/ada/ma.2.9 2 a friction-less assumption (which is indispensable, as actual costs correlated with practical marketapplications) to implement a hedging plan for any contingent claim of the european type.various studies have been conducted about the linear black-scholes model [9–15] though itadopts the unrealistic assumption of no transaction costs. several studies have been attempted toevaluate the price of european options [16–23], american options [24–28], asian options [29, 30],and barrier options [31] in a completely friction-less market. recently, the fractional black-scholesmodel [32–34] received some attention.contrastingly, the non-linear black-scholes pde, where the non-linear term denotes the pres-ence of transaction costs, is of great importance to our contemporary world over some time both interms of approach and applicability. several models [35] consider transaction costs: leland model,paras, and avellaneda model, boyle and vorst model, hodges and neuberger model, barles andsoner model, and rapm (risk-adjusted pricing methodology) model. if the transaction cost pa-rameters are equal to zero, all of these non-linear transaction cost models are unvarying with thelinear model.soner et al. [36] showed that there is no nontrivial hedging portfolio for option pricing withtransaction costs. they also suggested that the best hedging strategy is buying an asset andtaking on it for a certain period as a call or put option. leland [37] inaugurates the idea of usingtransaction costs at discrete times. he also indicated that the hedging error could be minimized ifthe length of re-balancing frequency approaches zero. later, boyle and vorst [38] demonstrated fur-ther in a discrete-time framework with a binomial tree model for the option prices with proportionaltransaction costs, and it is pretty accurate for possible parameter values. besides, dewynne etal. [39] considered path-dependent and exotic options with transaction costs. recently, asymptoticanalysis [40] and markov chain approximation [41] were also studied for pricing european optionswith transaction costs in some previous literature.on the other hand, few researchers [42–46, 49–51] paid their attention to solve the non-linearblack-scholes equation numerically. for example, the exponential time differencing (etd) method[44] was applied to solve the non-linear black–scholes model for pricing american options with ahighly stable and efficient transaction cost. lesmana and wang [45] developed the numerical methodbased on an upwind finite difference scheme for a non-linear parabolic pde, and they attemptedto pricing european options under transaction costs. ankudinova and ehrhardt [46] focused on thenon-linear black-scholes equation for european call options using several transaction cost modelsas well as crank–nicolson and rigal compact schemes. r. l. valkov [47] has solved the non-linearblack–scholes-bellman model numerically as well as discuss the monotonicity and consistency ofhis suggested scheme in considerable detail. a monotone finite volume spatial discretization and asecond-order predictor-corrector scheme in time are considered by radoslas valkov [48] to handlethe black–scholes equation with uncertain volatility and dividend. the applicability of implicit https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 3 numerical schemes for the valuation of contingent claims in non-linear black–scholes models hasbeen discussed by pascal heider [49]. he also studied the practical implications of the derivedstability criteria on relevant numerical examples. he claimed that if certain stability requirementsare satisfied, it is possible to construct convergent implicit algorithms for non-linear black–scholesequations. ekaterina dremkova and matthias ehrhardt [50] have solved non-linear black–scholesequations for american options with a non-linear volatility function using various compact finitedifference techniques to improve the order of the accuracy. the existence and uniqueness ofsolutions to the well-known non-linear black-scholes equation have been demonstrated by naoyukiishimura [51] for both in the classical and weak senses.however, in this paper, we work on approximating non-linear black-scholes pde for valuingeuropean options when there are transaction costs. for this, we organize the present research workas follows: we modify the original model into parabolic type pde exploiting the transformations [46]which are written in section 2. a brief description of different volatility models is given in section3 subsequently. section 4 is devoted to discretize the transformed parabolic type equation byusing some numerical schemes. stability and consistency analysis are included in sections 5 and6, respectively. in section 7, numerical examples are given to show the efficacy of the proposedschemes. subsequently, a general conclusion is drawn in section 8. finally, all relevant referencesare included. 2. the model equation this section considers a non-linear black-scholes pde and modifies it to a non-linear parabolictype equation with appropriate and available transformations, which would be easy to computenumerically. let us consider the non-linear black-scholes pde [46], ∂f ∂t + rs ∂f ∂s + 1 2 σ̃2s2 ∂2f ∂s2 − rf = 0, 0 < s < ∞,t ∈ (0,t ) (1) subject to the terminal and boundary conditions for european call and put options: f (s,t ) = max(s − k, 0),f (s,t) = 0 when s = 0,f (s,t) = s − ke−r(t−t), when s → ∞and f (s,t ) = max(k − s, 0),f (s,t) = ke−r(t−t), when s = 0,f (s,t) = 0, when s → ∞respectively. throughout this paper, we use the notations: f = f (s,t) = the option price, s =stock price, k = strike price, t = maturity time, r = interest rate, t = time in years, and σ̃ = σ̃ ( t,s, ∂f ∂s , ∂ 2f ∂s2 ) depends on the volatility model. now consider the transformations [46] as given below, y = ln ( k−1s ) ,τ = 1 2 σ2(t − t) and u(y,t) = k−1e−yf (s,t) and substituting these into equation (1) to obtain the following non-linear parabolic pde https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 4 ∂u ∂τ = 2r σ2 ∂u ∂y + ( σ̃ σ )2 ( ∂2u ∂y2 + ∂u ∂y ) ,ymin < y < ymax,τ ∈ ( 0, σ2 2 t ) (2) with the modified initial and boundary conditions for european call and put options: u(y, 0) = max ( 1 −e−y, 0 ) as y ∈ (−∞,∞),u(y,τ) = 0 as y →−∞,u(y,τ) = 1−e−(y+2rτ/σ2)as y →∞, and u(y, 0) = max (e−y − 1, 0) as y ∈ (−∞,∞),u(y,τ) = e−(y+2rτ/σ2) as y →−∞, u(y,τ) = 0 as y →∞, respectively. 3. volatility models this section concerns four stochastic volatility models to discretize the non-linear black-scholespde, whose solution provides the option price for transaction fees. we give a short description,but details are available in some previous literature [46]. leland volatility model (lvm). leland [37] developed a technique for replicating options in thepresence of transactions costs for a small time interval. he proposed that the option price isthe solution of the non-linear black-scholes equation (1) but with the adjusted volatility [46] asfollows: σ̃ = σ √ 1 + √ 2 π µ σ √ ∆t sign (fss) (3) where, σ is the original volatility, µ is the round-trip transaction cost per unit dollar of the trans-action, and ∆t is the transaction frequency. in this formula, both µ and ∆t are assumed to be smallwhile keeping the ratio µ√ ∆t of order one. boyle and vorst volatility model (bvvm). boyle and vorst [38] derived a method for calculatingoption prices in a discrete-time where option price meets to black-scholes price with the modifiedvolatility [46] given by σ̃ = σ √ 1 + µ σ √ ∆t sign (fss) (4) where, σ,µ, and ∆t represents the same meaning as leland. barles and soner volatility model (bsvm). barles and soner [43] evolved a model using theutility function approach of hodges and neuberger [52] along with asymptotic analysis of partialdifferential equations. for this case, the formula for the modified volatility [46] is given by σ̃ = σ √ 1 + er(t−t)a2s2fss (5) where, µ = a√∈ is the round-trip transaction cost per unit dollar of the transaction for someconstant a > 0 and ∈→ 0. https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 5 rapm volatility model (rapmvm). kratka [53] took the first step for this model and laterimproved by jandačka and ševčovič [54]. here the modified volatility is of the form [46] σ̃ = σ √ 1 + 3 × 3 √ c2m 2π sfss (6) where, m ≥ 0 is the transaction cost measure and c ≥ 0 is the risk premium measure. 4. derivations of computational schemes in this section, we derive five computational schemes, in detail, for equation (2) using twowell-known numerical methods. 4.1. dufort-frankel finite difference scheme. the dufort-frankel fd scheme [55] can be appliedto solve various kinds of problems which occur in finance. this scheme is a multi-step method, andrequires another scheme for simulating the first temporal vector. in this formulation ∂u ∂τ , ∂u ∂y , and ∂2u ∂y2are discretized by central difference and uj i is replaced by (uj+1 i + u j−1 i ) /2. thus, discretizingequation (2) by dufort-frankel fdm, we obtain 1 2∆τ ( u j+1 i −uj−1 i ) = ( σ̃ σ )2 [ 1 (∆y)2 ( u j i−1 − ( u j+1 i + u j−1 i ) + u j i+1 )] + ( σ̃ σ )2 [ 1 2∆y ( u j i+1 −uj i−1 )] + r σ2∆y ( u j i+1 −uj i−1 ) or, equivalently u j+1 i =u j−1 i + 2r(∆τ)(∆y) σ2 ( u j i+1 −uj i−1 ) + ∆τ ∆y ( σ̃ σ )2 ( u j i+1 −uj i−1 ) + 2(∆τ) (∆y)2 ( σ̃ σ )2 ( u j i−1 −u j+1 i −uj−1 i + u j i+1 ) which can be written as u j+1 i = aiu j i−1 + biu j i+1 + ciu j−1 i ; i = 0, 1, 2, . . . ,n− 1; j = 0, 1, 2, . . . ,m− 1 (7) where ai = [ (∆y)2 + 2(∆τ) ( σ̃ σ )2]−1 × [ (∆τ) ( σ̃ σ )2 (2 − ∆y) − (∆y)(∆τ) 2r σ2 ] , bi = [ (∆y)2 + 2(∆τ) ( σ̃ σ )2]−1 × [ (∆τ) ( σ̃ σ )2 (2 + ∆y) + (∆y)(∆τ) 2r σ2 ] , and ci = [ (∆y)2 + 2(∆τ) ( σ̃ σ )2]−1 × [ (∆y)2 − 2(∆τ) ( σ̃ σ )2] which is our proposed dufort-frankel finite difference scheme (dffds). https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 6 4.2. laasonen finite difference scheme. the laasonen finite difference scheme [55] can be ap-plied to solve linear and non-linear partial differential equations. this method metamorphosedpartial differential equations into a system of linear algebraic equations. in this formulation ∂u ∂τis approximated by a central differencing at a step ∆τ 2 , and ∂u ∂y , ∂ 2u ∂y2 are approximated by centraldifferences at time levels j + 1. now the discretized form of equation (2) is as follows 1 ∆τ ( u j+1 i −uj i ) = 1 2(∆y)2 ( σ̃ σ )2 [ 2 ( u j+1 i−1 − 2u j+1 i + u j+1 i+1 ) + ∆y ( u j+1 i+1 −uj+1 i−1 )] + r σ2∆y ( u j+1 i+1 −uj+1 i−1 ) after simplification, we get[ r∆τ ∆yσ2 + ∆τ 2(∆y)2 ( σ̃ σ )2 (∆y − 2) ] u j+1 i−1 + [ 1 + 2∆τ (∆y)2 ( σ̃ σ )2] u j+1 i − [ r∆τ ∆yσ2 + ∆τ 2(∆y)2 ( σ̃ σ )2 (∆y + 2) ] u j+1 i+1 = u j i the above equation reduces to diu j+1 i−1 + (1 + ei ) u j+1 i + fiu j+1 i+1 = u j i ; i = 0, 1, 2, . . . ,n− 1; j = 0, 1, 2, . . . ,m− 1 (8) where di = r∆τ ∆yσ2 + ∆τ 2(∆y)2 ( σ̃ σ )2 (∆y − 2),ei = 2∆τ (∆y)2 ( σ̃ σ )2 and fi = − r∆τ ∆yσ2 − ∆τ 2(∆y)2 ( σ̃ σ )2 (∆y + 2) 4.3. finite volume schemes. the finite volume scheme is a scheme of solving different kinds oftime-dependent or independent partial differential equations in algebraic equations. in this scheme,we divide the physical space into a finite number of control volumes. in this section, we describeit in a few lines, but details are available in the previous study [56] conducted by malalasekera etal. applying the finite volume integration in equation (2) over a control volume (cv) with a finite timestep ∆τ, we obtain ∫ τ+∆τ τ ∫ cv ∂u ∂τ dv dτ = ( 2r σ2 + ( σ̃ σ )2)∫ τ+∆τ τ ∫ cv ∂u ∂y dv dτ + ( σ̃ σ )2 ∫ τ+∆τ τ ∫ cv ∂2u ∂y2 dv dτ https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 7 after rearranging, we get∫ cv [∫ τ+∆τ τ ∂u ∂τ dτ ] dv = ( 2r σ2 + ( σ̃ σ )2)∫ τ+∆τ τ [∫ cv ∂u ∂y dv ] dτ + ( σ̃ σ )2 ∫ τ+∆ τ [∫ cv ∂2u ∂y2 dv ] dτ applying gauss’s divergence theorem, the above equation leads ( up −u0p ) ∆v = 1 2 ( 2r σ2 + ( σ̃ σ )2) a ∫ τ+∆τ τ (ue −uw ) dτ + ( σ̃ σ )2 ∫ τ+∆τ τ [( a ue −up δype ) − [( a up −uw δywp )]] dτ (9) for 0 ≤ θ ≤ 1, we assume ∫ τ+∆τ τ updτ = [ θup + (1 −θ)u0p ] ∆τ (10) applying equation (10) into equation (9) and dividing by we get ( up −u0p ) ∆y ∆τ = 1 2 ( 2r σ2 + ( σ̃ σ )2)[ θ (ue −uw ) + (1 −θ) ( u0e −u 0 w )] + ( σ̃ σ )2 θ ( ue −up δype − up −uw δywp ) + ( σ̃ σ )2 (1 −θ) ( u0e −u 0 p δype − u0p −u 0 w δywp ) (11) for convenience, we put δywp = δype = ∆y on the following three schemes. explicit scheme. substitution of θ = 0 into equation (11) gives the following explicit discretizedequation, ( up −u0p ) ∆y ∆τ = 1 2 ( 2r σ2 + ( σ̃ σ )2)( u0e −u 0 w ) + 1 ∆y ( σ̃ σ )2 ( u0e − 2u 0 p + u 0 w ) this equation may be re-writtens as up = αiu 0 w + (1 + βi ) u 0 p + γiu 0 e (12) where αi = ∆τ (∆y)2 ( σ̃ σ )2 − ∆τ 2∆y ( 2r σ2 + ( σ̃ σ )2) , βi = − 2∆τ (∆y)2 ( σ̃ σ )2 and γi = ∆τ (∆y)2 ( σ̃ σ )2 + ∆τ 2∆y ( 2r σ2 + ( σ̃ σ )2) which is the desired finite volume explicit scheme (fves). https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 8 crank-nicolson scheme. putting θ = 1 2 into equation (11), we get the following crank-nicolsondiscretized equation, ( up −u0p ) ∆y ∆τ = 1 4 ( 2r σ2 + ( σ̃ σ )2)( ue −uw + u0e −u 0 w ) + 1 2∆y ( σ̃ σ )2 ( ue − 2up + uw + u0e − 2u 0 p + u 0 w ) after simplification, we get the following equation λiuw + (1 + ξi ) up + ηiue = −λiu0π + (1 −ξi ) u 0 p −ηiu 0 e (13) where λi = ∆τ 4∆y ( 2r σ2 + ( σ̃ σ )2) − ∆τ 2(∆y)2 ( σ̃ σ )2 ,ξi = ∆τ (∆y)2 ( σ̃ σ )2 and ηi = − [ ∆τ 4∆y ( 2r σ2 + ( σ̃ σ )2) + ∆τ 2(∆y)2 ( σ̃ σ )2] which is the proposed finite volume crank-nicolson scheme (fvcns). fully implicit scheme. substitution of θ = 1 into equation (11) leads to the following form: ( up −u0p ) ∆y ∆τ = 1 2 ( 2r σ2 + ( σ̃ σ )2) (ue −uw ) + 1 ∆y ( σ̃ σ )2 (ue − 2up + uw ) and the reduced formula is then qiuw + (1 + ri ) up + siue = u 0 p (14) where qi = ∆τ 2∆y ( 2r σ2 + ( σ̃ σ )2) − ∆τ (∆y)2 ( σ̃ σ )2 , ri = 2∆τ (∆y)2 ( σ̃ σ )2 and si = − ∆τ 2∆y ( 2r σ2 + ( σ̃ σ )2) − ∆τ (∆y)2 ( σ̃ σ )2 which is our proposed finite volume fully implicit scheme (fvfis). 5. stability of the numerical schemes to test the stability of the derived schemes in section 4, with the help of the von-neumannstability method [55], let us consider a fourier component for uj i and u0p as u j i = ujeiθi and u0p = ujeiθi (15) where i = √−1, i.e., imaginary unit, uj is the amplitude at a time level j,θ(= r∆y) is the phaseangle, r is the wave number in the x-direction, and i represents the index of the node. https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 9 similarly, u j∓1 i±1 = u j∓1eiθ(i±1) u0w = u jeiθ(i−1) u0e = u jeiθ(i+1) up = u j+1eiθi uw = u j+1eiθ(i−1) ue = u j+1eiθ(i+1) (16) for convenience, let us suppose that g = uj+1 uj . thus, the stability requirement is |g|2 ≤ 1.applying equation (15) and equation (16) into equation (7), and dividing by eiθi , we get |g|2 = 1 4 {4∆τ (σ̃ σ )2 cos θ± √ a }2 + 4(∆τ)2(∆y)2 ( 2r σ2 + ( σ̃ σ )2)2 ( 1 − cos2 θ ) × ( (∆y)2 + 2∆τ ( σ̃ σ )2)−2 (17) where a =16(∆τ)2 ( σ̃ σ )4 cos2 θ− 4(∆τ)2(∆y)2 ( 2r σ2 + ( σ̃ σ )2)2 ( 1 − cos2 θ ) + 4(∆y)4 − 16(∆τ)2 ( σ̃ σ )4 + i16(∆τ)2(∆y) ( σ̃ σ )2 ( 2r σ2 + ( σ̃ σ )2) cos θ √ 1 − cos2 θ for extremum value of |g|2, solving d|g|2 d(cos θ) = 0 for cos θ, and substituting it into d2|g|2 d(cos θ)2 < 0. thenfrom equation (17), we cannot confirm that the maximum value of |g|2 would occur. however, theextreme values of cos θ must yet be investigated. for cos θ = 1, equation (17) gives |g|2 = 1, andthe stability requirement is satisfied. for cos θ = −1, equation (17) also yields |g|2 = 1 and, andthe stability requirement is satisfied. thus, the dffds proposed in equation (7) is unconditionallystable. similarly, we can show that lfds and fvfis wrote in equation (8) and equation (14), respectively,both are unconditionally stable. again, applying equation (15) and equation (16) into equation (12) and dividing by eiθi , we get g = 1 + 2∆τ (∆y)2 ( σ̃ σ )2 (cos θ− 1) + i ∆τ ∆y ( 2r σ2 + ( σ̃ σ )2) sin θ then we may obtain easily, |g|2 = { 1 + 2∆τ (∆y)2 ( σ̃ σ )2 (cos θ− 1) }2 + ( ∆τ ∆y )2 ( 2r σ2 + ( σ̃ σ )2)2 ( 1 − cos2 θ ) (18) for extremum value of |g|2 such that d|g|2 d(cos θ) = 0, we can find cos θ = 1 ∆τ × [ 2 ( σ̃ σ )2 − 4 ∆τ (∆y)2 ( σ̃ σ )4] × (2r σ2 + ( σ̃ σ )2)2 − 4 ∆τ (∆y)2 ( σ̃ σ )4−1 (19) https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 10 considering d2|g|2 d(cos θ)2 < 0, and substituting the value of cos θ from equation (19) into equation (18),which does not provide us the maximum value of |g|2. but, the extreme values of cos θ must beinvestigated. for cos θ = 1, equation (18) gives |g|2 = 1, and the stability requirement is satisfied.for cos θ = −1, equation (18) yields |g|2 = {1 − 4∆τ (∆y)2 ( σ̃ σ )2}2 and, imposing the requirement of |g|2 ≤ 1, yields, fves in equation (12) is conditionally stable and the condition is( σ̃ σ )2 ≤ (∆y)2 2∆τ (20) similarly, we can state that fvcns, equation (13) is also conditionally stable and the conditionis ( σ̃ σ )2 ≤ (∆y)2 ∆τ (21) 6. consistency of the numerical schemes for consistency, the finite difference equation (fde) approximation of a pde must reduce tothe original pde as the step sizes approach zero [55]. now expanding each u(y,τ) in a taylor series expansion about uj i , we get u j+1 i = u j i + ∆τ ∂u ∂τ + (∆τ)2 2! ∂2u ∂τ2 + (∆τ)3 3! ∂3u ∂τ3 + o(∆τ)4 (22) u j+1 i+1 =u j i + ∆τ ∂u ∂τ + ∆y ∂u ∂y + 1 2! ( ∆τ ∂ ∂τ + ∆y ∂ ∂y )2 u + 1 3! ( ∆τ ∂ ∂τ + ∆y ∂ ∂y )3 u + o [ (∆τ)4, (∆y)4 ] (23) u j+1 i−1 =u j i + ∆τ ∂u ∂τ − ∆y ∂u ∂y + 1 2! ( ∆τ ∂ ∂τ −∆y ∂ ∂y )2 u + 1 3! ( ∆τ ∂ ∂τ − ∆y ∂ ∂y )3 u + o [ (∆τ)4, (∆y)4 ] (24) applying equations (22), (23), and (24) into equation (8) yields (di + ei + fi ) u j i + (1 + di + ei + fi ) ∆τ ∂u ∂τ + (1 + di + ei + fi ) (∆τ)2 2 ∂2u ∂τ2 + (−di + fi ) ∆y ∂u ∂y + (−di + fi ) ∆τ∆y ∂2u ∂τ∂y + (di + fi ) (∆y)2 2 ∂2u ∂y2 + o [ (∆τ)3, (∆y)3 ] = 0 from which we get ∂u ∂τ + ∆τ 2 ∂2u ∂τ2 − ( 2r σ2 + ( σ̃ σ )2) ∂u ∂y − ∆τ ( 2r σ2 + ( σ̃ σ )2) ∂2u ∂y∂τ − ( σ̃ σ )2 ∂2u ∂y2 + o [ (∆τ)2, (∆y)2 ] = 0 https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 11 it is obvious that if ∆τ, ∆y → 0, then the original pde (2) is recovered. therefore, the laasonenfinite difference scheme, equation (8), is consistent. now according to lax’s equivalence theorem,[55], lfds is convergent for all values of the parameters. similar arguments hold for dffds andfvfis. on the other hand, fves and fvcns are also convergent if the conditions (20) and (21)respectively, are satisfied. table 1. call option prices using the leland volatility model. s0 exact finite difference schemes finite volume schemes(linear) dffds lfds fves fvfis fvcns37.00 0.00001 0.04734 0.04893 0.00000 0.00054 0.0005047.00 0.00182 0.30914 0.31368 0.00006 0.01340 0.0129757.00 0.05078 1.09349 1.09949 0.00036 0.10771 0.1058867.00 0.45226 2.93576 2.94472 0.00335 0.65191 0.6479577.00 1.97686 6.06630 6.07104 0.01709 2.26632 2.2623087.00 5.46222 10.56460 10.56947 1.25649 5.63768 5.6368897.00 11.17037 16.34912 16.34757 6.49278 11.07370 11.07714107.00 18.71972 23.33442 23.33541 16.15278 18.66210 18.66616117.00 27.48006 31.19401 31.19406 26.33380 27.42028 27.42359127.00 36.91158 39.65579 39.65486 36.72393 36.83405 36.83640137.00 46.67034 48.63231 48.63285 46.68664 46.59745 46.59938147.00 56.57397 57.89847 57.89955 56.61703 56.45705 56.45879157.00 66.53723 67.45340 67.45432 66.59903 66.46595 66.46767167.00 76.52370 77.08904 77.08984 76.50332 76.39766 76.39940177.00 86.51886 86.88858 86.88931 86.48557 86.40846 86.41023187.00 96.51716 96.75424 96.75478 96.48843 96.43330 96.43508197.00 106.51657 106.59706 106.59728 106.40288 106.36023 106.36202207.00 116.51636 116.67091 116.67078 116.55261 116.52319 116.52499217.00 126.51629 126.48888 126.48906 126.39973 126.37558 126.37738227.00 136.51627 136.46008 136.46065 136.39791 136.37870 136.38049237.00 146.51626 146.57401 146.57489 146.53847 146.52419 146.52597247.00 156.51626 156.41475 156.41484 156.38607 156.37369 156.37545257.00 166.51626 166.40053 166.39979 166.37904 166.36868 166.37039267.00 176.51626 176.52571 176.52411 176.51173 176.50347 176.50515 7. results and discussions in this section, we choose the same parameters: r = 0.1, σ = 0.2, k = 100, t = 1, µ = 0.05, ∆t = 0.01, a = 0.02, m = 0.01, and c = 30, as illustrated in the literature [46]. then wecalculate the call option values using the proposed schemes, described in previous section 4, fordifferent volatility models. we compare the approximate results with the exact value of the linearblack-scholes model and among themselves also. https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 12 figure 1. approximate results of equation (1) by using (a) boyle and vorst volatilitymodel, and (b) barles and soner volatility model. from table 1 and figure 8.7 (see appendix), we observe that fully implicit fvs and crank-nicolson fvs provide comparatively better results than the other methods. note that all of themethods provide poor results when the initial stock price is less than the strike price (here strikeprice, in comparison with the exact value of the linear black-scholes model.from table 8.3 in appendix 8 and figure 1 (a), we can make similar comments, but here thefves gives a very poor approximation than the other methods when the initial stock price is lessthan the strike price (k = 100). table 8.4 in appendix 8 and figure 1(b) show that all of themethods provide a closer approximation to the exact value of the linear black-scholes model forall of the initial stock price, whether it is greater than the strike price, k = 100. table 2. call option prices using rapm volatility model. s0 exact finite difference schemes finite volume schemes(linear) dffds lfds fves fvfis fvcns37.00 0.00001 0.09975 0.10198 0.00075 0.00054 0.0005047.00 0.00182 0.64320 0.64859 0.02029 0.01340 0.0129757.00 0.05078 2.01164 2.01581 0.16518 0.10771 0.1058867.00 0.45226 4.58788 4.59820 0.97606 0.65191 0.6479577.00 1.97686 8.35995 8.36458 3.28335 2.26632 2.2623087.00 5.46222 13.24558 13.25523 7.65193 5.63768 5.6368897.00 11.17037 19.14277 19.14296 13.90023 11.07370 11.07714107.00 18.71972 25.95302 25.96004 21.59995 18.66210 18.66616117.00 27.48006 33.49710 33.50138 30.05441 27.42028 27.42359127.00 36.91158 41.58132 41.58229 39.02457 36.83405 36.83640137.00 46.67034 50.16133 50.16454 48.32523 46.59745 46.59938147.00 56.57397 59.07941 59.08280 57.80665 56.45705 56.45879157.00 66.53723 68.31796 68.31997 67.49623 66.46595 66.46767167.00 76.52370 77.71942 77.72089 77.20105 76.39767 76.39941177.00 86.51886 87.32671 87.32812 87.03280 86.40846 86.41023187.00 96.51716 97.04281 97.04399 96.91429 96.43330 96.43509197.00 106.51657 106.79810 106.79881 106.75046 106.36023 106.36202207.00 116.51636 116.77932 116.77954 116.81690 116.52319 116.52498217.00 126.51629 126.56491 126.56478 126.62193 126.37555 126.37734227.00 136.51627 136.50685 136.50637 136.57895 136.37864 136.38041237.00 146.51626 146.59198 146.59115 146.67807 146.52410 146.52585247.00 156.51626 156.42589 156.42489 156.50633 156.37350 156.37521257.00 166.51626 166.40454 166.40336 166.47896 166.36838 166.37004267.00 176.51626 176.52231 176.52094 176.59035 176.50307 176.50468 https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 13 finally, from table 2 and the corresponding figure 8.8 in appendix 8, we may observe thatfor the rapm volatility model, the fvcns and fvfis give better approximation than the othernumerical schemes when the initial stock price is closer to and/or greater than the strike price.on the other hand, from figures 2, 3, 4, it is clear that fvfis and fvcns produce comparativelybetter results than the other schemes for all of the volatility models. figures 5, 6 depict the optionprices at various time periods (from initial time t = 0 to maturity time, t = t) with different initialstock values. the similar results of solution surface for option price by using barles and sonervolatility model and rapm volatility model are presented in appendix 8, see figures 8.9,8.10. figure 2. approximate results of equation (1) using (a) dufort-frankel finite dif-ference scheme, and (b) laasonen finite difference scheme. figure 3. approximate results of equation (1) using (a) finite volume explicitscheme, and (b) finite volume fully implicit scheme. figure 4. approximate results of equation (1) using finite volume crank-nicolsonscheme. https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 14 (a) lvm (dffds) (b) lvm (lds) (c) lvm (fves) (d) lvm (fvcns) (e) lvm (fvfis) figure 5. solution surface for option price by using leland volatility model. (a) bvvm (dffds) (b) bvvm (lds) (c) bvvm (fves) (d) bvvm (fvcns) (e) bvvm (fvfis) figure 6. solution surface for option price by using boyle and vorst volatilitymodel. 8. conclusion in this research work, we have derived some numerical schemes using the fvm and fdm tosolve the non-linear black-scholes pde for european option pricing with the transaction costs byexploiting the transformations available in the existing literature [46]. thus we have modified themodel equation accordingly to a non-linear parabolic pde. for the convergence of these schemes,stability and consistency have been shown rigorously. then these schemes have been applied tovarious volatility models. according to the visible results, as presented in the earlier sections, it https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 15 is noted that all of the proposed schemes provide the best approximation to the exact value of thelinear black-scholes model for all initial stock prices, regardless of whether they are closer to orgreater than the strike price; particularly in the case of barles and soner volatility model. wemay claim that the fvfis and fvcns approximate better than the other methods for all four-volatility models. thus, it is observed that the fvfis and fvcns are very effective and proficientin locating approximate solutions to non-linear black-scholes models. notice that the limitationof these schemes is that they may offer poor results sometimes when the initial stock price is lessthan the strike price compared to the exact value of the linear black-scholes model. finally, wemay conclude that the proposed schemes may be applied to other non-linear partial differentialequations to compute the numerical solutions with the desired accuracy. conflicts of interest. the authors declare no competing interests exist. funding statement. this research received no external funding. references [1] j. c. hull, options, futures, and other derivatives, 8th ed., pearson prentice 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finite difference schemes finite volume schemes(linear) dffds lfds fves fvfis fvcns37.00 0.00001 0.08130 0.08362 0.00000 0.00054 0.0005047.00 0.00182 0.44919 0.45475 0.00016 0.01340 0.0129757.00 0.05078 1.43187 1.43827 0.00137 0.10771 0.1058867.00 0.45226 3.52436 3.53389 0.00181 0.65191 0.6479577.00 1.97686 6.88032 6.88512 -0.01095 2.26632 2.2623087.00 5.46222 11.52515 11.53065 -0.03314 5.63768 5.6368897.00 11.17037 17.36380 17.36252 1.18226 11.07370 11.07714107.00 18.71972 24.30605 24.30805 14.49730 18.66210 18.66616117.00 27.48006 32.07120 32.07189 26.66264 27.42028 27.42359127.00 36.91158 40.41514 40.41449 37.98090 36.83405 36.83640137.00 46.67034 49.26375 49.26507 47.25031 46.59745 46.59938147.00 56.57397 58.41457 58.41649 56.80881 56.45705 56.45879157.00 66.53723 67.86188 67.86337 66.66025 66.46595 66.46767167.00 76.52370 77.41353 77.41510 76.50821 76.39767 76.39940177.00 86.51886 87.14056 87.14251 86.47640 86.40846 86.41023187.00 96.51716 96.94695 96.94884 96.47214 96.43330 96.43509197.00 106.51657 106.75019 106.75144 106.38908 106.36023 106.36202207.00 116.51636 116.78198 116.78255 116.54146 116.52319 116.52500217.00 126.51629 126.57938 126.58055 126.39087 126.37558 126.37738227.00 136.51627 136.53059 136.53250 136.39143 136.37869 136.38048237.00 146.51626 146.62463 146.62715 146.53436 146.52418 146.52596247.00 156.51626 156.45839 156.45983 156.38285 156.37367 156.37541257.00 166.51626 166.43684 166.43715 166.37675 166.36863 166.37034267.00 176.51626 176.55434 176.55347 176.51042 176.50341 176.50508 table 8.4. call option prices using barles and soner volatility model. s0 exact finite difference schemes finite volume schemes(linear) dffds lfds fves fvfis fvcns37.00 0.00001 0.00056 0.00060 0.00047 0.00054 0.0005047.00 0.00182 0.01448 0.01496 0.01255 0.01340 0.0129757.00 0.05078 0.11766 0.11964 0.10402 0.10771 0.1058867.00 0.45226 0.70853 0.71289 0.64387 0.65191 0.6479577.00 1.97686 2.42414 2.42846 2.25831 2.26632 2.2623087.00 5.46222 5.90671 5.90840 5.63756 5.63768 5.6368897.00 11.17037 11.38951 11.38681 11.08596 11.07370 11.07714107.00 18.71972 18.90018 18.89694 18.68219 18.66210 18.66616117.00 27.48006 27.57177 27.56903 27.44285 27.42028 27.42359127.00 36.91158 36.90926 36.90718 36.85630 36.83405 36.83640137.00 46.67034 46.63600 46.63420 46.61611 46.59745 46.59938147.00 56.57397 56.47483 56.47316 56.47209 56.45705 56.45879157.00 66.53723 66.47486 66.47318 66.47729 66.46595 66.46767167.00 76.52370 76.40243 76.40071 76.40662 76.39766 76.39940177.00 86.51886 86.41173 86.40997 86.41573 86.40846 86.41022187.00 96.51716 96.43561 96.43381 96.43938 96.43329 96.43508197.00 106.51657 106.36234 106.36053 106.36582 106.36022 106.36202207.00 116.51636 116.52509 116.52327 116.52827 116.52319 116.52499217.00 126.51629 126.37745 126.37563 126.38047 126.37557 126.37737227.00 136.51627 136.38055 136.37873 136.38340 136.37869 136.38048237.00 146.51626 146.52603 146.52421 146.52870 146.52418 146.52596247.00 156.51626 156.37561 156.37380 156.37798 156.37368 156.37543257.00 166.51626 166.37066 166.36886 166.37271 166.36866 166.37038267.00 176.51626 176.50552 176.50374 176.50725 176.50346 176.50514 https://doi.org/10.28924/ada/ma.2.9 eur. j. math. anal. 10.28924/ada/ma.2.9 20 (a) bsvm (dffds) (b) bsvm (lds) (c) bsvm (fves) (d) bsvm (fvcns) (e) bsvm (fvfis) figure 8.9. solution surface for option price by using barles and soner volatilitymodel. (a) rapm (dffds) (b) rapm (lds) (c) rapm (fves) (d) rapm (fvcns) (e) rapm (fvfis) figure 8.10. solution surface for option price by using rapm volatility model. https://doi.org/10.28924/ada/ma.2.9 1. introduction 2. the model equation 3. volatility models leland volatility model (lvm) boyle and vorst volatility model (bvvm) barles and soner volatility model (bsvm) rapm volatility model (rapmvm) 4. derivations of computational schemes 4.1. dufort-frankel finite difference scheme 4.2. laasonen finite difference scheme 4.3. finite volume schemes 5. stability of the numerical schemes 6. consistency of the numerical schemes 7. results and discussions 8. conclusion conflicts of interest funding statement references appendix ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 106-132doi: 10.28924/ada/ma.1.106 new iterative algorithm for solving constrained convex minimization problem and split feasibility problem austine efut ofem1,∗ , unwana effiong udofia2, donatus ikechi igbokwe3 1department of mathematics, university of uyo, uyo, nigeria ofemaustine@gmail.com 2department of mathematics and statistics, akwa ibom state university, ikot akpaden, mkpatenin, nigeria unwanaudofia.aksu@yahoo.com 3department of mathematics, michael okpara university of agriculture, umudike, nigeria igbokwedi@yahoo.com ∗correspondence: ofemaustine@gmail.com abstract. the purpose of this paper is to introduce a new iterative algorithm to approximate the fixedpoints of almost contraction mappings and generalized α-nonexpansive mappings. also, we show thatour proposed iterative algorithm converges weakly and strongly to the fixed points of almost contrac-tion mappings and generalized α-nonexpansive mappings. furthermore, it is proved analytically thatour new iterative algorithm converges faster than one of the leading iterative algorithms in the liter-ature for almost contraction mappings. some numerical examples are also provided and used to showthat our new iterative algorithm has better rate of convergence than all of s, picard-s, thakur andm iterative algorithms for almost contraction mappings and generalized α-nonexpansive mappings.again, we show that the proposed iterative algorithm is stable with respect to t and data dependentfor almost contraction mappings. some applications of our main results and new iterative algorithmare considered. the results in this article are improvements, generalizations and extensions of severalrelevant results existing in the literature. 1. introduction fixed point theory is concerned with solution of the equation t` = `, (1.1) where t could be a nonlinear operator defined on a metric space. any ` that solves (1.1) is calledthe fixed point of t and the collection all such elements is denoted by f (t ). fixed point theory is received: 10 sep 2021. key words and phrases. stability; almost contraction map; generalized α-nonexpansive mapping; data dependence;iterative algorithm; constrained convex minimization problem; split feasibility problem.106 https://adac.ee https://doi.org/10.28924/ada/ma.1.106 https://orcid.org/0000-0001-8064-2326 eur. j. math. anal. 1 (2021) 107 an area in nonlinear analysis that has become very attractive and interesting with a large numberof applications in various fields of mathematics and other branches of science. fixed point theoryhas remained not only a field with a huge development, but also a very helpful means for solvingvarious problems in different fields of mathematics. it is well known that fixed point theorems areused for proving the existence and uniqueness to various mathematical models like differential,integral and partial differential equations and variational inequalities problems etc., representingphenomena arising in different fields such as steady state temperature distribution, chemical equa-tions, neutron transport theory, economic theories, epidemics and flow of fluids. furthermore, itas also significant in the field of computer science, image processing, artificial intelligence, deci-sion making, population dynamics, computer science, operational research, industrial engineering,pattern recognition, medicine, group health underwriting, management and many others.existence theorem is concerned with establishing sufficient conditions in which the equation (1.1)will have solution, but does not necessarily show how to find such solution. on the other hand,iteration method of fixed points is concerned with approximation or computation of sequences whichconverge to the solution of (1.1). when existence of a fixed point of an operator is guaranteed,obtaining constructive technique for finding such a fixed point is also paramount.in 2003, berinde [6] introduced the concept of weak contraction mappings which is also knownas almost contraction mappings. he showed that the class of almost contraction mappings is moregeneral than the class of zamfirescu mappings [41] which includes contraction mappings, kannanmappings [22] and chatterjea mappings [10].throughout this paper, let ω denote a banach space and λ a nonempty closed convex subset of ω. let r stand for set of real numbers. definition 1.1. a mapping t : λ → λ is called almost contraction if there exists a constant γ ∈ (0, 1) and some constant l ≥ 0, such that ‖t`−tζ‖≤ γ‖`−ζ‖ + l‖`−t`‖, ∀`,ζ ∈ λ. (1.2) definition 1.2. a mapping t : λ → λ is said to be suzuki generalized nonexpansive if for all `,ζ ∈ λ, we have 1 2 ‖`−t`‖≤‖`−ζ‖ =⇒‖t`−tζ‖≤‖`−ζ‖. suzuki generalized nonexpansive mappings is also known as mappings satisfying condition (c).in [33], suzuki showed that the class of suzuki generalized nonexpansive mappings is more generalthan the class of nonexpansive mappings and obtained some fixed points and convergence theorems. definition 1.3. a mapping t : λ → λ is said to be α-nonexpansive if there exists α ∈ [0, 1) suchthat ‖t`−tζ‖2 ≤ α‖t`−ζ‖2 + α‖`−tζ‖2 + (1 − 2α)‖`−ζ‖2, eur. j. math. anal. 1 (2021) 108 for all `,ζ ∈ λ. the class of α-nonexpansive mappings was introduced in 2011 by aoyama and kohsaka [3]as generalization of nonexpansive mappings and further obtained some convergence results. itis worthy noting that nonexpansive mappings are continuous on their domains, but suzuki-typegeneralized nonexpansive mappings and α-nonexpansive mappings need not be continuous (see[33]). clearly, every nonexpansive mapping is an α-nonexpansive mapping with α = 0 (i.e., 0-nonexpansive) and every α-nonexpansive mapping with a nonempty fixed point set is quasinonex-pansive. definition 1.4. a mapping t : λ → λ is said to be generalized α-nonexpansive if there exists α ∈ [0, 1) such that 1 2 ‖`−t`‖ ≤ ‖`−ζ‖ implies ‖t`−tζ‖ ≤ α‖t`−ζ‖ + α‖tζ − `‖ + (1 − 2α)‖`−ζ‖ for all `,ζ ∈ λ. in [26], pant and shukla introduced a wider class of nonexpansive mappings in banach spacesknown as generalized α-nonexpansive mappings which contains the class of suzuki generalizednonexpansive mappings.it is well known that the case of contraction mappings is simple and carries most of the goodbehavior using picard iterative algorithm. but when we move to the case of nonexpansive mappings,the picard iterative algorithm need not converge to a fixed point. apparently, the conclusion ofbanach contraction principle fails for nonexpansive mappings even if λ is compact. as an example,one may consider a geometric rotation on the unit circle in the plane r2.the limitation of picard iterative algorithm gave many researchers in nonlinear analysis the roomto construct more efficient iterative algorithms for approximating the fixed points of nonexpansivemappings and other classes of mappings which are more general than the class of nonexpansivemappings.some notable iterative algorithms in the existing literature are: mann [24], ishikawa [21], noor[25], argawal et al. [2], abbas and nazir [1], sp [27], s* [20], cr [12], normal-s [28], picard-s [17],thakur [36], thakur new [37], m [39], m* [38], garodia and uddin [16], two-step mann [35] iterativealgorithms and many others.in 2007, the s iterative algorithm was introduced by argawal et al. [2] as follows: ψ0 ∈ λ, µs = (1 −βs)ψs + βstψs, ψs+1 = (1 −δs)tψs + δstµs, ∀s ≥ 1, (1.3) where {δs} and {βs} are sequences in [0,1]. eur. j. math. anal. 1 (2021) 109 in 2014, the picard-s iterative algorithm was introduced by gursoy and karakaya [17] as follows: u0 ∈ λ, ϕs = (1 −βs)us + βstus, %s = (1 −δs)tus + δstϕs, us+1 = t%s, ∀s ≥ 1, (1.4) where {δs} and {βs} are sequences in [0,1]. the authors showed with the aid of an example thatpicard-s iterative algorithm (1.4) converges at a rate faster than all of picard, mann, ishikawa,noor, sp, cr, s, s*, abbas and nazir, normal-s and two-step mann iterative algorithms forcontraction mappings.in 2016, thakur et al. [37] introduced the following three steps iterative algorithm: ω0 ∈ λ, ρs = (1 −βs)ωs + βstωs, vs = t ((1 −δs)ωs + δsρs), ωs+1 = tvs, ∀s ≥ 1, (1.5) where {δs} and {βs} are sequences in [0,1]. with the help of numerical example, they proved that(1.5) is faster than picard, mann, ishikawa, agarwal, noor and abbas iterative algorithm for suzukigeneralized nonexpansive mappings.in 2018, ullah and arshad [39] introduced m iterative algorithm as follows: m0 ∈ λ, cs = (1 −δs)ms + δstms, ds = tcs, ms+1 = tds, ∀s ≥ 1, (1.6) where {δs} is a sequence in [0,1]. numerically they showed that m iterative algorithm (1.2)converges faster than s iterative algorithm (1.3) and picard-s iterative algorithm (1.4) for suzukigeneralized nonexpansive mappings. also, they noted that the speed of convergence of picard-siterative algorithm (1.4) and thakur iterative algorithm (1.5) are almost same.motivated by the above results, in this paper, we construct a new four step iterative algorithmwhich outperforms the iterative algorithm (1.6) in terms of convergence rate for almost contractionmappings as follows:  `0 ∈ λ, gs = (1 −βs)`s + βst`s, ws = (1 −δs)t`s + δstgs, ζs = tws, `s+1 = tζs, ∀s ≥ 1, (1.7) where {δs} and {βs} are sequences in [0,1]. eur. j. math. anal. 1 (2021) 110 the purpose of this paper is to prove analytically that our new iterative algorithm convergesfaster than (1.6) for almost contraction mappings. in order to support our analytical proof, weuse some new examples to show that our iterative algorithm (1.7) converges faster than (1.6) anda number of other leading iterative algorithms in the literature. we also prove the weak andstrong convergence of new iterative algorithm (1.7) to the fixed points generalized α-nonexpansivemappings in a uniformly convex banach spaces. furthermore, we show that our new iterativealgorithm is t-stable and data dependent. finally, we use our new iterative algorithm (1.7) tosolve a constrained convex minimization problem and a split feasibility problem. 2. preliminaries the following definitions, propositions and lemmas will be useful in proving our main results. definition 2.1. a banach space ω is said to be uniformly convex if for each � ∈ (0, 2], there exists δ > 0 such that for `,ζ ∈ ω satisfying ‖`‖≤ 1, ‖ζ‖≤ 1 and ‖`−ζ‖ > �, we have ∥∥∥`+ζ2 ∥∥∥ < 1 −δ. definition 2.2. a banach space ω is said to satisfy opial’s condition if for any sequence {`s} in ω which converges weakly to ` ∈ ω implies lim sup s→∞ ‖`s − `‖ < lim sup s→∞ ‖`s −ζ‖, ∀ζ ∈ ω with ζ 6= `. definition 2.3. let {`s} be a bounded sequence in ω. for ` ∈ λ ⊂ ω, we put r(`,{`s}) = lim sup s→∞ ‖`s − `‖. the asymptotic radius of {`s} relative to λ is defined by r(λ,{`s}) = inf{r(`,{`s}) : ` ∈ λ}. the asymptotic center of {`s} relative to λ is given as: a(λ,{`s}) = {` ∈ λ : r(`,{`s}) = r(λ,{`s})}. in a uniformly convex banach space, it is well known that a(λ,{`s}) consist of exactly one point. definition 2.4. [5] let {as} and {bs} be two sequences of real numbers that converge to a and brespectively, and assume that there exists k = lim s→∞ ‖as −a‖ ‖bs −b‖ . then, (r1) if k = 0, we say that {as} converges faster to a than {bs} does to b.(r2) if 0 < k < ∞, we say that {as} and {bs} have the same rate of convergence. eur. j. math. anal. 1 (2021) 111 definition 2.5. [5] let {ηs} and {φs} be two fixed point iteration processes that converge to thesame point z, the error estimates ‖ηs −z‖ ≤ as, ∀s ≥ 1, ‖φs −z‖ ≤ bs, ∀s ≥ 1, are available where {as} and {bs} are two sequences of positive numbers converging to zero. thenwe say that {ηs} converges faster to z than {φs} does if {as} converges faster than {bs}. definition 2.6. [5] let t , t̃ : λ → λ be two operators. we say that t̃ is an approximate operatorfor t if for some � > 0, we have ‖t`− t̃`‖≤ �, ∀` ∈ λ. definition 2.7. [18] let {ys} be any sequence in λ. then, an iteration process `s+1 = f (t,ys),which converges to fixed point z, is said to be stable with respect to t , if for εs = ‖ys+1−f (t,ys)‖, ∀s ∈n, we have lim s→∞ εs = 0 ⇔ lim s→∞ ys = z. definition 2.8. [31] a mapping t : λ → λ is said to satisfy condition (i) if a nondecreasingfunction f : [0,∞) → [0,∞) exists with f (0) = 0 and for all r > 0 then f (r) > 0 such that ‖`−t`‖≥ f (d(`,f (t )))) for all ` ∈ λ, where d(`,f (t )) = infz∈f(t)‖`−z‖. proposition 2.9. [26] let λ be a nonempty subset of a banach space ω. suppose t : λ → λ is any mapping. then(i) if t is a suzuki generalized nonexpansive mapping, it follows that t is a generalized α-nonexpansive mapping.(ii) every generalized α-nonexpansive mapping with a nonempty fixed point set is quasinonexpansive mapping.(ii) if t is a generalized α-nonexpansive mapping, then f (t ) is closed. moreover, if ω is strictly convex and λ is convex, then f (t ) is also convex.(iv) if t is a generalized α-nonexpansive mapping, then the following inequality holds: ‖`−tζ‖≤ ( 3 + α 1 −α ) ‖`−t`‖ + ‖`−ζ‖, ∀ `,ζ ∈ λ. lemma 2.10. [26] let t be a self mapping on a subset λ of a banach space ω which satisfies opial’s condition. suppose t is a generalized α-nonexpansive mapping. if {`s} converges weakly to z and lim s→∞ ‖t`s − `s‖ = 0, then tz = z. that is, i −t is demiclosed at zero. lemma 2.11. [33] let t be a self mapping on a weakly compact convex subset λ of a banach space ω with the opial’s property. if t is a suzuki generalized nonexpansive mapping, then t has a fixed point. eur. j. math. anal. 1 (2021) 112 lemma 2.12. [40] let {‘s} and {λs} be nonnegative real sequences satisfying the following inequalities: ‘s+1 ≤ (1 −σs)‘s + λs, where σs ∈ (0, 1) for all s ∈n, ∞∑ s=0 σs = ∞ and lim s→∞ s σs = 0, then lim s→∞ ‘s = 0. lemma 2.13. [32] let {‘s} be a nonnegative real sequence and there exits an s0 ∈n such that for all s ≥ s0 satisfying the following condition: ‘s+1 ≤ (1 −σs)‘s + σsλs, where σs ∈ (0, 1) for all s ∈n, ∞∑ s=0 σs = ∞ and λs ≥ 0 for all s ∈n, then 0 ≤ lim sup s→∞ ‘s ≤ lim sup s→∞ λs. lemma 2.14. [29] suppose ω is a uniformly convex banach space and {ιs} is any sequence satisfying 0 < p ≤ ιs ≤ q < 1 for all s ≥ 1. suppose {`s} and {ζs} are any sequences of ω such that lim sup s→∞ ‖`s‖ ≤ x, lim sup s→∞ ‖ζs‖ ≤ x and lim sup s→∞ ‖ιs`s + (1 − ιs)ζs‖ = x hold for some x ≥ 0. then lim s→∞ ‖`s −ζs‖ = 0. 3. rate of convergence in this section, we will prove that our new iterative algorithm (1.7) converges faster than theiterative algorithm (1.6) for almost contraction mappings. theorem 3.1. let ω be a banach space and let λ be a nonempty closed convex subset of ω. let t : λ → λ be a mapping satisfying (1.2) with f (t ) 6= ∅. let {`s} be the iterative algorithm defined by (1.7) with sequences {δs}, {βs} ∈ [0, 1] such that ∞∑ s=0 δsβs = ∞, then {`s} converges strongly to a unique fixed point of t . proof. let z ∈ f (t ) and from (1.7), we have get ‖gs −z‖ = ‖(1 −βs)`s + βst`s −z‖ ≤ (1 −βs)‖`s −z‖ + βs‖t`s −z‖ ≤ (1 −βs)‖`s −z‖ + βsγ‖`s −z‖ = (1 − (1 −γ)βs)‖`s −z‖. (3.1) eur. j. math. anal. 1 (2021) 113 using (1.7) and (3.1), we have ‖ws −z‖ = ‖(1 −δs)t`s + δstgs −z‖ ≤ (1 −δs)‖t`s −z‖ + δs‖tgs −z‖ ≤ γ(1 −δs)‖`s −z‖ + γδs‖gs −z‖ ≤ γ(1 −δs)‖`s −z‖ + γδs(1 − (1 −γ)βs)‖`s −z‖ = γ(1 − (1 −γ)δsβs)‖`s −z‖. (3.2) from (1.7) and (3.2), we obtain ‖ζs −z‖ = ‖tws −z‖ ≤ γ‖ws −z‖ ≤ γ2(1 − (1 −γ)δsβs)‖`s −z‖. (3.3) using (1.7) and (3.3), we have ‖`s+1 −z‖ = ‖tζs −z‖ ≤ γ‖ζs −z‖ ≤ γ3(1 − (1 −γ)δsβs)‖`s −z‖. (3.4) from (3.4), we have the following inequalities: ‖`s+1 −z‖ ≤ γ3(1 − (1 −γ)δsβs)‖`s −z‖ ≤ γ3(1 − (1 −γ)δs−1βs−1)‖`s−1 −z‖... ‖`1 −z‖ ≤ γ3(1 − (1 −γ)δ0β0)‖`0 −z‖. (3.5) from (3.5), we get ‖`s+1 −z‖ ≤ ‖`0 −z‖γ3(s+1) s∏ t=0 (1 − (1 −γ)δtβt). (3.6) since γ ∈ (0, 1), δt,βt ∈ [0, 1] for all t ∈n, it follows that (1 − (1 −γ)δtβt) ∈ (0, 1). since fromclassical analysis we know that 1 − ` ≤ e−` for all ` ∈ [0, 1], thus from (3.6), we have ‖`s+1 −z‖≤ γ3(s+1)‖`0 −z‖ e (1−γ) s∑ t=0 δtβt . (3.7) if we take the limits of both sides of (3.7), we get lim s→∞ ‖`s −z‖ = 0. � eur. j. math. anal. 1 (2021) 114 theorem 3.2. let ω be a banach space and let λ be a nonempty closed convex subset of ω. let t : λ → λ be a mapping satisfying (1.2) with f (t ) 6= ∅. for given `0 = m0 ∈ λ, let {`s} and {ms} be the iterative algorithms defined by (1.7) and (1.6), respectively, with real sequences {δs} and {βs} in [0,1] such that δs ≤ δ < 1 and βs ≤ β < 1, for all s ∈n and for some δ, β > 0. then {`s} converges to z faster than {ms} does. proof. from (3.6) in theorem 3.1 together with the assumptions αs ≤ α < 1 and βs ≤ β < 1, forall s ∈n and for some α, β > 0, then we have ‖`s+1 −z‖ ≤ ‖`0 −z‖γ3(s+1) s∏ t=0 (1 − (1 −γ)αtβt) = ‖`0 −z‖γ3(s+1)(1 − (1 −γ)αβ)s+1. (3.8) similarly, from (1.6), we get ‖cs −z‖ = ‖(1 −δs)ms + δstms −z‖ ≤ (1 −δs)‖ms −z‖ + δs‖tms −z‖ ≤ (1 −δs)‖ms −z‖ + δsγ‖mn −z‖ = (1 − (1 −γ)δs)‖ms −z‖. (3.9) using (1.6) and (3.9), we get ‖ds −z‖ = ‖tcs −z‖ ≤ γ‖cs −z‖ ≤ γ(1 − (1 −γ)δs)‖ms −z‖. (3.10) finally, from (1.6) and (3.10), we obtain ‖ms+1 −z‖ = ‖tds −z‖ ≤ γ‖ds −z‖ ≤ γ2(1 − (1 −γ)δs)‖ms −z‖. (3.11) from (3.11), we have the following inequalities: ‖ms+1 −z‖ ≤ γ2(1 − (1 −γ)δs)‖ms −z‖ ≤ γ2(1 − (1 −γ)δs−1)‖ms−1 −z‖... ‖m1 −z‖ ≤ γ2(1 − (1 −γ)δ0)‖m0 −z‖. (3.12) eur. j. math. anal. 1 (2021) 115 from (3.12), we get ‖ms+1 −z‖ ≤ ‖m0 −z‖γ2(s+1) s∏ t=0 (1 − (1 −γ)δt). since δs ≤ δ < 1 and βs ≤ β < 1, for all s ∈n and for some δ, β > 0, then we have ‖ms+1 −z‖ ≤ ‖m0 −z‖γ2(s+1) s∏ t=0 (1 − (1 −γ)δt) = ‖m0 −z‖γ2(s+1)(1 − (1 −γ)δ)s+1. set as = ‖`0 −z‖γ3(s+1)(1 − (1 −γ)δ)s+1, and bs = ‖`0 −z‖γ2(s+1)(1 − (1 −γ)δ)s+1. (3.13) hence, as bs = ‖`0 −z‖γ3(s+1)(1 − (1 −γ)δβ)s+1 ‖m0 −z‖γ2(s+1)(1 − (1 −γ)δ)s+1 → 0 as s →∞. this implies that our new iterative algorithm (1.7) converges faster to z than m iterative algorithm(1.6). � in order to support analytical prove in theorem 3.2 and demonstrate the advantage of our newiterative algorithm (1.7), we give the following example. example 3.3. let ω = < and λ = [1, 50]. let t : λ → λ be a mapping defined by t (`) = √ `2 − 8` + 40. obviously, 5 is the fixed point of t . take δs = βs = 34 , with an initial value of `1 = 50. by writing all the codes in matlab (r2015a) for example 3.3, we obtain the following com-parison table 1 and figure 1. eur. j. math. anal. 1 (2021) 116 table 1. comparison of convergence behaviour of our new iterative algorithm withs, picard-s, thakur and m iterative algorithms.step s picard-s thakur m new1 50.00000000 50.00000000 50.00000000 50.00000000 50.000000002 44.16905011 40.46668490 40.46648707 39.77487312 36.794280913 38.40054569 31.13624438 31.13566491 29.79220887 24.079581494 32.71513008 22.15533283 22.15389446 20.25245189 12.593214715 27.14503094 13.88761070 13.88380778 11.71208997 5.609365616 21.74399379 7.46589475 7.45557218 6.06597569 5.003558697 16.60935306 5.14776230 5.14203305 5.02641919 5.000015698 11.93484164 5.00348330 5.00331403 5.00042732 5.000000009 8.12786414 5.00007676 5.00007301 5.00000684 5.0000000010 5.84725921 5.00000169 5.00000161 5.00000011 5.0000000011 5.12789697 5.00000004 5.00000004 5.00000000 5.0000000012 5.01483168 5.00000000 5.00000000 5.00000000 5.0000000013 5.00164168 5.00000000 5.00000000 5.00000000 5.00000000 iteration number s 2 4 6 8 10 12 14 s e q u e n ce v a lu e s 5 10 15 20 25 30 35 40 45 50 new iteration m iteration thakur iteration picard-s iteration s iteration figure 1. graph corresponding to table 1. eur. j. math. anal. 1 (2021) 117 4. convergence results in this section, we will prove the weak and strong convergence of our new iterative algorithm (1.7)for generalized α–nonexpansive mappings in the framework of uniformly convex banach spaces.firstly, we will state and prove the following lemmas which will be useful in obtaining our mainresults. lemma 4.1. let ω be a banach space and λ be a nonempty closed convex subset of ω. let t : λ → λ be a generalized α–nonexpansive mapping with f (t ) 6= ∅. if {`s} is the iterative algorithm defined by (1.7), then lim s→∞ ‖`s −z‖ exists for all z ∈ f (t ). proof. let z ∈ f (t ). by proposition 2.9(ii), we know that every suzuki generalized nonexpansivemapping with f (t ) 6= ∅ is quasi-nonexpansive mapping. then, from (1.7), we have ‖gs −z‖ = ‖(1 −βs)`s + βst`s −z‖ ≤ (1 −βs)‖`s −z‖ + βs‖t`s −z‖ ≤ (1 −βs)‖`s −z‖ + βs‖`s −z‖ = ‖`s −z‖. (4.1) using (1.7) and (4.1), we obtain ‖ws −z‖ = ‖(1 −δs)t`s + δstgs −z‖ ≤ (1 −δs)‖t`s −z‖ + δs‖tgs −z‖ ≤ (1 −δs)‖`s −z‖ + δs‖gs −z‖ ≤ (1 −δs)‖`s −z‖ + δs‖`s −z‖ = ‖`s −z‖. (4.2) again, using (1.7) and (4.2), we get ‖ζs −z‖ = ‖tws −z‖ ≤ ‖ws −z‖ ≤ ‖`s −z‖. (4.3) lastly, from (1.7) and (4.3), we have ‖`s −z‖ = ‖tζs −z‖ ≤ ‖ζs −z‖ ≤ ‖`s −z‖. (4.4) this implies that {‖`s −z‖} is bounded and nondecreasing for all z ∈ f (t ). hence, lim s→∞ ‖`s −z‖exists. � eur. j. math. anal. 1 (2021) 118 lemma 4.2. let ω be a uniformly convex banach space and λ be a nonempty closed convex subset of ω. let t : λ → λ be a generalized α–nonexpansive mapping. suppose {`s} is the iterative algorithm defined by (1.7). then, f (t ) 6= ∅ if and only if {`s} is bounded and lim s→∞ ‖t`s−`s‖ = 0. proof. suppose f (t ) 6= ∅ and let z ∈ f (t ). then, by lemma 4.1, lim s→∞ ‖`s −z‖ exists and {`s} isbounded. put lim s→∞ ‖`s −z‖ = x. (4.5) from (4.4) and (4.5), we obtain lim sup s→∞ ‖gs −z‖≤ lim sup s→∞ ‖`s −z‖ = x. (4.6) from proposition 2.9(ii), we know that every generalized α–nonexpansive mapping with f (t ) 6= ∅is quasi-nonexpansive mapping. so that we have lim sup s→∞ ‖t`s −z‖≤ lim sup s→∞ ‖`s −z‖ = x. (4.7) again, using (1.7), we get ‖`s+1 −z‖ = ‖tζs −z‖ ≤ ‖ζs −z‖ = ‖tws −z‖ ≤ ‖ws −z‖ = ‖(1 −δs)t`s + δstgs −z‖ ≤ (1 −δs)‖t`s −z‖ + δs‖tgs −z‖ ≤ (1 −δs)‖`s −z‖ + δs‖gs −z‖ = ‖`s −z‖−δs‖`s −z‖ + δs‖gs −z‖. (4.8) from (4.8), we have ‖`s+1 −z‖−‖`s −z‖ δs ≤‖gs −z‖−‖`s −z‖. (4.9) since δs ∈ [0, 1], then from (4.9), we have ‖`s+1 −z‖−‖`s −z‖≤ ‖`s+1 −z‖−‖`s −z‖ δs ≤‖gs −z‖−‖`s −z‖, which implies that ‖`s+1 −z‖≤‖gs −z‖. therefore, from (4.5), we obtain x ≤ lim inf s→∞ ‖gs −z‖. (4.10) eur. j. math. anal. 1 (2021) 119 from (4.6) and (4.10) we obtain x = lim s→∞ ‖gn −z‖ = lim s→∞ ‖(1 −βs)`s + βst`s −z‖ = lim s→∞ ‖(1 −βs)(`s −z) + βs(t`s −z)‖ = lim s→∞ ‖βs(t`s −z) + (1 −βs)(`s −z)‖. (4.11) from (4.5), (4.7), (4.11) and lemma 2.14, we obtain lim s→∞ ‖t`s − `s‖ = 0. (4.12) conversely, assume that {`s} is bounded and lim s→∞ ‖t`s−`s‖ = 0. let z ∈ a(λ,{`s}), by definition2.3 and proposition 2.9(iv), we have (tz,{`s}) = lim sup s→∞ ‖`s −tz‖ ≤ lim sup s→∞ ( (3 + α) (1 −α) ‖t`s − `s‖ + ‖`s −z‖ ) = lim sup s→∞ ‖`s −z‖ = r(z,{`s}). (4.13) this implies that z ∈ a(λ,{`s}). since ω is uniformly convex, a(λ,{`s}) is singleton, thus wehave tz = z. � theorem 4.3. let ω, λ, t be same as in lemma 4.2. suppose tat ω satisfies opial’s condition and f (t ) 6= ∅. then, the sequence {`s} defined by (1.7) converges weakly to a fixed point of t . proof. let z ∈ f (t ), then by lemma 4.1, we have lim s→∞ ‖`s − z‖ exists. now we show that {`s}has weak sequential limit in f (t ). let ` and ζ be weak limits of the subsequences {`sj} and {`sk}of {`s}, respectively. by lemma 4.2, we have lim s→∞ ‖t`s − `s‖ = 0 and from lemma 2.10, i −t isdemiclosed at zero. it follows that (i −t )` = 0 implies ` = t`, similarly tζ = ζ.next we show uniqueness. suppose ` 6= ζ, then by opial’s property, we obtain lim s→∞ ‖`s − `‖ = lim sj→∞ ‖`sj − `‖ < lim sj→∞ ‖`sj −ζ‖ = lim s→∞ ‖`s −ζ‖ = lim sk→∞ ‖`sk −ζ‖ < lim sk→∞ ‖`sk − `‖ = lim s→∞ ‖`s − `‖, (4.14) which is a contradiction, so ` = ζ. hence, {`s} converges weakly to a fixed point of t . � eur. j. math. anal. 1 (2021) 120 theorem 4.4. let ω, λ, t be same as in lemma 4.2. then, the iterative algorithm {`s} defined by (1.7) converges strongly to a point of f (t ) if and only if lim inf s→∞ d(`s,f (t )) = 0, where d(`s,f (t )) = inf{‖`−z‖ : z ∈ f (t )}. proof. necessity is obvious. assume that lim inf s→∞ d(`s,f (t )) = 0. from lemma 4.1, we have lim s→∞ ‖`s − z‖ exists for all z ∈ f (t ), it follows that lim inf s→∞ d(`s,f (t )) exists. but by hypothesis, lim inf s→∞ d(`s,f (t )) = 0, thus lim s→∞ d(`s,f (t )) = 0. next we prove that {`s} is a cauchy sequencein λ. since lim inf s→∞ d(`s,f (t )) = 0, then given ε > 0, there exists s0 ∈n such that, for all s,n ≥ s0,we have d(`s,f (t )) ≤ � 2 , d(`n,f (t )) ≤ � 2 . thus, we have ‖`s − `n‖ ≤ ‖`s −z‖ + ‖`n −z‖ ≤ d(`s,f (t )) + d(`n,f (t )) ≤ � 2 + � 2 = �. hence {`s} is a cauchy sequence in λ. since λ is closed, therefore there exists a point `1 ∈ λsuch that lim s→∞ `s = `1. since lim s→∞ d(`s,f (t )) = 0, it implies that lim s→∞ d(`1,f (t )) = 0. hence, `1 ∈ f (t ) since f (t ) closed. � theorem 4.5. let ω, λ, t be same as in lemma 4.2. if t satisfies condition (i), then the iterative algorithm {`s} defined by (1.7) converges strongly to a fixed point of t . proof. we have shown in lemma 4.2 that lim s→∞ ‖t`s − `s‖ = 0. (4.15) using condition (i) in definition 2.8 and (4.15), we get lim s→∞ f (d(`s,f (t ))) ≤ lim s→∞ ‖t`s − `s‖ = 0, (4.16) i.e., lim s→∞ f (d(`s,f (t ))) = 0. since f : [0,∞) → [0,∞) is a nondecreasing function satisfying f (0) = 0, f (r) > 0 for all r ∈ (0,∞), we have lim s→∞ d(`s,f (t )) = 0. (4.17) from theorem 4.4, then sequence {`s} converges strongly to a point of f (t ). � eur. j. math. anal. 1 (2021) 121 5. numerical result in this section, we provide an example of generalized α-nonexpansive mapping which is notsuzuki generalized nonexpansive mapping. with the aid of the provided example, we will provethat our new iterative algorithm (1.7) outperforms a number of iterative algorithms in the existingliterature in terms of convergence. example 5.1. let λ = [0,∞) be endowed with the usual norm | · | and let t : λ → λ be definedas: t` = { 0, if ` ∈ [0, 1 5 ), 3` 4 , if ` ∈ [1 5 ,∞). (5.1) firstly, we show that t does not satisfy condition (c). to see this, let ` = 1 15 and ζ = 1 5 , then 1 2 |`−t`| = 1 30 < 2 15 = |`−ζ|. but |t`−tζ| = 3ζ 4 = 3 20 > 2 15 = |`−ζ|. hence, t does not satisfy condition (c), which implies that t is not a suzuki generalized nonex-pansive mapping.now we show that t is a generalized α-nonexpansive mapping with α = 1 3 (i.e., generalized 1 3 -nonexpansive). we consider the following cases: case (a): when `,ζ ∈ [0, 1 5 ), we have 1 3 |t`−ζ| + 1 3 |`−tζ| + 1 3 |`−ζ| ≥ 0 = |t`−tζ|. case (b): when `,ζ ∈ [1 5 ,∞), we obtain 1 3 |t`−ζ| + 1 3 |`−tζ| + 1 3 |`−ζ| = 1 3 ∣∣∣∣3`4 −ζ ∣∣∣∣ + 13 ∣∣∣∣`− 3ζ4 ∣∣∣∣ + 13|`−ζ| ≥ 1 3 ∣∣∣∣(3`4 −ζ ) + ( `− 3ζ 4 )∣∣∣∣ + 13|`−ζ| = 7 12 |`−ζ| + 1 3 |`−ζ| = 11 12 |`−ζ| ≥ 3 4 |`−ζ| = |t`−tζ|. eur. j. math. anal. 1 (2021) 122 case (c): when ` ∈ [1 5 ,∞) and ζ ∈ [0, 1 5 ), we get 1 3 |t`−ζ| + 1 3 |`−tζ| + 1 3 |`−ζ| = 1 3 ∣∣∣∣3`4 −ζ ∣∣∣∣ + 13|`| + 13|`−ζ| ≥ 1 3 ∣∣∣∣3`4 −ζ ∣∣∣∣ + 13|`−ζ| ≥ 7` 12 = |t`−tζ|. hence, t is generalized α-nonexpansive mapping with α = 1 3 (i.e., generalized 1 3 -nonexpansive)with f (t ) = {0}.with the aid of matlab (r2015a), we obtain the following comparison table 2 and figure 2 forvarious iterative algorithms with control sequences δs = 0.65, βs = 0.8 and initial guess `1 = 50. table 2. comparison of convergence behaviour of our new iterative algorithm withs, picard-s, thakur and m iterative algorithms.step s picard-s thakur m new1 50.00000000 50.00000000 50.00000000 50.00000000 50.000000002 32.62500000 24.46875000 24.46875000 23.55468750 18.351562503 21.28781250 11.97439453 11.97439453 11.09646606 6.735596924 13.89029766 5.85996932 5.85996932 5.22747581 2.472174565 9.06341922 2.86772249 2.86772249 2.46263118 0.000000006 5.91388104 1.40339169 1.40339169 1.16013016 0.000000007 3.85880738 0.00000000 0.00000000 0.00000000 0.000000008 2.51787182 0.00000000 0.00000000 0.00000000 0.000000009 1.64291136 0.00000000 0.00000000 0.00000000 0.00000000 iteration number s 1 2 3 4 5 6 7 8 9 s e q u e n ce v a lu e s 0 5 10 15 20 25 30 35 40 45 50 new iteration m iteration thakur iteration picard-s iteration s iteration figure 2. graph corresponding to table 2. from the above table 2 and figure 2, it is clear that our new iterative algorithm (1.7) outperformsa number of existing iterative algorithms. eur. j. math. anal. 1 (2021) 123 6. stability result our aim in this section is to show that our new iterative algorithm (1.7) is t–stable. theorem 6.1. let ω be a banach space and λ be a nonempty closed convex subset of ω. let t be a mapping satisfy (1.2). let {`s} be the iterative algorithm defined by (1.7) with sequences δs and βs ∈ [0, 1] such that ∑∞ s=0δsβs = ∞. then the iterative algorithm (1.7) is t–stable. proof. let {ys} ⊂ ω be an arbitrary sequence in λ and suppose that the sequence iterativelygenerated by (1.7) is `s+1 = f (g,ys) converging to a unique point z and that εs = ‖ys+1−f (t,ys)‖.to prove that (1.7) is t-stable, we have to show that lim s→∞ εs = 0 ⇔ lim s→∞ ys = z.let lim s→∞ εs = 0. then from (1.7) and (1.6), we obtain ‖ys+1 −z‖ = ‖ys+1 − f (t,ys) + f (t,ys) −z‖ ≤ ‖ys+1 − f (t,ys)‖ + ‖f (t,ys) −z‖ = εs + ‖f (t,ys) −z‖ = εs + ‖t (t ((1 −δs)tys + δst ((1 −βs)ys + βstys))) −z‖ = γ3(1 − (1 −γ)δsβs)‖ys −z‖ + εs. (6.1) for all s ≥ 1, put θs = ‖ys −z‖, σs = (1 −γ)δsβs ∈ (0, 1), λs = εs. since lim s→∞ εs = 0, this implies that λsσs = εs(1−γ)δsβs → 0 as s →∞. apparently, all the conditionsof lemma 2.12 are fulfilled. hence, from lemma 2.12 we have lim s→∞ ys = z.conversely, let lim s→∞ ys = z. the we have εs = ‖ys+1 − f (t,ys)‖ = ‖ys+1 −z + z − f (t,ys)‖ ≤ ‖ys+1 −z‖ + ‖f (t,ys) −z‖ ≤ ‖ys+1 −z‖ + γ3(1 − (1 −γ)δsβs)‖ys −z‖. (6.2) from (6.2), it follows that lim s→∞ εs = 0. hence, our new iterative algorithm (1.7) is stable withrespect to t . � 7. data dependence result in this section, we obtain data dependence result for the mapping t satisfying (1.2) by utilizingour new iterative algorithm (1.7). eur. j. math. anal. 1 (2021) 124 theorem 7.1. let t̃ be an approximate operator of a mapping t satisfying (1.2). let {`s} be an iterative sequence generated by (1.7) for t and define an iterative algorithm as follows:  ˜̀ 0 ∈ λ, g̃s = (1 −βs)˜̀s + βst̃ ˜̀s, w̃s = (1 −δs)t̃ ˜̀s + δst̃ g̃s, ζ̃s = t̃w̃s, ˜̀ s+1 = t̃ ζ̃s, ∀s ≥ 1, (7.1) where {δs} and {βs} are sequences in [0, 1] satisfying the following conditions:(i) 1 2 ≤ δsβs, ∀s ∈n,(ii) ∞∑ s=0 δsβs = ∞. if tz = z and t̃ z̃ = z̃ such that lim s→∞ ˜̀ s = z̃, we have ‖z − z̃‖≤ 7� 1 −γ , where � > 0 is a fixed number. proof. using (1.7), (1.2) and (7.1), we have ‖`s+1 − ˜̀s+1‖ = ‖tζs − t̃ ζ̃s‖ = ‖tζs −tζ̃s + tζ̃s − t̃ ζ̃s‖ ≤ ‖tζs −tζ̃s‖ + ‖tζ̃s − t̃ ζ̃s‖ ≤ γ‖ζs − ζ̃s‖ + l‖ζs −tζs‖ + �. (7.2) from (1.7), (1.2) and (7.1), we have ‖ζs − ζ̃s‖ = ‖tws − t̃w̃s‖ = ‖tws −tw̃s + tw̃s − t̃w̃s‖ ≤ ‖tws −tw̃s‖ + ‖tw̃s − t̃w̃s‖ ≤ γ‖ws − w̃s‖ + l‖ws −tws‖ + �. (7.3) putting (7.3) into (7.2), we have ‖`s+1 − ˜̀s+1‖ ≤ γ2‖ws − w̃s‖ + γl‖ws −tws‖ +γ� + l‖ζs −tζs‖ + �. (7.4) eur. j. math. anal. 1 (2021) 125 again, using (1.7), (1.2) and (7.1), we get ‖ws − w̃s‖ = (1 −δs)‖t`s − t̃ ˜̀s‖ + δs‖tgs − t̃ g̃s‖ ≤ (1 −δs){‖t`s −t ˜̀s‖ + ‖t ˜̀s − t̃ ˜̀s‖} +δs{‖tgs −tg̃s‖ + ‖tg̃s − t̃ g̃s‖} ≤ (1 −δs){γ‖`s − ˜̀s‖ + l‖`s −t`s‖ + �} +δs{γ‖gs − g̃s‖ + l‖gs −tgs‖ + �}. (7.5) using (1.7), (1.2) and (7.1), we get ‖gs − g̃s‖ ≤ (1 −βs)‖`s − ˜̀s‖ + βs‖t`s − t̃ ˜̀s‖ ≤ (1 −βs)‖`s − ˜̀s‖ + βs{‖t`s −t ˜̀s‖ + ‖t ˜̀s − t̃ ˜̀s‖} ≤ (1 −βs)‖`s − ˜̀s‖ + βs{γ‖`s − ˜̀s‖ + l‖`s −t`s‖ + �} = [1 − (1 −γ)βs]‖`s − ˜̀s‖ + βsl‖`s −t`s‖ + βs� (7.6) using (7.6) and (7.5), we have ‖ws − w̃s‖ ≤ (1 −δs){γ‖`s − ˜̀s‖ + l‖`s −t`s‖ + �} +δs{γ[1 − (1 −γ)βs]‖`s − ˜̀s‖ + γβ‖`s −t`s‖ + γβs�} = γ[1 − (1 −γ)δsβs]‖`s − ˜̀s‖ + (1 −δs)l‖`s −t`s‖ +(1 −δs)� + γδsβsl‖`s −t`s‖ + γδsβs�. (7.7) substituting (7.7) into (7.4), we obtain ‖`s+1 − ˜̀s+1‖ ≤ γ3[1 − (1 −γ)δsβs]‖`s − ˜̀s‖ + γ2(1 −δs)l‖`s −t`s‖ +γ2(1 −δs)� + γ3δsβsl‖`s −t`s‖ + γ3δsβs� +γl‖ws −tws‖ + γ� + l‖ζs −tζs‖ + �. (7.8) since γ,γ2,γ3 ∈ (0, 1) and δs,βs ∈ [0, 1], then (7.8) becomes ‖`s+1 − ˜̀s+1‖ ≤ [1 − (1 −γ)δsβs]‖`s − ˜̀s‖ + l‖`s −t`s‖ +δsβsl‖`s −t`s‖ + l‖ws −tws‖ +l‖ζs −tζs‖ + δsβs� + 3�. (7.9) by our assumption (i) that 1 2 ≤ δsβs, we have 1 −δsβs ≤ δsβs ⇒ 1 = 1 −δsβs + δsβs ≤ δsβs + δsβs = 2δsβs. eur. j. math. anal. 1 (2021) 126 this yields ‖`s+1 − ˜̀s+1‖ ≤ [1 − (1 −γ)δsβs]‖`s − ˜̀s‖ + 3δsβsl‖`s −t`s‖ +2δsβsl‖ws −tws‖ + 2δsβsl‖ζs −tζs‖ + 7δsβs� = (1 − (1 −γ)δsβs)‖`s − ˜̀s‖ +δsβs(1 −γ) × { 3l‖`s −t`s‖ + 2l‖ws −tws‖ (1 −γ) + 2l‖ζs −tζs‖ + 7� (1 −γ) } . (7.10) set θs = ‖`s − ˜̀s‖ σs = (1 −γ)δsβs ∈ (0, 1) λs = { 3l‖`s −t`s‖ + 2l‖ws −tws‖ + 2l‖ζs −tζs‖ + 7� (1 −γ) } from theorem 3.1, we know that lim s→∞ `s = z and since tz = z, it follows that lim s→∞ ‖`s −t`s‖ = lim s→∞ ‖ws −tws‖ = lim s→∞ ‖ζs −gζs‖ = 0. using lemma 2.13, we get 0 ≤ lim sup s→∞ ‖`s − ˜̀s‖≤ lim sup s→∞ 7� (1 −γ) . (7.11) since by theorem 3.1, we have that lim s→∞ `s = z and from our hypothesis lim s→∞ ˜̀ s = z̃, it followsfrom (7.11) that ‖z − z̃‖≤ 7� (1 −γ) . this completes the proof. � 8. some applications in this section, we will prove that the sequence generated by our new iterative algorithm (1.7)converges strongly to solutions of the constrained convex minimization problem and split feasibilityproblem.now, we present the definitions of some operators that will we be important in proving our mainresults. let h be a hilbert space and let c be a nonempty closed and convex subset of h. definition 8.1. let t : c → c be a mapping. then t is said to be: (i) nonexpansive, if ‖t`−tζ‖≤‖`−ζ‖, for all `,ζ ∈ c; eur. j. math. anal. 1 (2021) 127 (ii) lipschitz continuous, if there exists l > 0 such ‖t`−tζ‖≤ l‖`−ζ‖, for all `,ζ ∈ c; (iii) monotone if, 〈t`−tζ,`−ζ〉≥ 0, for all `,ζ ∈ c; (8.1) (iv) $-strongly monotone if there exists $ > 0, such that 〈`−ζ,t`−tζ〉≥ $‖`−ζ‖, for all `,ζ ∈ c. (8.2) for any ` ∈ h, we define the map pc : h → c satisfying ‖`−pc`‖≤‖`−ζ‖, for all ζ ∈ c. pc is called the metric projection of h onto c. it is well known that pc is nonexpansive. 8.1. application to constrained convex minimization problem.consider the following constrained convex minimization problem: minimize {f (`) : ` ∈ c}, (8.3) where f : c → r is a real-valued function. the minimization problem (8.3) is consistent if it hasa solution. throughout this paper, we shall use γ to stand for the solution set of the problem(8.3). it is worthy noting that f is (fréchect) differentiable, the gradient-projection method (gpm)generates a sequence {`s} by using the recursive formula:{ `0 ∈ c, `s+1 = pc(`s −λ∇f (`s)), for all s ≥ 1. (8.4) in more general form, (8.4) can be written as: { `0 ∈ c, `s+1 = pc(`s −λs∇f (`s)), for all s ≥ 1, (8.5) where λ and λs are positive real numbers.it is well known that if ∇f is $-strongly monotone and l-lipschitzian with $,l > 0, then theoperator t = pc(i −λ∇f ) (8.6) is a contraction; thus the sequence {`s} in (8.4) converges in norm to the unique minimizer of (8.3).from [14, 30], we know that z ∈ c solve the minimization problem (8.3) if and only if z solvesthe following fixed point equation: z = pc(i −λ∇f )z, (8.7) eur. j. math. anal. 1 (2021) 128 where λ > 0 is any fixed positive number. the operator t = pc(i − λ∇f ) is well known tobe nonexpansive (see [14, 30] and the references therein). several authors have have considereddifferent iterative algorithm for constrained convex minimization problems (see [4, 9, 13, 19, 34] andthe references therein). we now give our main results theorem 8.2. let c be a nonempty closed convex subset of a real hilbert space h. supposed that the minimization problem (8.3) is consistent and let γ denote the solution set. supposed that the gradient ∇f is l-lipschitzian with constant l > 0. let {`s} be the sequence generated iteratively by  `0 ∈ c, gs = (1 −βs)`s + βspc(i −λ∇f )`s ws = (1 −δs)pc(i −λ∇f )`s + δspc(i −λ∇f )gs ζs = pc(i −λ∇f )ws `s+1 = pc(i −λ∇f )ζs, ∀s ≥ 1. (8.8) where {δs}, {βs} are sequences in [0,1] and λ ∈ ( 0, l 2 ) . then the sequence {`s} converges strongly to a minimizer z of (8.3). 8.2. application to split feasibility problem.for modeling inverse problems which emanate from phase retrieval and medical image reconstruc-tion, in 1994, censor and elfving [11] firstly introduced the following split feasibility problem (sfp)in finite-dimensional hilbert spaces.let c and q be nonempty closed convex subsets of the hilbert spaces h1 and h2, respectivelyand a : h1 → h2 be a bounded linear operator. then the split feasibility problem (sfp) isformulated to find z ∈ c such that az ∈ q. (8.9) sfp has many applications, it has been found that sfp can been used in many areas such asimage restoration, computer tomograph, radiation therapy treatment planning. there exists someiterative several iterative methods for solving split feasibility problems, see, for instance [8, 15, 30].in 2002, byrne [8] applied the forward-backward method, a type of projection gradient methodto approximate (8.9). the so called cq-iterative procedure is defined as follows: `s+1 = pc[i −γa∗(1 −pq)a]`n, ∀ n ≥ 1, (8.10) where γ ∈ (0, 2‖a‖2) with λ being the spectral radius of the of operator a∗a, pc and pq denotethe projections onto sets c and q, respectively, and a∗ : h∗2 → h∗1 is the adjoint of a.we assume that the solution set γ of the sfp (8.10) is nonempty, let γ = {` ∈ c : a` ∈ q} = c ∩a−1q, then γ is closed, convex and nonempty set. eur. j. math. anal. 1 (2021) 129 lemma 8.3. [15] let operator t = pc[i −γa∗(i −pq)a], where γ ∈ ( 0, 2‖a‖2 ) . then, t is said to be a nonexpansive map. since by our assumption γ 6= ∅, then it is clear that any z ∈ c solves (8.9) if and only if it solvesthe fixed point equation: t = pc[i −γa∗(i −pq)a]z = z, z ∈ c. thus, f (t ) = γ = c ∩a−1q, i.e., the solution set γ is equal the set of fixed point of the map t .for more explicit explanation, the reader can see [42, 43].now, to prove our main results in this part, we will consider the following scheme: `0 ∈ c, gs = (1 −βs)`s + βspc[i −γa∗(i −pq)a]`s ws = (1 −δs)pc[i −γa∗(i −pq)a]`s + δspc[i −γa∗(i −pq)a]gs ζs = pc[i −γa∗(i −pq)a]ws `s+1 = pc[i −γa∗(i −pq)a]ζs, (8.11) for all s ≥ 1, where {δs}, {βs} are sequences in [0,1] and γ ∈ (0, 2‖a‖2). theorem 8.4. let {`s} be the sequence iteratively generated by (8.11). then, {`s} converses weakly to an element in γ. proof. since t = pc[i −γa∗(i −pq)a] is a nonexpansive map and by proposition 2.9 we knowthat every generalized α-nonexpansive map is nonexpansive map with α = 0 (i.e., 0-nonexpansive),so the conclusion follows from theorem 4.3. � theorem 8.5. if {`s} is the sequence generated by the iterative scheme (8.11). then {`s} converges strongly the an element in γ if and only if lim inf s→∞ d(`s, γ) = 0. proof. since t = pc[i − γa∗(i − pq)a] is nonexpansive map, then the conclusion of the prooffollows from theorem 4.4. � theorem 8.6. if t = pc[i − γa∗(i − pq)a] satisfies condition (i) and {`s} is the sequence iteratively defined by (8.11), then {`s} converges strongly to a point in γ. proof. the result follows from theorem 4.5. � 9. conclusion in this paper, we have shown numerically and analytically that our new iterative algorithm (1.7)has a better rate of convergence than m iterative algorithm and some other well known existingiterative algorithms in the literature for almost contraction mapping and generalized α-nonexpansivemappings. also, it is shown that our new iterative algorithm (1.7) is t–stable and data dependent eur. j. math. anal. 1 (2021) 130 which make it reliable. as some applications of our new iterative algorithm (1.7), it is used tofind the solutions of constrained convex minimization problem and split feasibility problem. now,owing to the fact that the class of generalized α-nonexpansive mappings which is considered inour paper is more general than the class of suzuki generalized nonexpansive mappings which hasbeen considered by ullah and arshad [39] for m iteration, it implies that our results generalizeand improve the results in ullah and arshad [39] and several other related results existing in theliterature. references [1] m. abbas and t. nazir, a new faster iteration process applied to constrained minimization and feasibility problems,mat. vesn. 66(2014), 223–234.[2] r. p. agarwal, d. o. regan and d. r. sahu, iterative construction of fixed points of nearly asymptotically nonex-pansive mappings, j. nonlinear convex anal. 8(2007), 61–79.[3] k. 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(basel). 23 (1972), 292–298. https://doi.org/10.1186/1029-242x-2014-280 eur. j. math. anal. 1 (2021) 132 [42] h.k. xu, a variable krasnosel’skii-mann algorithm and the multiple-set split feasibility problem, inverse probl.22(6) (2006), 2021–2034.[43] h.k. xu, iterative methods for the split feasibility problem in infinite-dimensional hilbert spaces. inverse probl. 26(2010), 105018. 17 pp. 1. introduction 2. preliminaries 3. rate of convergence 4. convergence results 5. numerical result 6. stability result 7. data dependence result 8. some applications 8.1. application to constrained convex minimization problem 8.2. application to split feasibility problem 9. conclusion references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 86-105doi: 10.28924/ada/ma.1.86 some properties on the [p,q]-order of meromorphic solutions of homogeneous and non-homogeneous linear differential equations with meromorphic coefficients mansouria saidani, benharrat belaïdi∗ department of mathematics, laboratory of pure and applied mathematics, university of mostaganem (umab), b. p. 227 mostaganem, algeria saidaniman@yahoo.fr, benharrat.belaidi@univ-mosta.dz ∗correspondence: benharrat.belaidi@univ-mosta.dz abstract. in the present paper, we investigate the [p,q]-order of solutions of higher order lineardifferential equations ak (z) f (k) +ak−1 (z) f (k−1) + · · ·+a1 (z) f ′ +a0 (z) f =0 and ak (z) f (k) +ak−1 (z) f (k−1) + · · ·+a1 (z) f ′ +a0 (z) f =f (z) ,where a0 (z) , a1 (z) , ...,ak (z) 6≡ 0 and f (z) 6≡ 0 are meromorphic functions of finite [p,q]-order.we improve and extend some results of the authors by using the concept [p,q]-order. 1. introduction and main results in this paper, we assume that the reader is familiar with the fundamental results and the standardnotations of the nevanlinna’s value distribution theory of meromorphic functions (see [7] , [9] , [14] , [24]) . in addition, for any integers p ≥ q ≥ 1 and a meromorphic function f in the whole complexplane, we will use ρ[p,q] (f ) , µ[p,q] (f ) to denote respectively the [p,q]-order and the lower [p,q]-order, λ[p,q] (f −a) (or λ[p,q] (f −a)) to denote the [p,q]-convergence exponent of the sequence ofdistinct a-points (or of a-points) and λ[p,q](1f ) to denote the [p,q]-exponent of convergence of thepoles, we refer the reader to see [12] , [15] , [16] and [25] . in particular for q = 1, ρ[p,1] (f ) = ρp (f )is the iterated p-order, µ[p,1] (f ) = µp (f ) is the iterated lower p-order, λ[p,1] (f −a) = λp (f ,a)(or λ[p,1] (f −a) = λp (f ,a)) is the iterated convergence exponent of the sequence of distinct a-points (or of a-points), λ[p,1](1f ) = λp (1f ) is the iterated exponent of convergence of the poles, see [7] , [11] , [13] , [14] and [24] for notations and definitions. received: 6 sep 2021. key words and phrases. linear differential equations; meromorphic functions; [p,q]-order; [p,q]-exponent of conver-gence of zeros. 86 https://adac.ee https://doi.org/10.28924/ada/ma.1.86 https://orcid.org/0000-0002-6635-2514 eur. j. math. anal. 1 (2021) 87 several authors have investigated the growth of solutions of second order and higher orderhomogeneous and non-homogeneous linear differential equations with analytic, entire or meromor-phic coefficients, see ([1−3], [6], [8], [11], [13−16], [18] , [20−21], [23], [25]). in the recent years,many authors have studied the complex linear differential equations f (k) + ak−1 (z) f (k−1) + · · · + a1 (z) f ′ + a0 (z) f = 0, (1.1) f (k) + ak−1 (z) f (k−1) + · · · + a1 (z) f ′ + a0 (z) f = f (z) , (1.2)where a0 (z) 6≡ 0, a1 (z) , ...,ak−1 (z) and f (z) 6≡ 0 are meromorphic functions of finite iterated p-order. in [2] , belaïdi considered the growth of meromorphic solutions of equations (1.1) and (1.2) with meromorphic coefficients of finite iterated p−order and obtained some results whichimprove and generalize some previous results. theorem a ([2]) let h ⊂ [0, +∞) be a set with a positive upper density, and let aj (z) (j = 0, 1, ..., k − 1) be meromorphic functions with finite iterated p-order. if there exist positive constants σ > 0,α > 0 such that ρ = max { ρp ( aj ) : j = 1, ...,k − 1 } < σ and |a0 (z) | ≥ expp (αrσ) as |z| = r ∈ h, r → +∞, then every meromorphic solution f 6≡ 0 of equation (1.1) satisfies µp (f ) = ρp(f ) = +∞, ρp+1(f ) ≥ σ. furthermore, if λp ( 1 f ) < ∞, then i (f ) = p + 1 and σ ≤ ρp+1 (f ) ≤ ρp (a0) . theorem b ([2]) let h ⊂ [0, +∞) be a set with a positive upper density, and let aj (z) (j = 0, 1, ...,k − 1) and f (z) 6≡ 0 be meromorphic functions with finite iterated p-order. if there exist positive constants σ > 0,α > 0 such that |a0 (z) | ≥ expp (αrσ) as |z| = r ∈ h, r → +∞, and ρ = max { ρp ( aj ) (j = 1, ...,k − 1),ρp (f ) } < σ, then every meromorphic solution of equation (1.2) with λp ( 1 f ) < σ satisfies λp (f ) = λp(f ) = ρp(f ) = ∞, λp+1 (f ) = λp+1(f ) = ρp+1(f ). furthermore, if λp ( 1 f ) < min{µp (f ) ,σ} , then i (f ) = p + 1 and λp+1 (f ) = λp+1(f ) = ρp+1 (f ) ≤ ρp (a0) . recently, in [18] the authors have studied the growth of solutions of the equations (1.1) and (1.2) when as(z) to dominate all other coefficients and they got some results about ρp+1 (f ) asfollows. theorem c ([18]) let h ⊂ (1, +∞) be a set with a positive upper logarithmic density (or ml (h) = +∞), and let aj (z) (j = 0, 1, ...,k − 1) be meromorphic functions with finite iterated p-order. eur. j. math. anal. 1 (2021) 88 if there exist positive constants σ > 0,α > 0 and an integer s, 0 ≤ s ≤ k − 1, such that |as (z) | ≥ expp (αrσ) as |z| = r ∈ h, r → +∞, and ρ = max { ρp ( aj ) (j 6= s) } < σ, then every non-transcendental meromorphic solution f 6≡ 0 of (1.1) is a polynomial with deg f ≤ s − 1 and every transcendental meromorphic solution f of (1.1) with λp ( 1 f ) < µp (f ) satisfies i (f ) = p + 1 µp (f ) = ρp(f ) = +∞ and σ ≤ ρp+1 (f ) ≤ ρp (as) . theorem d ([18]) let h ⊂ (1, +∞) be a set with a positive upper logarithmic density (or ml (h) = +∞), and let aj (z) (j = 0, 1, ...,k−1) and f (z) 6≡ 0 be meromorphic functions with finite iterated p-order. if there exist positive constants σ > 0,α > 0 and an integer s, 0 ≤ s ≤ k − 1, such that |as (z) | ≥ expp (αrσ) as |z| = r ∈ h, r → +∞, and max { ρp ( aj ) (j 6= s), ρp (f ) } < σ, then every non-transcendental meromorphic solution f of (1.2) is a polynomial with deg f ≤ s−1 and every transcendental meromorphic solution f of (1.2) with λp ( 1 f ) < min{σ,µp(f )} satisfies i (f ) = p + 1 λp (f ) = λp(f ) = ρp(f ) = µp (f ) = +∞ and σ ≤ λp+1 (f ) = λp+1(f ) = ρp+1 (f ) ≤ ρp (as) .thus, the following question arises: can we have the same properties as in theorems c andd for the solutions of equations ak (z) f (k) + ak−1 (z) f (k−1) + · · · + a1 (z) f ′ + a0 (z) f = 0 (1.3) and ak (z) f (k) + ak−1 (z) f (k−1) + · · · + a1 (z) f ′ + a0 (z) f = f (z) , (1.4)when the coefficients aj (j = 0, 1, ...,k) are of [p,q]−order? in this paper, we proceed this wayand we obtain the following results. theorem 1.1 let h ⊂ (1, +∞) be a set with a positive upper logarithmic density (or ml (h) = +∞) and let aj (z) (j = 0, 1, ...,k) with ak (z) 6≡ 0 be meromorphic functions with finite [p,q]order. if there exist a positive constant σ > 0 and an integer s, 0 ≤ s ≤ k, such that for sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ −ε) logq r } as |z| = r ∈ h, r → +∞ and ρ = max { ρ[p,q] ( aj ) (j 6= s) } < σ, then every non-transcendental meromorphic solution f 6≡ 0 of (1.3) is a polynomial with deg f ≤ s−1 and every transcendental meromorphic solution f of (1.3) with λ[p,q] ( 1 f ) < µ[p,q] (f ) satisfies ρ[p,q](f ) = µ[p,q] (f ) = +∞, σ ≤ ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . remark 1.1 putting ak (z) ≡ 1 and q = 1 in theorem 1.1, we obtain theorem c. eur. j. math. anal. 1 (2021) 89 corollary 1.1 under the hypotheses of theorem 1.1, suppose further that ϕ is a transcendental meromorphic function satisfying ρ[p+1,q] (ϕ) < σ. then, every transcendental meromorphic solution f of equation (1.3) with λ[p,q] ( 1 f ) < µ[p,q] (f ) satisfies σ ≤ λ[p+1,q] (f −ϕ) = λ[p+1,q] (f −ϕ) = ρ[p+1,q] (f −ϕ) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . considering the non-homogeneous linear differential equation (1.4), we obtain the followingresults. theorem 1.2 let h ⊂ (1, +∞) be a set with a positive upper logarithmic density (or ml (h) = +∞), and let aj (z) (j = 0, 1, ...,k) with ak (z) 6≡ 0 and f (z) 6≡ 0 be meromorphic functions with finite [p,q]-order. if there exist a positive constant σ > 0 and an integer s, 0 ≤ s ≤ k, such that for sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ −ε) logq r } as |z| = r ∈ h, r → +∞ and max { ρ[p,q] ( aj ) (j 6= s), ρ[p,q] (f ) } < σ, then every non-transcendental meromorphic solution f of (1.4) is a polynomial with deg f ≤ s − 1 and every transcendental meromorphic solution f of (1.4) with λ[p,q] ( 1 f ) < min { σ,µ[p,q](f ) } satisfies λ[p,q] (f ) = λ[p,q](f ) = ρ[p,q](f ) = µ[p,q] (f ) = +∞ and σ ≤ λ[p+1,q] (f ) = λ[p+1,q](f ) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . remark 1.2 putting ak (z) ≡ 1 and q = 1 in theorem 1.2, we obtain theorem d. corollary 1.2 let aj (z) (j = 0, 1, ...,k) , f (z) , h satisfy all the hypotheses of theorem 1.2, and let ϕ be a transcendental meromorphic function satisfying ρ[p+1,q] (ϕ) < σ. then, every transcendental meromorphic solution f with λ[p,q] ( 1 f ) < min{σ,µ[p,q] (f )} of equation (1.4) satisfies σ ≤ λ[p+1,q] (f −ϕ) = λ[p+1,q] (f −ϕ) = ρ[p+1,q] (f −ϕ) ≤ ρ[p,q] (as) . remark 1.3 in [17, 19] , the authors have studied the growth and the oscillation of solutionsof equations (1.3) and (1.4) when the coefficients aj (z) (j = 0, 1, ...,k) and f (z) are entirefunctions of iterated p-order or of [p,q]-order. however, in the present paper the coefficients aj (z) (j = 0, 1, ...,k) and f (z) are meromorphic functions with reduction of the hypotheses in theorems1.1 and 1.2. so, this article may be understood as an extension and an improvement of [17, 19] . eur. j. math. anal. 1 (2021) 90 2. some auxiliary lemmas in order to prove our theorems, we need the following definition, proposition and lemmas. thelebesgue linear measure of a set e ⊂ [0, +∞) is m (e) = ∫ e dt, and the logarithmic measure of a set f ⊂ [1, +∞) is ml (f ) = ∫ f dt t . the upper density of e ⊂ [0, +∞) is given by dens (e) = lim sup r→∞ m (e ∩ [0, r]) r and the upper logarithmic density of the set f ⊂ [1, +∞) is defined by log dens (f ) = lim sup r−→+∞ ml (f ∩ [1, r]) log r . proposition 2.1 ([2]) for all h ⊂ (1, +∞) the following statements hold: (i) if ml (h) = +∞, then m (h) = +∞; (ii) if dens (h) > 0, then m (h) = +∞; (iii) if log dens (h) > 0, then ml (h) = +∞. lemma 2.1 ([5]) let f be a transcendental meromorphic function in the plane, and let α > 1 be a given constant. then, there exist a set e1 ⊂ (1, +∞) that has a finite logarithmic measure, and a constant b > 0 depending only on α and (i, j) ((i, j) positive integers with i > j) such that for all z with |z| = r 6∈ [0, 1] ∪e1, we have∣∣∣∣∣f (i)(z)f (j)(z) ∣∣∣∣∣ ≤ b ( t (αr,f ) r (logα r) log t (αr,f ) )i−j . lemma 2.2 ([4]) let p ≥ q ≥ 1 be integers and g be an entire function such that ρ[p,q] (g) < +∞. then, there exist entire functions u(z) and v(z) such that g (z) = u(z)ev(z), ρ[p,q] (g) = max { ρ[p,q] (u) ,ρ[p,q] ( ev(z) )} and ρ[p,q] (u) = lim sup r→+∞ logp n ( r, 1 g ) logq r . moreover, for any given ε > 0, we have |u(z)| ≥ exp { −expp {( ρ[p,q] (u) + ε ) logq r }} (r /∈ e2) , where e2 ⊂ (1, +∞) is a set of r of finite linear measure. eur. j. math. anal. 1 (2021) 91 lemma 2.3 let p ≥ q ≥ 1 be integers. suppose that f is a meromorphic function such that ρ[p,q] (f ) < +∞. then, there exist entire functions u1 (z) , u2 (z) and v (z) such that f (z) = u1 (z) e v(z) u2 (z) (2.1) and ρ[p,q](f ) = max { ρ[p,q](u1),ρ[p,q](u2), ρ[p,q](e v(z)) } . (2.2) moreover, for any given ε > 0, we have exp { −expp { (ρ(p,q) (f ) + ε) logq r }} ≤ |f (z)| ≤ expp+1 { (ρ(p,q) (f ) + ε) logq r } (r /∈ e3) , (2.3) where e3 ⊂ (1, +∞) is a set of r of finite linear measure. proof. when p ≥ q = 1, the lemma is due to tu and long [21]. thus, we assume that p > q > 1 or p = q > 1. by hadamard factorization theorem, we can write f as f (z) = g(z) d(z) , where g (z) and d (z) are entire functions satisfying µ[p,q] (g) = µ[p,q] (f ) = µ ≤ ρ[p,q] (f ) = ρ[p,q] (g) < +∞ and λ[p,q] (d) = ρ[p,q] (d) = λ[p,q] ( 1 f ) < µ. by lemma 2.2, there exist entire functions u(z) and v(z) such that g (z) = u(z)ev(z), ρ[p,q] (g) = max { ρ[p,q] (u) ,ρ[p,q] ( ev(z) )} . so, there exist entire functions u(z), v(z) and d (z) such that f (z) = u(z)ev(z) d (z)and ρ[p,q](f ) = max { ρ[p,q] (u) ,ρ[p,q](d),ρ[p,q] ( ev(z) )} . thus (2.1) and (2.2) hold. set f (z) = u1(z)ev(z) u2(z) , where u1 (z) , u2 (z) are the canonical productsformed with the zeros and poles of f respectively. by the definition of [p,q]-order, for sufficientlylarge r and any given ε > 0, we have |u1 (z)| ≤ expp+1 {( ρ[p,q] (u1) + ε 3 ) logq r } , |u2 (z)| ≤ expp+1 {( ρ[p,q] (u2) + ε 3 ) logq r } . (2.4) since max {ρ[p,q](u1),ρ[p,q](u2), ρ[p,q](ev(z))} = ρ[p,q](f ), then we obtain |u1 (z)| ≤ expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } , (2.5) |u2 (z)| ≤ expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } , (2.6) eur. j. math. anal. 1 (2021) 92∣∣∣ev(z)∣∣∣ ≤ expp+1{(ρ[p,q] (f ) + ε 3 ) logq r } . (2.7) by lemma 2.2, there exists a set e3 ⊂ (1, +∞) of r with a finite linear measure such that for anygiven ε > 0, we have |u1 (z)| ≥ exp { −expp {( ρ[p,q] (u1) + ε 3 ) logq r }} ≥ exp { −expp {( ρ[p,q] (f ) + ε 3 ) logq r }} , (r /∈ e3) , (2.8) |u2 (z)| ≥ exp { −expp {( ρ[p,q] (u2) + ε 3 ) logq r }} ≥ exp { −expp {( ρ[p,q] (f ) + ε 3 ) logq r }} , (r /∈ e3) . (2.9) then, by using (2.5) , (2.7) and (2.9), we obtain for sufficiently large r /∈ e3 and any given ε > 0 |f (z)| = |u1 (z)| ∣∣ev(z)∣∣ |u2 (z)| ≤ expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } exp { −expp {( ρ[p,q] (f ) + ε 3 ) logq r }} ≤ expp+1 {( ρ[p,q] (f ) + ε ) logq r } . (2.10) on the other hand, we have ρ[p−1,q] (v) = ρ[p,q](ev(z)) ≤ ρ[p,q] (f ) and ∣∣ev(z)∣∣ ≥ e−|v(z)|. makinguse of the definition of [p,q]-order, we obtain |v (z)| ≤ m(r,v) ≤ expp {( ρ(p−1,q) (v) + ε 3 ) logq r } ≤ expp {( ρ[p,q] (f ) + ε 3 ) logq r } . then, for sufficiently large r and any given ε > 0, we have∣∣∣ev(z)∣∣∣ ≥ e−|v(z)| ≥ exp {−expp {(ρ[p,q] (f ) + ε 3 ) logq r }} . (2.11) by (2.6) , (2.8) and (2.11), we can easily obtain |f (z)| = |u1 (z)| ∣∣ev(z)∣∣ |u2 (z)| ≥ exp { −expp {( ρ[p,q] (f ) + ε 3 ) logq r }} exp { −expp {( ρ[p,q] (f ) + ε 3 ) logq r }} expp+1 {( ρ[p,q] (f ) + ε 3 ) logq r } . = exp { −3 expp {( ρ[p,q] (f ) + ε 3 ) logq r }} ≥ exp { −expp {( ρ[p,q] (f ) + ε ) logq r }} . thus, we complete the proof of lemma 2.3. lemma 2.4 under the assumptions of theorem 1.1 or theorem 1.2, we have ρ[p,q] (as) = β ≥ σ. eur. j. math. anal. 1 (2021) 93 proof. assume that ρ[p,q] (as) = β < σ. according to the hypotheses of theorems 1.1 or 1.2, thereexists a positive constant σ > 0 such that for sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ −ε) logq r } (2.12) as |z| = r ∈ h, r → +∞, where h ⊂ (1, +∞) is a set with a positive upper logarithmic density (by proposition 2.1, we have ml (h) = +∞). by lemma 2.3, we can find a set e3 ⊂ (1, +∞) thathas finite linear measure (and so of finite logarithmic measure) such that when |z| = r /∈ e3, wehave for any given ε (0 < 2ε < σ−β) |as (z) | ≤ expp+1 { (β + ε) logq r } . (2.13) by (2.12) and (2.13) , we obtain for |z| = r ∈ h re3, r → +∞ expp+1 { (σ −ε) logq r } ≤ |as (z) | ≤ expp+1 { (β + ε) logq r } and by ε (0 < 2ε < σ−β) this is a contradiction. hence ρ[p,q] (as) = β ≥ σ. lemma 2.5 (wiman-valiron, [10] , [22]) let f be a transcendental entire function, and let z be a point with |z| = r at which |f (z)| = m (r, f ). then the estimation f (j) (z) f (z) = ( νf (r) z )j (1 + o (1)) (j ≥ 1 is an integer ) holds for all |z| outside a set e4 of r of finite logarithmic measure, where νf (r) is the central index of f . lemma 2.6 ([12]) let f be an entire function of [p,q]-order and let νf (r) be the central index of f . then ρ[p,q] (f ) = lim sup r→+∞ logp νf (r) logq r , µ[p,q] (f ) = lim inf r→+∞ logp νf (r) logq r . the following two lemmas were given in [4] without proof, so for the convenience of the reader,we prove them. lemma 2.7 let f (z) = g(z) d(z) be a meromorphic function, where g (z) , d (z) are entire functions satisfying µ[p,q] (g) = µ[p,q] (f ) = µ ≤ ρ[p,q] (f ) = ρ[p,q] (g) ≤ +∞ and λ[p,q] (d) = ρ[p,q] (d) = β = λ[p,q] ( 1 f ) < µ. then, there exists a set e5 ⊂ (1, +∞) of finite logarithmic measure such that for all |z| = r /∈ [0, 1] ∪e5 and |g (z) | = m (r,g) , we have f (n) (z) f (z) = ( νg (r) z )n (1 + o (1)) , n ∈n, where νg (r) denote the central index of g. eur. j. math. anal. 1 (2021) 94 proof. by mathematical induction, we obtain f (n) = g(n) d + n−1∑ j=0 g(j) d ∑ (j1...jn) cjj1...jn ( d′ d )j1 ×···× ( d(n) d )jn , (2.14) where cjj1...jn are constants and j + j1 + 2j2 + · · · + njn = n. hence f (n) f = g(n) g + n−1∑ j=0 g(j) g ∑ (j1...jn) cjj1...jn ( d′ d )j1 ×···× ( d(n) d )jn . (2.15) from lemma 2.5, there exists a set e4 ⊂ (1, +∞) with finite logarithmic measure such that for apoint z satisfying |z| = r /∈ e4 and |g (z)| = m (r,g), we have g(j)(z) g(z) = ( νg (r) z )j (1 + o (1)) (j = 1, 2, ...,n) , (2.16) where νg (r) is the central index of g. substituting (2.16) into (2.15) yields f (n) (z) f (z) = ( νg (r) z )n [(1 + o (1)) + n−1∑ j=0 ( νg (r) z )j−n (1 + o (1)) ∑ (j1...jn) cjj1...jn ( d′ d )j1 ×···× ( d(n) d )jn . (2.17) since ρ[p,q] (d) = β < µ, then for any given ε (0 < 2ε < µ−β) and sufficiently large r , we have t (r,d) ≤ expp {( β + ε 2 ) logq r } by using lemma 2.1, for α = 2, there exist a set e1 ⊂ (1, +∞) with ml(e1) < ∞ and a constant b > 0, such that for all z satisfying |z| = r /∈ [0, 1] ∪e1, we have∣∣∣∣∣d(m) (z)d (z) ∣∣∣∣∣ ≤ b [t (2r,d)]m+1 ≤ b[expp {(β + ε2) logq (2r)}]m+1 ≤ expp { (β + ε) logq r }m , m = 1, 2, ...,n. (2.18) by lemma 2.6 and µ[p,q] (g) = µ[p,q] (f ) = µ, it follows that νg (r) > expp { (µ−ε) logq r } for sufficiently large r . thus, by using j1 + 2j2 + · · · + njn = n− j, we obtain∣∣∣∣∣∣ ( νg (r) z )j−n ( d′ d )j1 ×···× ( d(n) d )jn∣∣∣∣∣∣ ≤ [ expp { (µ−ε) logq r } r ]j−n × [ expp { (β + ε) logq r }]n−j = [ r expp { (β + ε) logq r } expp { (µ−ε) logq r } ]n−j → 0 (2.19) as r → +∞, where |z| = r /∈ [0, 1] ∪e5, e5 = e1 ∪e4 and |g (z)| = m (r,g) . from (2.17) and (2.19), we obtain our assertion. eur. j. math. anal. 1 (2021) 95 lemma 2.8 let f (z) = g(z) d(z) be a meromorphic function, where g (z), d (z) are entire functions satisfying µ[p,q] (g) = µ[p,q] (f ) = µ ≤ ρ[p,q] (f ) = ρ[p,q] (g) ≤ +∞ and λ[p,q] (d) = ρ[p,q] (d) = λ[p,q] ( 1 f ) < µ. then, there exists a set e6 ⊂ (1, +∞) of finite logarithmic measure such that for all |z| = r /∈ [0, 1] ∪e6 and |g (z) | = m (r,g), we have∣∣∣∣ f (z)f (s) (z) ∣∣∣∣ ≤ r2s, (s ∈n) . proof. by lemma 2.7, there exists a set e5 of finite logarithmic measure such that the estimation f (s)(z) f (z) = ( νg (r) z )s (1 + o (1)) (s ≥ 1 is an integer) (2.20) holds for all |z| = r /∈ [0, 1] ∪e5 and |g (z)| = m (r,g), where νg (r) is the central index of g. onthe other hand, by lemma 2.6, for any given ε (0 < ε < 1), there exists r > 1 such that for all r > r, we have νg (r) > expp { (µ−ε) logq (r) } . (2.21) if µ = +∞, then µ−ε can be replaced by a large enough real number m. set e6 = [1,r] ∪e5, lm (e6) < +∞. hence from (2.20) and (2.21), we obtain∣∣∣∣ f (z)f (s) (z) ∣∣∣∣ = ∣∣∣∣ zνg (r) ∣∣∣∣s 1|1 + o (1)| ≤ rs(expp {(µ−ε) logq (r)})s ≤ r2s,where |z| = r /∈ [0, 1] ∪e6, r → +∞ and |g (z)| = m (r,g) . lemma 2.9 ([6]) let ϕ : [0, +∞) →r and ψ : [0, +∞) →r be monotone nondecreasing functions such that ϕ(r) ≤ ψ(r) for all r /∈ (e7 ∪ [0, 1]) , where e7 is a set of finite logarithmic measure. let α > 1 be a given constant. then, there exists an r1 = r1(α) > 0 such that ϕ(r) ≤ ψ(αr) for all r > r1. lemma 2.10 ([19]) let f (z) = g(z) d(z) be a meromorphic function, where g (z), d (z) are entire functions. if 0 ≤ ρ[p,q] (d) < µ[p,q] (f ) , then µ[p,q] (g) = µ[p,q] (f ) and ρ[p,q] (g) = ρ[p,q] (f ) . moreover, if ρ[p,q] (f ) = +∞, then ρ[p+1,q] (g) = ρ[p+1,q] (f ) . lemma 2.11 assume that k ≥ 2 and a0, a1, ...,ak 6≡ 0, f are meromorphic functions. let ρ = max { ρ[p,q] ( aj ) (j = 0, 1, ...,k),ρ[p,q] (f ) } < ∞ and let f be a meromorphic solution of infinite [p,q]-order of equation (1.4) with λ[p,q] ( 1 f ) < µ[p,q] (f ) . then, ρ[p+1,q](f ) ≤ ρ. proof. let f be a meromorphic solution of infinite [p,q]-order of equation (1.4) with λ[p,q](1f ) < µ[p,q] (f ) . so, we can use hadamard factorization theorem and write f as f (z) = g(z)d(z) , where g(z)and d(z) are entire functions satisfying µ[p,q] (g) = µ[p,q] (f ) = µ ≤ ρ[p,q] (f ) = ρ[p,q] (g) ≤ +∞ eur. j. math. anal. 1 (2021) 96 and λ[p,q] (d) = ρ[p,q] (d) = λ[p,q](1f ) < µ. by lemma 2.3, there exists a set e3 ⊂ (1, +∞)of r with a finite linear measure such that for all |z| = r /∈ e3 and any given ε (0 < 2ε < µ[p,q] (f ) −ρ[p,q] (d)), we have |aj (z) | ≤ expp+1 { (ρ(p,q) ( aj ) + ε) logq r } ≤ expp+1 { (ρ + ε) logq r } , j = 0, 1, ...,k − 1, (2.22) |ak (z) | ≥ exp { −expp { (ρ(p,q) (ak) + ε) logq r }} ≥ exp { −expp { (ρ + ε) logq r }} (2.23) and |f (z)| ≤ expp+1 { (ρ(p,q) (f ) + ε) logq r } ≤ expp+1 { (ρ + ε) logq r } . (2.24) by (2.24), for all z satisfying |z| = r /∈ e3 at which |g(z)| = m(r,g) and any given ε ( 0 < 2ε < µ[p,q] (f ) −ρ[p,q] (d) ) , we obtain∣∣∣∣f (z)f (z) ∣∣∣∣ = |f (z)||g(z)| |d (z)| ≤ expp+1 { (ρ[p,q] (d) + ε) logq r } expp+1 { (ρ + ε) logq r } expp+1 { (µ[p,q] (f ) −ε) logq r } ≤ expp+1 { (ρ + ε) logq r } . (2.25) by lemma 2.7, there exists a set e5 ⊂ (1, +∞) of finite logarithmic measure such that for all |z| = r /∈ [0, 1] ∪e5 and |g (z) | = m (r,g) , we have f (j) (z) f (z) = ( νg (r) z )j (1 + o (1)) , j = 1, ...,k. (2.26) we can rewrite (1.4) as∣∣∣∣∣f (k) (z)f (z) ∣∣∣∣∣ ≤ 1|ak (z) | |a0 (z) | + ∣∣∣∣f (z)f (z) ∣∣∣∣ + k−1∑ j=1 |aj (z) | ∣∣∣∣∣f (j) (z)f (z) ∣∣∣∣∣  . (2.27) by substituting (2.22) , (2.23) , (2.25) and (2.26) into (2.27), we obtain∣∣∣∣νg (r)z ∣∣∣∣k |1 + o (1)| ≤ 1exp {−expp {(ρ + ε) logq r}}× 1 + k−1∑ j=1 ∣∣∣∣νg (r)z ∣∣∣∣j |1 + o (1)|  expp+1{(ρ + ε) logq r} + expp+1 { (ρ + ε) logq r }) = 2 + k−1∑ j=1 ∣∣∣∣νg (r)z ∣∣∣∣j |1 + o (1)|  exp {2 expp {(ρ + ε) logq r}} . hence |νg (r)| |1 + o (1)| ≤ (k + 1) r |1 + o (1)|exp { 2 expp { (ρ + ε) logq r }} (2.28) eur. j. math. anal. 1 (2021) 97 holds for all z satisfying |z| = r /∈ [0, 1] ∪e3 ∪e5 and |g (z) | = m (r,g) , r → +∞. by (2.28),we get lim sup r→+∞ logp+1νg (r) logq r ≤ ρ + ε. (2.29) since ε > 0 is arbitrary, by (2.29) and lemma 2.6, we obtain ρ[p+1,q] (g) ≤ ρ. since ρ[p,q] (d) < µ[p,q] (f ) , so by lemma 2.10, we have ρ[p+1,q] (g) = ρ[p+1,q] (f ) . thus, ρ[p+1,q] (f ) ≤ ρ. therefore,lemma 2.11 is proved. lemma 2.12 ([19]) let aj (z) (j = 0, 1, ...,k) , ak (z) (6≡ 0) ,f (z) (6≡ 0) be meromorphic functions and let f be a meromorphic solution of (1.4) of infinite [p,q]-order satisfying the following condition b = max { ρ[p+1,q] (f ) , ρ[p+1,q] ( aj ) (j = 0, 1, ...,k) } < ρ[p+1,q] (f ) . then λ[p+1,q](f ) = λ[p+1,q](f ) = ρ[p+1,q] (f ) . lemma 2.13 let h ⊂ (1, +∞) be a set with a positive upper logarithmic density (or infinite logarithmic measure), and let aj (z) (j = 0, 1, ...,k) with ak (z) 6≡ 0 and f (z) 6≡ 0 be meromorphic functions with finite [p,q]-order. if there exist a positive constant σ > 0 and an integer s, 0 ≤ s ≤ k, such that for sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ −ε) logq r } as |z| = r ∈ h, r → +∞ and max { ρ[p,q] ( aj ) (j 6= s), ρ[p,q] (f ) } < σ, then every transcendental meromorphic solution f of equation (1.4) satisfies ρ[p,q](f ) ≥ σ. proof. assume that f is a transcendental meromorphic solution of equation (1.4) with ρ[p,q](f ) < σ.from (1.4) , we have as = f f (s) − k∑ j=0 j 6=s aj f (j) f (s) . (2.30) since max {ρ[p,q](aj) (j 6= s), ρ[p,q] (f )} < σ and ρ[p,q] (f ) < σ, then from (2.30) we obtainthat ρ1 = ρ[p,q] (as) ≤ max { ρ[p,q] ( aj ) (j 6= s), ρ[p,q] (f ) , ρ[p,q] (f ) } < σ. by lemma 2.3, for any ε (0 < 2ε < σ−ρ1) , there exists a set e3 ⊂ (1, +∞) with a finite linearmeasure such that |as (z)| ≤ expp+1 { (ρ(p,q) (as) + ε) logq r } = expp+1 { (ρ1 + ε) logq r } (2.31) holds for all z satisfying |z| = r /∈ e3. from the hypotheses of lemma 2.13, there exists a set hwith log densh > 0 (or ml (h) = +∞) such that |as (z)| ≥ expp+1 { (σ −ε) logq r } (2.32) eur. j. math. anal. 1 (2021) 98 holds for all z satisfying |z| = r ∈ h, r → +∞. by (2.31) and (2.32), we conclude that for all zsatisfying |z| = r ∈ h re3, r → +∞, we have expp+1 { (σ −ε) logq r } ≤ expp+1 { (ρ1 + ε) logq r } and by ε (0 < 2ε < σ−ρ1) this is a contradiction as r → +∞. consequently, any transcendentalmeromorphic solution f of equation (1.4) satisfies ρ[p,q] (f ) ≥ σ. lemma 2.14 ([23]) let p ≥ q ≥ 1 be integers. let f be a meromorphic function for which ρ[p,q] (f ) = β < +∞, and let k ≥ 1 be an integer. then for any ε > 0, m ( r, f (k) f ) = o ( expp−1 { (β + ε) logq r }) , holds outside of a possible exceptional set e8 of finite linear measure. lemma 2.15 let a0,a1, ...,ak 6≡ 0,f 6≡ 0 be finite [p,q]-order meromorphic functions. if f is a meromorphic solution with ρ[p,q] (f ) = +∞ and ρ[p+1,q] (f ) = ρ < +∞ of equation (1.4) , then λ[p,q] (f ) = λ[p,q](f ) = ρ[p,q](f ) = +∞ and λ[p+1,q] (f ) = λ[p+1,q](f ) = ρ[p+1,q](f ) = ρ. proof let f be a meromorphic solution of (1.4) with infinite [p,q]-order and ρ[p+1,q] (f ) = ρ < +∞. note first that by definition, we have λ[p+1,q] (f ) ≤ λ[p+1,q] (f ) ≤ ρ[p+1,q] (f ) . then, itremains to show that ρ[p+1,q] (f ) ≤ λ[p+1,q] (f ) ≤ λ[p+1,q] (f ) . we rewrite (1.4) as 1 f = 1 f ( ak (z) f (k) f + ak−1 (z) f (k−1) f + · · · + a1 (z) f ′ f + a0 (z) ) . (2.33) by using lemma 2.14 and (2.33), for |z| = r outside a set e8 of a finite linear measure and anygiven ε > 0, we get m ( r, 1 f ) ≤ m ( r, 1 f ) + k∑ j=1 m ( r, f (j) f ) + k∑ j=0 m ( r,aj ) + o (1) ≤ m ( r, 1 f ) + k∑ j=0 m ( r,aj ) + o ( expp { (ρ + ε) logq r }) . (2.34) on the other hand, by (1.4), if f has a zero at z0 of order α (α > k), and a0, a1, ..., ak are allanalytic at z0, then f must have a zero at z0 of order at least α−k. hence, n ( r, 1 f ) ≤ kn ( r, 1 f ) + n ( r, 1 f ) + k∑ j=0 n ( r,aj ) eur. j. math. anal. 1 (2021) 99 and n ( r, 1 f ) ≤ kn ( r, 1 f ) + n ( r, 1 f ) + k∑ j=0 n ( r,aj ) . (2.35) therefore, by (2.34) and (2.35), for all sufficiently large r /∈ e8 and any given ε > 0, we have t (r, f ) = t (r, 1 f ) + o (1) ≤ t (r,f ) + k∑ j=0 t ( r,aj ) + kn ( r, 1 f ) + o ( expp { (ρ + ε) logq r }) . (2.36) noting c = max {ρ[p,q](aj) (j = 0, 1, ...,k),ρ[p,q] (f )} . then, by using the definition of the [p,q]-order, for the above ε and sufficiently large r , we have t (r,f ) ≤ expp { (c + ε) logq r } , (2.37) t ( r,aj ) ≤ expp { (c + ε) logq r } , j = 0, 1, ...,k. (2.38) replacing (2.37) and (2.38) into (2.36) , for r /∈ e8 sufficiently large and any given ε > 0, weobtain t (r, f ) ≤ kn ( r, 1 f ) + (k + 2) expp { (c + ε) logq r } + o ( expp { (ρ + ε) logq r }) . (2.39) hence, for any f with ρ[p,q] (f ) = +∞ and ρ[p+1,q](f ) = ρ, by (2.39) , we have λ[p,q] (f ) ≥ ρ[p,q] (f ) = +∞, λ[p+1,q] (f ) ≥ ρ[p+1,q] (f ) , so ρ[p+1,q] (f ) ≤ λ[p+1,q] (f ) ≤ λ[p+1,q] (f ) . and the fact that λ[p+1,q] (f ) ≤ λ[p+1,q] (f ) ≤ ρ[p+1,q] (f ) , we obtain λ[p+1,q] (f ) = λ[p+1,q] (f ) = ρ[p+1,q] (f ) = ρ. 3. proof of theorem 1.1 assume that f 6≡ 0 is a rational solution of (1.3). first, we will prove that f must be a polynomialwith deg f ≤ s − 1. for this, if f is a rational function, which has a pole at z0 of degree m ≥ 1,or f is a polynomial with deg f ≥ s, then f (s)(z) 6≡ 0. by (1.3) and lemma 2.4, we obtain σ ≤ ρ[p,q](as) = ρ[p,q](asf (s)) = ρ[p,q] −  k∑ j=0, j 6=s ajf (j)  ≤ max j=0,1,...,k, j 6=s { ρ[p,q] ( aj )} which is a contradiction. therefore, f must be a polynomial with deg f ≤ s − 1. eur. j. math. anal. 1 (2021) 100 now, we assume that f is a transcendental meromorphic solution of (1.3) such that λ[p,q](1f ) < µ[p,q](f ). by lemma 2.3, for any given ε (0 < 2ε < σ−ρ) , there exists a set e3 ⊂ (1, +∞) witha finite linear measure (and so of finite logarithmic measure) such that |aj (z) | ≤ expp+1 { (ρ + ε) logq r } , j = 0, 1, ...,k, j 6= s (3.1) holds for all z satisfying |z| = r /∈ e3. in view of lemma 2.8, there exists a set e6 ⊂ (1, +∞) offinite logarithmic measure such that |z| = r /∈ [0, 1] ∪e6, |g (z) | = m (r,g) and for r sufficientlylarge, we have ∣∣∣∣ f (z)f (s) (z) ∣∣∣∣ ≤ r2s (s ≥ 1 is an integer) . (3.2) according to lemma 2.1, there exist a set e1 ⊂ (1, +∞) with ml(e1) < ∞ and a constant b > 0,such that for all z satisfying |z| = r /∈ [0, 1] ∪e1, we have∣∣∣∣∣f (j) (z)f (z) ∣∣∣∣∣ ≤ b [t (2r, f )]k+1 , j = 1, 2, ...,k, j 6= s. (3.3) from the hypotheses of theorem 1.1, there exists a set h ⊂ (1, +∞) with ml (h) = +∞, suchthat for all z satisfying |z| = r ∈ h, r → +∞ and sufficiently small ε > 0, we have |as (z) | ≥ expp+1 { (σ −ε) logq r } . (3.4) now, by rewriting equation (1.3) in the form |as| ≤ ∣∣∣∣ ff (s) ∣∣∣∣ |a0| + k∑ j=1 j 6=s ∣∣aj∣∣ ∣∣∣∣∣f (j)f ∣∣∣∣∣  (3.5) and substituting (3.1) , (3.2) , (3.3) and (3.4) into (3.5), for all z satisfying |z| = r ∈ hr([0, 1]∪ e1 ∪e3 ∪e6), r → +∞, we obtain expp+1 { (σ −ε) logq r } ≤ bkr2s expp+1 { (ρ + ε) logq r } [t (2r, f )] k+1 . since 0 < 2ε < σ−ρ, then we have exp { (1 −o (1)) expp { (σ −ε) logq r }} ≤ bkr2s [t (2r, f )]k+1 . (3.6) from (3.6) and lemma 2.9, for any given γ > 1 and sufficiently large r > r, we get exp { (1 −o (1)) expp { (σ −ε) logq r }} ≤ bk (γr)2s [t (2γr,f )]k+1 which gives ρ[p,q](f ) = µ[p,q] (f ) = +∞, σ ≤ ρ[p+1,q] (f ) . (3.7) by using lemma 2.4, we have max { ρ[p,q] ( aj ) : j = 0, 1, ...,k } = ρ[p,q] (as) = β < +∞. eur. j. math. anal. 1 (2021) 101 since f is of infinite [p,q]-order meromorphic solution of equation (1.3) satisfying λ[p,q](1f ) < µ[p,q] (f ), then by lemma 2.11, we obtain ρ[p+1,q] (f ) ≤ max { ρ[p,q] ( aj ) : j = 0, 1, ...,k } = ρ[p,q] (as) . (3.8) by (3.7) and (3.8) , we conclude that µ[p,q] (f ) = ρ[p,q] (f ) = +∞ and σ ≤ ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . 4. proof of corollary 1.1 assume that ϕ is a transcendental meromorphic function such that ρ[p+1,q] (ϕ) < σ. noting g = f − ϕ, then ρ[p+1,q] (g) = ρ[p+1,q] (f ), so by theorem 1.1, σ ≤ ρ[p+1,q] (g) ≤ ρ[p,q] (as) . bysubstituting f = g + ϕ into (1.3), we obtain ak (z) g (k) + ak−1 (z) g (k−1) + · · · + a1 (z) g′ + a0 (z) g = − ( ak (z) ϕ (k) + ak−1 (z) ϕ (k−1) + · · · + a1 (z) ϕ′ + a0 (z) ϕ ) = g (z) . (4.1) it is clear that the right side g of equation (4.1) is non-zero, because by theorem 1.1, ϕ is not asolution of equation (1.3). moreover, the [p + 1,q]-order of g satisfies ρ[p+1,q] (g) ≤ max { ρ[p+1,q] (ϕ) , ρ[p+1,q] ( aj ) (j = 0, 1, ...,k) } < σ, which implies max { ρ[p+1,q] (g) , ρ[p+1,q] ( aj ) (j = 0, 1, ...,k) } < σ ≤ ρ[p+1,q] (g) . then by lemma 2.12, we obtain σ ≤ λ[p+1,q] (g) = λ[p+1,q] (g) = ρ[p+1,q] (g) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) ,that is σ ≤ λ[p+1,q] (f −ϕ) = λ[p+1,q] (f −ϕ) = ρ[p+1,q] (f −ϕ) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . 5. proof of theorem 1.2 assume that f is a rational solution of (1.4). first, we will prove that f must be a polynomial with deg f ≤ s − 1. for this, if f is a rational function, which has a pole at z0 of degree m ≥ 1, or f isa polynomial with deg f ≥ s, then f (s)(z) 6≡ 0. by (1.4) and lemma 2.4, we obtain σ ≤ ρ[p,q](as) = ρ[p,q](asf (s)) = ρ[p,q] f − k∑ j=0 j 6=s aj (z) f (j)  ≤ max j=0,1,...,k, j 6=s { ρ[p,q] ( aj ) , ρ[p,q] (f ) } , eur. j. math. anal. 1 (2021) 102 which is a contradiction. therefore, f must be a polynomial with deg f ≤ s − 1. now, we assume that f is a transcendental meromorphic solution of (1.4) such that λ[p,q](1f ) < µ[p,q](f ). from lemma 2.13, we know that f satisfies ρ[p,q] (f ) ≥ σ. by the hypothesis λ[p,q](1f ) < min{µ[p,q](f ),σ} and hadamard factorization theorem, we can write f as f (z) = g(z)d(z), where g (z)and d (z) are entire functions satisfying µ[p,q](g) = µ[p,q](f ) = µ ≤ ρ[p,q](g) = ρ[p,q](f ), ρ[p,q](d) = λ[p,q] ( 1 f ) = β < min{µ[p,q](f ),σ}. the definition of the lower [p,q]-order assures us that |g (z)| = m(r,g) ≥ expp+1 { (µ[p,q] (g) −ε) logq r } . (5.1) putting ρ1 = max { ρ[p,q] ( aj ) (j 6= s) ,ρ[p,q] (f ) } < σ. then, by lemma 2.3 and (5.1), for any given ε satisfying 0 < 2ε < min{σ −ρ1,µ[p,q] (g) −ρ[p,q] (d)}, there exists a set e3 ⊂ (1, +∞) with a finite logarithmic measure such that for all z satisfying |z| = r /∈ e3 at which |g (z) | = m(r,g), we obtain∣∣∣∣f (z)f (z) ∣∣∣∣ = |f (z)||g(z)| |d (z)| ≤ expp+1 { (ρ[p,q] (d) + ε) logq r } expp+1 { (ρ1 + ε) logq r } expp+1 { (µ[p,q] (g) −ε) logq r } ≤ expp+1 { (ρ1 + ε) logq r } . (5.2) by using the same arguments as in the proof of theorem 1.1, for any given ε ( 0 < 2ε < min{σ −ρ1,µ[p,q] (g) −ρ[p,q] (d)} ) and all z satisfying |z| = r ∈ hr(e1 ∪e3 ∪e6) , r → +∞ at which |g (z) | = m(r,g), we have (3.2) , (3.3) , (3.4) hold and |aj (z) | ≤ expp+1 { (ρ1 + ε) logq r } , j = 0, 1, ...,k, j 6= s. (5.3) by (1.4) , we have |as| ≤ ∣∣∣∣ ff (s) ∣∣∣∣ |a0| + k∑ j=1 j 6=s ∣∣aj∣∣ ∣∣∣∣∣f (j)f ∣∣∣∣∣ + ∣∣∣∣ff ∣∣∣∣  . (5.4) hence, by substituting (3.2) , (3.3) , (3.4) , (5.2) and (5.3) into (5.4) , for all z satisfying |z| = r ∈ h r (e1 ∪e3 ∪e6), r → +∞, at which |g (z) | = m (r,g) and any given ε ( 0 < 2ε < min{σ −ρ1,µ[p,q] (g) −ρ[p,q] (d)} ) , we obtain expp+1 { (σ −ε) logq r } ≤ r2s ( expp+1 { (ρ1 + ε) logq r } eur. j. math. anal. 1 (2021) 103 + k∑ j=1,j 6=s expp+1 { (ρ1 + ε) logq r } b [t (2r, f )] k+1 + expp+1 { (ρ1 + ε) logq r }) ≤ b (k + 1) r2s [t (2r, f )]k+1 expp+1 { (ρ1 + ε) logq r } . (5.5) since 0 < 2ε < σ −ρ1, then we can use lemma 2.9 with (5.5) such that for any given γ > 1 andsufficiently large r > r, we obtain exp { (1 −o (1)) expp { (σ −ε) logq r }} ≤ b (k + 1) (γr)2s [t (2γr,f )]k+1 which gives ρ[p,q](f ) = µ[p,q] (f ) = +∞, ρ[p+1,q](f ) ≥ σ. (5.6) making use of lemma 2.4, we have max { ρ[p,q] ( aj ) (j = 0, 1, ...,k) ,ρ[p,q] (f ) } = ρ[p,q] (as) = β < +∞. by lemma 2.11 and since f is of infinite [p,q]-order meromorphic solution of equation (1.4)satisfying λ[p,q](1f ) < µ[p,q] (f ) , we get ρ[p+1,q] (f ) ≤ max { ρ[p,q] ( aj ) (j = 0, 1, ...,k) ,ρ[p,q] (f ) } = ρ[p,q] (as) . (5.7) since f 6≡ 0, then by lemma 2.15, we have λ[p,q] (f ) = λ[p,q](f ) = µ[p,q] (f ) = ρ[p,q](f ) = +∞ (5.8) and σ ≤ λ[p+1,q] (f ) = λ[p+1,q](f ) = ρ[p+1,q](f ). (5.9) by (5.7) , (5.8) and (5.9) , we conclude that λ[p,q] (f ) = λ[p,q](f ) = µ[p,q] (f ) = ρ[p,q](f ) = +∞ and σ ≤ λ[p+1,q] (f ) = λ[p+1,q](f ) = ρ[p+1,q](f ) ≤ ρ[p,q] (as) . 6. proof of corollary 1.2 assume that ϕ is a transcendental meromorphic function such that ρ[p+1,q] (ϕ) < σ. noting h = f − ϕ, then ρ[p+1,q] (h) = ρ[p+1,q] (f ), so by theorem 1.2, σ ≤ ρ[p+1,q] (h) ≤ ρ[p,q] (as) . bysubstituting f = h + ϕ into (1.4), we obtain ak (z) h (k) + ak−1 (z) h (k−1) + · · · + a1 (z) h′ + a0 (z) h = f (z) − ( ak (z) ϕ (k) + ak−1 (z) ϕ (k−1) + · · · + a1 (z) ϕ′ + a0 (z) ϕ ) = ψ (z) . (6.1) eur. j. math. anal. 1 (2021) 104 it is clear that the right side ψ of the equation (6.1) is non-zero, because by theorem 1.2, ϕ isnot a solution of equation (1.4). moreover, the [p + 1,q]-order of ψ verifies ρ[p+1,q] (ψ) ≤ max { ρ[p+1,q] (ϕ) , ρ[p+1,q] ( aj ) (j = 0, 1, ...,k) } < σ, which leads to max { ρ[p+1,q] (ψ) , ρ[p+1,q] ( aj ) (j = 0, 1, ...,k) } < σ ≤ ρ[p+1,q] (h) . therefore, by lemma 2.12, we obtain σ ≤ λ[p+1,q] (h) = λ[p+1,q] (h) = ρ[p+1,q] (h) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) , that is σ ≤ λ[p+1,q] (f −ϕ) = λ[p+1,q] (f −ϕ) = ρ[p+1,q] (f −ϕ) = ρ[p+1,q] (f ) ≤ ρ[p,q] (as) . acknowledgements. this paper was supported by the directorate-general for scientific researchand technological development (dgrsdt). references [1] b. belaïdi, growth and oscillation theory of [p,q]-order analytic solutions of linear differential equations in theunit disc. j. math. anal. 3 (2012), 1–11. http://www.ilirias.com/jma/repository/docs/jma3-1-1.pdf.[2] b. belaïdi, iterated order of meromorphic solutions of homogeneous and non-homogeneous linear differentialequations. romai j. 11 (2015), 33–46. http://rj.romai.ro/arhiva/2015/1/rjv11n1.pdf.[3] b. belaïdi, differential polynomials generated by meromorphic solutions of [p,q]-order to complex linear differentialequations. rom. j. math. comput. sci. 5 (2015), 46–62. http://rjm-cs.ro/belaidi-2015.pdf.[4] a. ferraoun and b. belaïdi, on the (p,q)−order of solutions of some complex linear differential equations. commun.optim. theory, 2017 (2017), article id 17, 1–23. https://doi.org/10.23952/cot.2017.17.[5] g. g. gundersen, estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. j.london math. soc. 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https://scindeks-clanci.ceon.rs/data/pdf/2217-5539/2017/2217-55391702103s.pdf https://pubs.ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817 https://pubs.ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817 https://doi.org/10.1080/1726037x.2017.1413065 https://www.math.u-szeged.hu/ejqtde/p453.pdf https://www.math.u-szeged.hu/ejqtde/p453.pdf https://doi.org/10.1155/2013/243873 https://doi.org/10.1155/2013/243873 1. introduction and main results 2. some auxiliary lemmas 3. proof of theorem 1.1 4. proof of corollary 1.1 5. proof of theorem 1.2 6. proof of corollary 1.2 references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 4doi: 10.28924/ada/ma.3.4 parameter estimation for spdes driven by cylindrical stable processes jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte,376 fretwell bldg, 9201 university city blvd. charlotte, nc 28223-0001, usacorrespondence: j.bishwal@uncc.edu abstract. we consider infinite dimensional extension of affine models with heavy tails in finance. westudy several estimators of the drift parameter in the stochastic partial differential equation drivenby cylindrical stable processes. we consider several sampling schemes. we also consider randomsampling scheme, e.g, when the solution process is observed at the arrival times of a poisson process.we obtain the consistency and the asymptotic normality of the estimators. 1. introduction parameter estimation in stochastic partial differential equations is a very young area of researchin view of its applications in finance, physics, biology and oceanography. loges [32] initiated thestudy of parameter estimation in infinite dimensional stochastic differential equations. when thelength of the observation time becomes large, he obtained consistency and asymptotic normality ofthe maximum likelihood estimator (mle) of a real valued drift parameter in a hilbert space valuedsde. koski and loges [28] extended the work of loges [32] to minimum contrast estimators. koskiand loges [27] applied the work to a stochastic heat flow problem. martingale estimation functionfor discretely observed diffusions was studied in bibby and srensen [2]. bishwal [6] studied a newestimating function for discretely sampled diffusions by removing the stochastic integral in girsanovlikelihood. bishwal [7] contains asymptotic theory on likelihood method and bayesian method fordrift estimation of finite and infinite dimensional stochastic differential equations. bishwal [12]studied applications of levy processes in stochastic volatility models in finance.huebner, khasminskii and rozovskii [23] started statistical investigation in spdes. they gavetwo contrast examples of parabolic spdes in one of which they obtained consistency, asymptoticnormality and asymptotic efficiency of the mle as noise intensity decreases to zero under thecondition of absolute continuity of measures generated by the process for different parameters (the received: 12 apr 2022. key words and phrases. stochastic partial differential equations; space-time colored noise; cylindrical stable process;stable random field; super levy process; poisson sampling; martingale estimating function; quasi likelihood estimator;stable ornstein-uhlenbeck process; stable black-scholes model; stable cox-ingersoll-ross model; consistency; asymp-totic normality. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 2 situation is similar to the classical finite dimensional case) and in the other they obtained theseproperties as the finite dimensional projection becomes large under the condition of singularity ofthe measures generated by the process for different parameters. the second example was extendedby huebner and rozovskii [24] and the first example was extended by huebner [22] to mle forgeneral parabolic spdes where the partial differential operators commute and satisfy differentorder conditions in the two cases.huebner [21] extended the problem to the ml estimation of multidimensional parameter. lototskyand rozovskii [33] studied the same problem without the commutativity condition. small noiseasymptotics of the nonparmetric estimation of the drift coefficient was studies by ibragimov andkhasminskii [29].based on continuous observations, usually there can be two asymptotic settings in spde: 1) t → ∞ 2) n → ∞ where t is the length of the observations and n is the number of fouriercoefficients of the spde solution.in a bayesian approach, using the first setting, bishwal [3] proved the bernstein-von misestheorem and asymptotic properties of regular bayes estimator of the drift parameter in a hilbertspace valued sde when the corresponding ergodic diffusion process is observed continuously overa time interval [0,t ]. the asymptotics are studied as t → ∞ under the condition of absolutecontinuity of measures generated by the process. results are illustrated for the example of anspde.using the second setting, bishwal [5] proved the bernstein-von mises theorem and spectralasymptotics of bayes estimators for parabolic spdes when the number of fourier coefficientsbecomes large. in this case, the measures generated by the process for different parameters aresingular.bishwal [10] studied bernstein-von mises theorem and small noise bayesian asymptotics for par-abolic stochastic partial differential equations. bishwal [9] studied hypothesis testing for fractionalstochastic partial differential equations with applications to neurophysiology and finance.in this paper we study the asymptotic properties of the quasi maximum likelihood estimatorwhen we have observations of finite-dimensional projections at poisson arrival time points. theasymptotic setting is only the large number of observations at random time points which are thearrivals of a poisson process.the rest of the paper is organized as follows: section 2 contains model, assumptions andpreliminaries. in section 3 we prove estimation results with additive noise. section 4 and 5, weprovide estimation results with multiplicative noise. in section 6, we give several examples. 2. model and preliminaries let h be a real separable hilbert space with inner product 〈·〉 and norm | · |. by l(h) we denotethe banach space of bounded linear operators from h into h endowded with the operator norm https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 3 ‖ · ‖l(h). we fix an orthonormal basis (en) in h. through the basis (en) we will often identify hin l2. more generally, for a given sequence ρ = (ρn) of real numbers we set l2ρ = {(xn) ∈r ∞ : ∑ n≥1 x2nρ 2 n < ∞}. where r∞ = rn. the space l2ρ becomes a separable hilbert space with the inner product: 〈x,y〉 =∑ n≥1 xnynρ 2 n for x = (xn),y = (yn) ∈ l2ρ . let us fix θ0, the unknown true value of the parameter θ.let (ω,f,p ) be a complete probability space and z(t,x) be a process on this space with valuesin the schwarz space of distributions d′(g) such that for φ,ψ ∈ c∞0 (g),‖φ‖−1l2(g) 〈w (t, ·),φ(·)〉is a one dimensional stable process.this process is usually referred to as the cylindrical α-stable process (c.s.p.), α ∈ (0, 2).we assume that there exists a complete orthonormal system {hi}∞i=1 in l2(g)) such that forevery i = 1, 2, . . . ,hi ∈ zm,20 (g) ∩c∞(g) and λθhi = βi (θ)hi, and lθhi = µi (θ)hi for all θ ∈ θ where lθ is a closed self adjoint extension of aθ, λθ := (k(θ)i −lθ)1/2m,k(θ) is a constant andand the spectrum of the operator λθ consists of eigenvalues {βi (θ)}∞i=1 of finite multiplicities and µi = −β2mi + k(θ).a levy process (zt) with values in h is an h-valued process defined on some stochastic basis (ω,f, (ft)t≥0,p ) having stationary independent increments, cadlag trajectories such that z0 = 0,p-a.s. one has that e[ei〈zt,s〉] = exp(−tψ(s)), s ∈ h where ψ : h → c is sazonov continuous, negative definite function such that ψ(0) = 0. thefunction ψ is called the exponent of (zt).the exponent ψ can be expressed by the infinite dimensional levy-khintchine formula ψ(s) = 1 2 〈qs,s〉− i〈a,s〉− ∫ h ( ei〈s,y〉 − 1 − i〈s,y〉 1 + |y|2 ) ν(dy), s ∈ h where q is the non-negative trace class operator on h, a ∈ h and ν is the levy measure or thejump intensity measure associated to (zt).cylindrical α-stable process (c.s.p.) is a levy process taking values in the hilbert space u = l2ρ ,with a properly chosen weight ρ.consider the linear spde dxt = θaxtdt + dzt,x ∈ hc.s.p. z(t) is a cylindrical α-stable process, α ∈ (0, 2) which can be expanded in the series z(t) = ∞∑ i=1 γizi (t)hi where {zi (t)}∞i=1 are independent, real valued, one dimensional, normalized, symmetric, α-stableprocesses and (γi ) is a given sequence of, possibly unbounded, positive numbers, and hi is a fixed https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 4 orthonormal basis in h. the latter series converges p-a.s. in h−α for α > d/2. indeed ‖z(t)‖2−α = ∞∑ i=1 γ2i z 2 i (t)‖hi‖ 2 −α = ∞∑ i=1 z2i (t)β −2α i and the later series converges p-a.s.for any j ∈n,t ≥ 0, e[eizj (t)h] = e−t|h| α . stable one-dimensional density :a one-dimensional, normalized, symmetric α-stable distribution µα, α ∈ (0, 2] has characteristicfunction µ̂α(s) = e −|s|α,s ∈r.the density of µα with respect to lebesgue measure will be denoted by pα. this even functionis known in closed form only if α = 1 or 2. the precise asymptotic behavior of the density pα,α ∈ (0, 2) is as follows:for any α ∈ (0, 2), there exists cα such that pα(x) ∼ cα xα+1 as x →∞. stable measures on hilbert space :a random variable ξ on h is called α-stable (α ∈ (0, 2]) if for any n there exists avector an ∈ h such that for any independent copies ξ1,ξ2, . . . ,ξn of ξ, the random variable n−1/α(ξ1 + ξ2, . . . + ξn) −an has the same distribution as ξ. a borel probability measure µ on his said to be α-stable if it is the distribution of a stable random variable with vales in h. stable ou process: dxt = −θxtdt + σdzt, x0 = x0the solution is xt = e −θtx0 + ∫ t 0 e−θ(t−s)σdzs.the stochastic integral can be defined as the limit in probability of riemann sums.let yt = ∫ t 0 e−θ(t−s)σdzs.then e[eihyt ] = exp [ −σα|h|α ∫ t 0 e−αθsds ] = e−|h| αcα(t) where cα(t) = σ ( 1 −e−αθt αθ ) . we show that the process x is stochastically continuous.first we show that y is stochastically continuous, i.e., lim h→0+ sup t≥0 p (|yt+h −yt| > �) = 0 https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 5 note that for any t ≥ 0, h ≥ 0, yt+h −yt = ∫ t+h t e(t+h−s)adzs + e ha ∫ t 0 e(t−s)adzs − ∫ t 0 e(t−s)adzs = ehayt −yt + ∫ t+h t e(t+h−s)adzs let us choose p ∈ (0,α). we have p (|yt+h −yt| > �) ≤ p ( |ehayt −yt| > � 2 ) + p (∣∣∣∣∫ t+h t e(t+h−s)adzs ∣∣∣∣ > �2 ) ≤ 2p e|ehayt −yt|p �p + 2p e| ∫h 0 esadzs|p �p = i1(t,h) + i2(h).but e|yt|p ≤ cp ( ∞∑ n=1 1 −e−αθt αθ )p/α and so [i2(h)]α/p → 0 as h → 0. concerning i1, by khintchine inequality |ehayt −yt| = ∑ n≥1 ∣∣(e−θh − 1)y nt ∣∣2 1/2 ≤ cp ẽ ∣∣∣∣∣∣ ∑ n≥1 rn(e −θh − 1)y nt ∣∣∣∣∣∣ p1/p . where ẽ denotes expectation w.r.t. to the measure p̃ p̃ (rn = 1) = p̃ (rn = 1) = 1/2 where arademacher sequence (rn) with rn : ω̃ → {−1, 1} is defined on the probability space (ω̃,f̃, p̃ ).hence e|ehayt −yt|p ≤ cppẽe ∑ n≥1 ∣∣(e−θh − 1)y nt ∣∣2 1/2 ≤ cp ∑ n≥1 ∣∣(1 −e−θht)βn∣∣α (1 −e−θht) αθ p/α ≤ cp αp/α ∑ n≥1 ∣∣(1 −e−θht)βn∣∣α θ p/α since lim h→0+ ∑ n≥1 ∣∣(1 −e−θht)βn∣∣α θ p/α = 0, we get lim h→0+ sup t≥0 2p e|ehayt −yt|p �p = 0. since e|yt|p ≤ cp ∑ n≥1 |βn|α (1 −e−θht) αθ p/α , hence lim h→0 ∑ n≥1 |βn|α (1 −e−θht) αθ p/α = 0 https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 6 hence lim h→0+ 2p e| ∫h 0 esadzs|p �p → 0. thus lim h→0+ sup t≥0 i1(t,h) = 0 this proves stochastic continuity of yt. using the stochastic continuity and ft-adaptedness of x, we conclude that the process x hasa predictable version. time change: let l be a one dimensional α-stable process, α ∈ (0, 2). then there exists an α-stable process, α ∈ (0, 2) z = (zt) such that∫ t 0 e−θsdls = z(u(t)) where u(t) = 1 −e−αθt αθ . recall that u ∈ c∞([0,∞]) with u′(t) 6= 0, t ≥ 0.in the limiting gaussian case of α = 2, it becomes time change for brownian motion. infinite dimensional stable ou process dxnt = −θx n t dt + σdz n t , x n 0 = xn,n ∈nwith x = (xn) ∈ l2 = h. the solution is a stochastic process x = xxt with values in r∞ withcomponents xxt = e −θtxn + ∫ t 0 e−θ(t−s)σdzns .(the stochastic integral can be defined as the limit in probability of riemann sums.) xxt = ∞∑ n=1 xnt en = e tax + za(t) where za(t) = ∫ t 0 e(t−s)adzs = ∞∑ n=1 (∫ t 0 e−θ(t−s)σdzns ) en. the process xxt is an ft-adapted irreducible markov process and its transition semigroup isstrong feller.let y nt = z n a(t) = ∫ t 0 e−θ(t−s)σdzns ,n ∈n, t ≥ 0.then e[eihy n t ] = exp [ −σα|h|α ∫ t 0 e−αθsds ] = e−|h| αcαn (t) https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 7 where cn(t) = σ ( 1 −e−αθt αθ )1/α . it follows that e[eihy n t ] = e[eihcn(t)ln ], h ∈r where (ln) are independent α-stable random variables having the same law µα. thus xxt is ft-adapted.the markov property easily follows from the identity za(t + h) −ehaza(t) = ∫ t+h t e(t+h−s)adzs, t,h ≥ 0. if the cylindrical levy process z takes values in hilbert space h, the by the kotelenez regularityresults trajectories of the process x are cadlag with values in h. moments of the process the ou process is stochastically continuous and trajectories in lp([0,t ]; h) for any 0 < p < αa.s. set yt := za(t). then we have e|yt|p ≤ c̃pσp ( ∞∑ n=1 1 −e−αθt αθ )p/α where c̃p depends on p. moments of the stochastic integral suppose (zt) is an α-stable levy process with 0 ≤ α ≤ 2 and y(t) is a predictable processsatisfying ∫t 0 |y(t)|αdt < ∞. then for any 0 < r < α, there exists a constant c such that e [ sup t≤t ∣∣∣∣∫ t 0 y(s)dzs ∣∣∣∣r] ≤ e [(∫ t 0 |y(t)|αdt )r/α] . equivalence of transition probabilities assume sup n≥1 e−γntγ 1/α n βn = ct < ∞, e ∫ t 0 ∑ n≥1 |y nt | 2 p/2 dt < ∞. let pα be the density of the one dimensional stable measure. then the laws µxt and µyt of xxt and x y t respectively are equivalent for any t > 0, x,y ∈ h,α ∈ (0, 2). moreover, the density dµxtdµxt of https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 8 µxt with respect to µyt is given by dµxt dµ y t = lim n→∞ n∏ k=1 pα ( zk−e−θtxk c(t) ) pα ( zk−e−θtyk c(t) ). the corresponding mle is denoted as θ̂n. priola et al. [38] obtained exponential convergence to the invariant measure, in the total variationnorm, for solutions to sdes driven by α-stable noises in finite and infinite dimensions using twoapproaches: lyapounov’s function approach by harris and doeblin’s coupling argument. in bothapproaches irreducibility and uniform strong feller property play crucial role.first we consider the method of moments estimation in modified tempered stable-ornstein-uhlenbeck model. masuda and uehara [36] studied two-step estimation in ergodic levy drivensde dxt = a(θ,xt)dt + b(β,xt−)dzt, x0 = x0.masuda [35] studied multi-step estimation in stable ou model: dxt = −θxtdt + σdzt, x0 = x0. for the least squares estimator (lse) θ̃n of θ, hu and long [20] obtained( t log n )1/α (θ̃n −θ0) →d s′α s′′+ α/2where sα is stable distribution of order β.while in gaussian ou case, for different parts θ > 0, θ < 0 and θ = 0, lan, lamn and labfhold respectively (see bishwal [11]), in stable case entirely different phenomena occur.the solution of the sde is given by xt = e −θ(t−s)xs + σ ∫ t s e−θ(t−s)dzu,t ≥ 0. due to the stable integral property, l (∫ t s e−θ(t−s)dzu ) = sα(κ∆(θ)) where κ∆(θ) = { 1 −e−θ∆ θα }1/α ∼ ∆1/α. for each j ≤ n, the transition probability is given by l(xtj |xtj−1 = x) = δx exp(−θ∆) ? sα(κ∆(θ)). lamn holds for θ ∈r when t is fixed. n1/α−1/2(θ̂n −θ) →d mn(0, iθ(t )−1). where iθ(t ) is the fisher information of the process. https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 9 we study estimation in mts-ou sv model. the inverse gaussian-ou and gamma-ou modelsare special cases.an infinitely divisible distribution is said to be α-modified tampered stable distribution (α-mts)distribution if its levy triplet is given by σ2 = 0, ν(dx) = c λα+ 1 2 + kα+ 1 2 λ+x xα+ 1 2 ix>0 + λ α+ 1 2 + kα+ 1 2 λ−x xα+ 1 2 ix<0 dx, γ = µ + c ( γ( 1 2 −α) 2α+ 1 2 (λ2α−1+ −λ 2α−1 − ) −λ α−1 2 + kα−1 2 (λ+) + λ α−1 2 − kα−1 2 (λ−) ) where c > 0,λ+,λ− > 0, µ ∈ r, α ∈ (−∞, 1)\{12} and kp(x) is the modified bessel functionof second kind. we denote the mts random variable by x ∼ mts(α,c,λ+,λ−,µ). the levymeasure ν(dx) is called the mts levy measure with parameter (α,c,λ+,λ−).the mts distribution is obtained by taking a symmetric α-stable distribution with α ∈ (0, 1)and multiplying by a levy measure with √|x|λα+ 12 k α+ 1 2 (λ|x|) on each half of the real axis. themeasure can be extended to the case α ≤ 0. if α = 1 2 , then γ may not be defined, so it is removed.the mts distribution was introduced by kim, rachev and chung [25].the tails of the α-mts distribution are thinner than those of the 2α-stable and fatter (heavier)than those of the 2α-ts distribution. at the zero neighborhood, all three have the same asymptoticbehavior.if λ+ > λ−, then the distribution is skewed to the left. if λ+ < λ−, then the distribution is skewed to the right. if λ+ = λ−, then the distribution is symmetric. c controls the kurtosis of the distribution. if c increases, the peakedness of the distributionincreases.as α decreases, the distribution has fatter tails and increased peakedness. the levy processcorresponding to the mts distribution has finite activity if α < 0 and infinite activity if α > 0. ithas finite variation if α < 1 2 and infinite variation if α > 1 2 .with proper choice of c and µ, mts distribution has zero mean and unit variance, and thedistribution is called standard mts distribution and denoted x ∼ stdmts(α,λ+,λ−).cgmy process proposed in carr et al. [14] is a tempered stable process. in order to obtain aclosed form solution of the european option price, cgmy used the generalised fourier transformof the distribution of the stock price under the assumption of markov property.the stochastic volatility model is given by dyt = (µ + βxt) dt + √ xt dwt + ρ dzt dxt = −θ xt dt + dzt where µ is the drift parameter, β is the risk premium, θ > 0 is the drift of the volatility and zt isa mts process. https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 10 we estimate θ from the observations of {yt} at the time points tk = k∆, k = 0, 1, 2, . . . ,n, ∆ > 0. cm(z) := dm dum log φts(u)|u=0for the tempered stable distribution ts(b,δ,γ) where 0 < b < 1,δ > 0,γ ≥ 0, the m-th cumulantis given by cm(z) = −δ(−2)mγ(b−m)/bb(b− 1) . . . (b− (m− 1))for γ > 0. when γ = 0, it is positive b-stable distribution for which the moments of only order k < b exist. for b = 1/2, ts distribution reduces to inverse gaussian (ig) distribution.the infinite divisibility of this distribution allows one to construct the corresponding levy process.a levy process z = (zt)t≥0 is said to be a tempered stable process if z1 follows a temperedstable distribution. the tempered stable process is of finite activity if α < 0 and infinite activity if 0 < α < 2. the tempered stable process is of finite variation if 0 < α < 1 and infinite variation if 1 < α < 2.the mts-garch model is given by log st st−1 = rt −dt + λtσt −g(σt; α,λ+,λ−) + σt�t σ2t = (α0 + α1σ 2 t−1� 2 t−1 + β1σ 2 t−1) ∧ρ, �0 = 0 α0,α1,β1 ≥ 0, α1 + β1 < 1, 0 < ρ < λ2+, �t ∼ stdmts(α,λ+,λ−), rt is the risk-free rate, dtis the dividend rate, λt is the market price of risk, g is the characteristic exponent of the laplacetransform for the distribution stdmts(α,λ+,λ−), i.e., g(x; α,λ+,λ−) = log(e(exp(x�t)).the characteristic function of z is given by φz(u) = exp(iuµ + gr(u; α,c,λ+,λ−) + gi(u; α,c,λ+,λ−)) where for u ∈r, gr(u; α,c,λ+,λ−) = 2 −α+3 2 √ πcγ ( 1 − α 2 )[ (λ2+ + u 2) α 2 −λα+ + (λ 2 − + u 2) α 2 −λα− ] , gi(u; α,c,λ+,λ−) = iuc2− α+1 2 γ ( 1 −α 2 )[ λα−1+ f ( 1, 1 −α 2 ; 3 2 , ;− u2 λ2+ ) −λα−1− f ( 1, 1 −α 2 ; 3 2 , ;− u2 λ2− )] where f is the hyper-geometric function. the value of gi for symmetric mts distribution is alwayszero.the m-th cumulant is given by cm(z) = µ if m = 1, cm(z) = 2 m−α+3 2 ( m− 1 2 ) !cγ ( m−α 2 ) (λα−m+ −λ α−m − ) if m = 3, 5, 7, . . . cm(z) = 2 −α+3 2 √ π ( m! m 2 ! ) cγ ( m−α 2 ) (λα−m+ + λ α−m − ) if m = 2, 4, 6, . . . https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 11 the mean, variance, skewness and excess kurtosis are given by e(z) = c1(z) = µ + 2 −α+1 2 cγ ( 1 −α 2 ) (λα−1+ −λ α−1 − ), v (z) = c2(z) = 2 −α+1 2 √ πcγ ( 1 − α 2 ) (λα−2+ + λ α−2 − ), s(z) = c3(z) c2(z) 3/2 = 2 α+9 4 γ ( 3−α 2 ) (λα−3+ −λ α−3 − ) π3/4c1/2(γ( 1−α 2 )(λα−2+ + λ α−2 − )) 3/2 , κ(z) = c4(z) c2(z) 2 = 3 · 2 α+3 2 cγ ( 2 − α 2 ) (λα−4+ + λ α−4 − )√ πc(γ( 1−α 2 )(λα−2+ + λ α−2 − )) 2 . if α ∈ (0, 2)\{1}, the levy measure of α-stable, α-ts and α-mts have the same asymptotic behav-ior at the zero neighborhood. however, the tails of the levy measures for the α-mts distributionare thinner than those of α-stable and heavier than those of α-ts distribution.when z is a ig process, the moment estimators of ρ and θ are given by θ̂n := γȳ ∆δρ̂n , ρ̂n := γ(γs2y − ∆δ) 2ȳwhere ȳ := 1 n n∑ j=1 yj, yj := yj∆ −y(j−1)∆, s2y := 1 n n∑ j=1 (yj − ȳ)2 = 1 n n∑ j=1 y2j − (ȳ) 2. when z is a gamma process, the moment estimators are given by θ̂n := 1 n2 [∑n i=1(yi∆ −y(i−1)∆) ]2 1 n2 ∑n i=1(yi∆ −y(i−1)∆)2 − ∆ n [∑n i=1(yi∆ −y(i−1)∆) ] 2a3(a + 1) b4∆ , ρ̂n := 1 n2 ∑n i=1(yi∆ −y(i−1)∆) 2 − ∆ n [∑n i=1(yi∆ −y(i−1)∆) ] 1 n2 [∑n i=1(yi∆ −y(i−1)∆) ] b3∆ 2a2(a + 1) . for the mts-ou model, the estimating functions are given by c1(y1) = λρ∆c1(z), c2(y1) = ∆c1(z) + 2λρ 2∆c1(z), c3(y1) = ∆c1(z) + 2λρ 2∆c2(z), c4(y1) = ∆c1(z) + 2λρ 2∆c3(z)which give e(y1) = c1(y1) = µ + 2 −α+1 2 cγ ( 1 −α 2 ) (λα−1+ −λ α−1 − ), v (y1) = c2(y1) = 2 −α+1 2 √ πcγ ( 1 − α 2 ) (λα−2+ + λ α−2 − ).this gives the moment estimators for the sou model θ̂n := 1 n2 [∑n i=1(yi∆ −y(i−1)∆) ]2 1 n2 ∑n i=1(yi∆ −y(i−1)∆)2 − ∆ n [∑n i=1(yi∆ −y(i−1)∆) ] https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 12 × [2− α+1 2 cγ ( 1 −α 2 ) (λα−1+ −λ α−1 − )] 2[2− α+1 2 √ πcγ ( 1 − α 2 ) (λα−2+ + λ α−2 − )]2∆ −1. ρ̂n := 1 n2 ∑n i=1(yi∆ −y(i−1)∆) 2 − ∆ n [∑n i=1(yi∆ −y(i−1)∆) ] 1 n2 [∑n i=1(yi∆ −y(i−1)∆) ] × [2− α+1 2 cγ ( 1 −α 2 ) (λα−1+ −λ α−1 − )2 −α+1 2 √ πcγ ( 1 − α 2 ) (λα−2+ + λ α−2 − )] −12−1∆. let ϑ = (ρ,θ) and ϑ̂n = (ρ̂n, θ̂n). by using theorem 2.2 in masuda [34] (see also theorem 4.1 vander vaart [41]), we obtain the strong consistency and asymptotic normality of the mm estimators: proposition 2.1 for fixed ∆ > 0 as n →∞, (a) ϑ̂n → ϑ0 a.s. as n →∞. (b) √ n(ϑ̂n −ϑ0) →d n2(0, (j−1(ϑ0)) as n →∞. where j(ϑ0) is the fisher information. 3. spdes with additive noise consider the parabolic spde duθ(t,x) = θuθ(t,x) + ∂2 ∂x2 uθ(t,x)dt + dz(t,x), t ≥ 0, x ∈ [0, 1] (3.1) u(0,x) = u0(x) ∈ l2([0, 1]), (3.2) uθ(t, 0) = uθ(t, 1), t ∈ [0,t ]. (3.3) here θ ∈ θ ⊆ r is the unknown parameter to be estimated on the basis of the observations ofthe field uθ(t,x),t ≥ 0, x ∈ [0, 1].let s3 and s4 be independent stable random variables, s3 is positive α/2-stable with distri-bution sα/2(σ1, 1, 0) and s4 is symmetric α-stable random variable with distribution sα(σ2, 0, 0), σ1 = c −2/α α/2 , σ2 = c −1/α α , cα = ( ∫∞ 0 x−α sin xdx)−1 = [γ(1 −α) cos(πα/2)]−1. in this case, in the limiting distribution, s3 and s4 are independent stable random variables witha rate faster than the cylindrical brownian motion case.for x ∈ [0, 1], we observe the process {ut,t ≥ 0} at times {t0,t1,t2, . . .}. we assume thatthe sampling instants {ti, i = 0, 1, 2, . . .} are generated by a poisson process on [0,∞), i.e., t0 = 0,ti = ti−1 + αi, i = 1, 2, ... where αi are i.i.d. positive random variables with a commonexponential distribution f (x) = 1 − exp(−λx). note that intensity parameter λ > 0 is theaverage sampling rate which is assumed to be known. it is also assumed that the sampling process https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 13 ti, i = 0, 1, 2, ... is independent of the observation process {xt,t ≥ 0}. we note that the probabilitydensity function of tk+i − tk is independent of k and is given by the gamma density fi (t) = λ(λt) i−1 exp(−λt)it/(i − 1)!, i = 0, 1, 2, .... (3.4) where it = 1 if t ≥ 0 and it = 0 if t < 0.consider the fourier expansion of the process uθ(t,x) = ∞∑ t=1 uθi (t)φi (x) (3.5) corresponding to some orthogonal basis {φi (x)}∞i=1. note that uθi (t), i ≥ 1 are independent onedimensional stable ornstein-uhlenbeck processes duθi (t) = µ θ i u θ i (t)dt + β −α i dzi (t) (3.6) uθi (0) = u θ 0i,recall that µi (θ) = k(θ) −β2mi . thus duθi (t) = (k(θ) −β 2m i )u θ i (t)dt + β −α i dzi (t) (3.7) the random field u(t,x) is observed at discrete times t and discrete positions x. equivalently, thefourier coefficients uθi (t) are observed at discrete time points.define ρ := ρ(λ,θ) = λ λ−κ(θ) + β2m i . the quasi-likelihood estimator is the solution of the estimating equation: g∗n(θ) = 0 (3.8) where g∗n(θ) = β2αi λ(ρ(λ,θ)) 2 ρ(λ, 2θ) n∑ i=1 uti−1 ( (uti−1θρ(λ,θ)) 2 + λ )−1 (uti −ρ(λ,θ)uti−1 ). (3.9) we call the solution of the estimating equation the quasi-likelihood estimator. there is no explicitsolution for this equation.the optimal estimating function for estimation of the unknown parameter θ is gn(θ) = β 2α i n∑ i=1 uti−1 [uti −ρ(λ,θ)uti−1 ]. (3.10) the martingale estimation function (mef) estimator of ρ is the solution of gn(θ) = 0 (3.11)and is given by ρ̂n := ∑n i=1 uti−1uti∑n i=1 u 2 ti−1 . https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 14 we do the parameter estimation in two steps: the rate λ of the poisson process can be estimatedgiven the arrival times ti , therefore it is done at a first step. since we observe total number ofarrivals n of the poisson process over the t intervals of length one, the mle of λ is given by λ̂n := n t . theorem 3.1 we have λ̂n → λ a.s. as n →∞. √ n(λ̂n −λ) →d n(0, eλ(1 −e−λ)) as n →∞. proof. let vi be the number of arrivals in the interval (i − 1, i]. then vi, i = 1, 2, . . . ,n are i.i.d.poisson distributed with parameter λ. since φ is continuous, we have i{0}(vi ) = i{0}(u(ti )) a.s. i = 1, 2, . . . ,n. note that 1 n n∑ i=1 i{0}(uti ) → a.s. e(i{0}v1) = p (v1 = 0) = e −λ as n →∞. lln and clt and delta method applied to the sequence i{0}(uti ), i = 1, 2, . . . ,n give the results. the clt result above allows us to construct confidence interval for the jump rate λ. corollary 3.1 a 100(1 −α)% confidence interval for λ is given by[ n t −z1−α 2 √ 1 n − 1 t , n t + z1−α 2 √ 1 n − 1 t ] where z1−α 2 is the (1 − α 2 )-quantile of the standard normal distribution. we obtain the strong consistency and asymptotic normality of the mef estimator. theorem 3.2 when α = 2, we have ρ̂n →a.s. ρ as n →∞ √ n(ρ̂n −ρ) →d n(0, λ−i (1 −e−ρ)) as n →∞. proof: by using the fact that every stationary mixing process is ergodic, it is easy to show that if utis a stationary ergodic o-u markov process and ti is a process with nonnegative i.i.d. incrementswhich is independent of ut, then {uti, i ≥ 1} is a stationary ergodic markov process. hence {uti, i ≥ 1} is a stationary ergodic markov process. https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 15 observe that uθi (t) := vi is a stationary ergodic markov chain and vi ∼ n(0,σ2) where σ2 isthe variance of u0. thus by slln for zero mean square integrable martingales, we have 1 n n∑ i=1 uti−1uti → a.s. e(ut0ut1 ) = ρe(u 2 t0 ) 1 n n∑ i=1 u2ti−1 → a.s. e(u2t0 ) thus ∑n i=1 uti−1uti∑n i=1 u 2 ti−1 →a.s. ρ. further, √ n(ρ̂n −ρ) = n−1/2 ∑n i=1 uti−1 (uti −θuti−1 ) n−1 ∑n i=1 u 2 ti−1 . since e(ut1ut2|ut1 ) = θu 2 t1it follows by lemma 3.1 in bibby and srensen [2] n−1/2 n∑ i=1 uti−1 (uti −θuti−1 ) converges in distribution to normal distribution with mean zero and variance equal to e[(ut1ut2 ) −e(ut1ut2|ut1 )] 2 = 1 −e2(θ−βiδ){2(βi −θ)(βi + 1)}−1. applying delta method the result follows. in the next step, we use the estimator of λ to estimate θ.note that 1 ρ̂n = ∑n i=1 u 2 ti−1∑n i=1 uti−1uti . thus 1 + β2m1 −κ(θ) λ = ∑n i=1 u 2 ti−1∑n i=1 uti−1utiwhich gives β2m1 −κ(θ) λ = ∑n i=1 u 2 ti−1∑n i=1 uti−1uti − 1 = − ∑n i=1 uti−1 [uti −uti−1 ]∑n i=1 uti−1utinow replace λ by its estimator mle λ̂n. β2m1 −κ(θ) = − ∑n i=1 uti−1 [uti −uti−1 ] t n ∑n i=1 uti−1utithus θ̂n = κ −1 ( β2mi + ∑n i=1 uti−1 [uti −uti−1 ] t n ∑n i=1 uti−1uti ) . https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 16 since the function κ−1(·) is a continuous function, by application of delta method, the followingresult is a consequence of theorem 3.2. theorem 3.3 when α = 2, a) θ̂n → θ a.s. as n →∞ b) √ n(θ̂n −θ) →d n(0, (κ′(θ))−2λ2(1 −e−2λ −1(κ(θ)−β2m1 ))) as n →∞. in the second stage, we substitute λ by its estimator λ̂n. theorem 3.4 when 0 < α < 2, a) θ̂n →a.s. θ as n →∞ b) n(α−1)/α 2 (θ̂n −θ) →d ( κ′(θ))−2λ2(1 −e−2λ −1(κ(θ)−β2m1 ) )1/α s4 s3 as n →∞. where s4 and s3 are independent stable random variables. in the second stage, we substitute λ by its estimator λ̂n. the limit distribution is normal onlyin the gaussian case α = 2. 4. spdes with linear multiplicative noise consider the spde with multiplicative noise: duθ(t,x) = (a0 + θa1)u θ(t,x)dt + muθ(t,x)dz(t,x), t ≥ 0, x ∈ [0, 1] (4.1) where m is a known linear operator.equation (4.1) is called diagonalizable if a0,a1 and m have point spectrum and a commonsystem of eigenfunction {hj, j ≥ 1}. denote by ρk,νk and µk, the eigenvalues of the operators a0,a1 and m respectively.then uθ(t,x) = ∑ j≥1 uj,thj. the fourier coefficients have the dynamics duk(t) = (θνk + ρk)uk(t)dt + σkuk(t)dzk(t), k ≥ 1 which is the stable black-scholes model whose solution is geometric stable process.let θνk + ρk =: µk(θ), ṽk,t := ln(uk,t/uk,0).conditional characteristic function (ccf) estimator is given by µ̂k(θ) = ṽk,t t 2(α−1)/α 2 + σ2k b1t −((α−1)2+1)/α2 . https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 17 since µk(θ) is strictly monotone function of θ, by invariance principle of ccfe, under invertibletransformations, we can find the ccfe of the parameter θ θ̂k,t = ṽk,t νkt 2(α−1)/α2 + σ2k b1νkt −((α−1)2+1)/α2 − ρk νkwhich can be represented as θ̂k,t = θ0 + σkmt νkt 2(α−1)/α2where mt is a square-integrable martingale. due to the lln for martingales, we have strongconsistency.note that in the standard black-scholes case where α = 2, σk = σ, the mle of the driftcoefficient of the geometric brownian motion is given by θ̂t = ln(ut/u0) t + σ2 2 = θ0 + σ wt t . due to the law of iterated logarithm for brownian motion, the mle is strongly consistent as t →∞. theorem 4.1 when 0 < α < 2, a) θ̂k,t is an unbiased estimator of θ.b) θ̂k,t → θ a.s. as t →∞.c) t (α−1)/α 2 (θ̂k,t −θ) →d ( σ2k ν2 k )1/α s4 s3 as t →∞ where s4 and s3 are independent stable random variables.d) if in addition, lim k→∞ ∣∣∣∣σkνk ∣∣∣∣ = 0,then for every fixed t > 0, θ̂k,t → θ a.s. as k →∞and ∣∣∣∣νkσk ∣∣∣∣ (θ̂k,t −θ) →d (t (α−1)/α2)1/α s4s3 as k →∞. remark: the parabolicity condition and the mle consistency condition in general are notconnected. in terms of operator’s order, parabolicity states that the order of operator m from thediffusion term is smaller than half of the order of operators a0 and a1 from deterministic part.the consistency condition assumes that the order of the operator m from the diffusion part doesnot exceed the order of the operator a1 from the deterministic part that contains the parameter ofinterest θ. https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 18 5. spdes with nonlinear multiplicative noise consider the spde with multiplicative noise: duθ(t,x) = (a0 + θa1)u θ(t,x)dt + muθ(t,x)dz(t,x), t ≥ 0, x ∈ [0, 1] (5.1) where m is a known nonlinear operator.equation (5.1) is called diagonalizable if a0,a1 and m have point spectrum and a commonsystem of eigenfunction {hj, j ≥ 1}. denote by ρk,νk and µk, the eigenvalues of the operators a0,a1 and m respectively.then uθ(t,x) = ∞∑ j=1 uj,thj. we consider stable cir model as example. here s1 and s2 are dependent stable randomvariables unlike the linear case where s3 and s4 are independent stable random variables. the existence and pathwise uniqueness of solutions to the sdes with non-lipschitz coefficientdriven by spectrally positive levy processes were studied in fu and li [17].consider the nonlinear spde dx(t,x) = θ 2 xxx (t,x)dt + √ x(t,x)dw (t,x) where w (t,x) is a space-time white noise. konno and shiga [26] studied the existence and weakuniqueness of the above equation as a martingale problem for the associated super-brownian mo-tion. the pathwise uniqueness of nonnegative solution still remains open. the main difficultycomes from the unbounded drift coefficient and non-lipschitz diffusion coefficient. wang et al. [42]studied proved a comparison theorem and showed that the solution of the nonlinear spde is distri-bution function valued. they also established pathwise uniqueness. as application they obtainedwell-posedness of martingale problems for two classes of measure-valued diffusions: interactingsuper-brownian motions and interacting fleming-viot processes. he et al. [18] obtained pathwiseunique solution to nonlinear spde with super levy process, which is a combination of space-time gaussian white noises and poisson random measures which is a generalization of work ofxiong [43] where the result for a super-brownian motion with binary branching mechanism wasobtained. using an extended yamada-watanabe argument, xiong [43] established strong exis-tence and uniqueness of the solution to the spde. super-brownian motion (smb), also calledthe dawson-watanabe process introduced by sawson and watanabe is a measure valued processarising as the limit of empirical measure process of a branching particle system. sbm satisfiesa martingale problem. when the state space is r, sbm has a density w.r.t. lebesgue measureand this density valued process x(t,x) satisfies the above spde. when the space r is s single https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 19 point, the spde becomes an sde which is cir diffusion dxt = √xtdwt whose uniqueness isestablished using the yamada-watanabe argument. xiong and yang (2019) studied existence andpathwise uniqueness to an spde with hölder continuous coefficient driven by α-stable colorednoise. the existence of the solution is shown by considering the weak limit of a sequence of sdesystem which is obtained by replacing the laplacian operator in the spde by its discrete version.the pathwise uniqueness is shown by using a backward doubly stochastic differential equation totake care of the laplacian. in the case of d = 1, the pathwise uniqueness of a nonnegative solutionto the corresponding equation was established by yang and zhou [45] for 1 < α < √5 − 1 andpathwise uniqueness for √5 − 1 < α < 2 is still open.consider spde model with multiplicative noise and mean reversion, where the j-th fouriercoefficient is the stable cox-ingersoll-ross (scir) model: duj,t = (a−θuj,t)dt + σu 1/α j,t−dzj,t, j ≥ 1 (5.2) where a is the mean reverting level and θ is mean reverting speed. recall that for α = 2, for every j ≥ 1, the process zj,t is a standard brownian motion, this is the famous cox-ingersoll-ross (cir)model used for modeling interest rate, which is also used a stochastic volatility process in hestonmodel. note that there are brownian cir models with additive compound poisson type jumps.when 1 < α < 2, zj,t is stable process with levy measure να(dz) = 1{z>0}dz αγ(−α)zα+1 . (5.3) the discontinuous scir model captures the heavy tailed property in the sense of infinite variance.there is empirical evidence from high frequency data available in support of application of purejump models in financial modeling.the scir model has the unique stationary distribution µ with laplace transform given by lµ(λ) = ∫ ∞ 0 e−λxµ(dx) = exp { − ∫ λ 0 αa αθ + σαzα−1 dz } , λ ≥ 0. (5.4) applying itô’s formula, for t ≥ r ≥ 0, we obtain uj,t = e −θ(t−r)uj,r + a ∫ t r e−θ(t−s)ds + σ ∫ t r e−θ(t−s)u 1/α j,s−dzj,s, j ≥ 1. (5.5) let the process be observed at {kh,k = 0, 1, . . . ,n} from a single realization {uj,t,t ≥ 0} for fixed h. for simplicity, we take h = 1. this equation can be considered as a first order autoregressive(ar(1)) equation uj,k = ρ + γuj,k−1 + �j,k, j ≥ 1 (5.6) where γ = e−θ, ρ = aθ−1(1 −γ) and �j,k = σ ∫ k k−1 e−θ(k−s)u 1/α j,s−dzj,s, k ≥ 1, j ≥ 1. (5.7) https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 20 for b ∈b(r+), let s2,j,n(b) = n∑ k=1 uj,k−1�kib(|uj,k−1�j,k|), s1,j,n(b) = n∑ k=1 u2j,k−1ib(uj,k−1), j ≥ 1. (5.8) it is easy to see that �j,k = uj,k −e(uj,k|fk−1), k ≥ 1, j ≥ 1. (5.9)is a sequence of martingale differences for every fixed j.let s1,j,n := s1,j,n(0,∞), s2,j,n := s2,j,n(0,∞) and recall that γ = e−θ.then θ̂j,n −θ = s2,j,n s1,j,n (5.10) where θ̂n is the conditional least squares estimator (clse) which minimizes n∑ k=1 �2j,k = n∑ k=1 [uj,k −e(uj,k|fk−1)]2 = n∑ k=1 [uj,k −ρ−γuj,k−1]2 (5.11) and are given by γ̂j,n = ∑n k=1 uj,k−1 ∑n k=1 uj,k −n ∑n k=1 uj,k−1uj,k ( ∑n k=1 uj,k−1) 2 −n ∑n k=1 u 2 j,k−1 , ρ̂j,n = 1 n n∑ k=1 uj,k − γ̂n 1 n n∑ k=1 uj,k−1, θ̂j,n = − log γ̂j,n, âj,n = ρ̂nθ̂n 1 − γ̂n . let (s1,s2) have the characteristic function given by e[exp{iλ1s1 + iλ2s2}] := exp { − σα θ2γ(−α) ∫ ∞ 0 e ( 1 − exp{iλ1y2 + iλ2y(α+1)/αvj,1} ) × e ( exp { ie−2θλ1y 2 1 −e−2θ + ie−θ(α+1)/αλ2y (α+1)/αvj,2 (1 −eθ(α+1))1/α }) dy yα+1 } (5.12) and vj,k := σ ∫ k k−1 e−θ(k−s)e−θ(s−k+1)/αdzj,s, k = 1, 2, j ≥ 1 (5.13) which are i.i.d. with the same distribution as σ ( e−θ − 1 (α− 1)θ )1/α zj,1 which is regularly varying with index α. the limit distribution is normal only in the gaussian case α = 2. following li and ma [31] it can be shown that for every fixed j, if we have 1 < α < (1 +√5)/2,then we have as n →∞ (d−2n s1,j,n,c −1 n s2,j,n) d→(s1,s2) on r2 https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 21 where dn = n1/α and cn = n(α+1)/α2 = d(α+1)/αn .for the stable spde model, we have the following result on the consistency and the limitdistribution of the clse: theorem 5.1 if we have 1 < α < (1 + √5)/2, then for every fixed j ≥ 1a) θ̂j,n →p θ as n →∞.b) n(α−1)/α 2 (θ̂j,n −θ) →d ( σ2 ν2 j )1/α s2 s1 as n →∞. c) if in addition, limj→∞ ∣∣νj∣∣ = ∞, then for every fixed n ≥ 1, θ̂j,t →p θ as j →∞ and ∣∣νj∣∣ (θ̂j,n −θ) →d σ(n−(α−1)/α2)1/α s2 s1 as j →∞. where s2 and s1 are defined in (5.12). remarks1) the limit distribution in the case (1 + √5)/2 < α < 2 is still open.2) the process (xj) is exponentially ergodic and hence strongly mixing.3) for the gaussian case (α = 2), the limit results are based on ergodic theory and martingaleconvergence theorem. for the non-gaussian case (1 < α < 2), limit results are obtained by thetheory of regular variation and convergence of point processes.4) let 0 < α < 2 and let zt be a one dimensional α-stable process with levy measure ν(dz).then as n →∞, np (n−1/αzt ∈ ·) →v tν(·). 6. examples (a) consider the linear stochastic heat equation with additive noise du(t,x) = θuxx (t,x)dt + dz(t,x) for 0 ≤ t ≤ t and x ∈ (0, 1) and θ > 0 with periodic boundary conditions.here 2m = m1 = 2 and µj = −θπ2j2,γ > 1/2. the eigenfunctions are hj(x1, . . . ,xn) = ( √ 2/π)d(sin(n1x1), . . . , sin(ndxd)), x = (x1, . . . ,xn) ∈ rd, j = (n1, . . . ,nd) ∈ nd. the corre-sponding eigenvalues are −νj where νj = (n21 + . . . + n2d).as n →∞,h → 0, nh1+α/ log n → 0, nh2α−1 log n →∞, nh2−α/2+ρ →∞ for some ρ > 0,( n log n )1/α h1/α(θ̂n −θ0) →d 2θ0(αθ0)−1/α s4 s3 https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 22 where s3 and s4 are independent stable random variables, s3 is positive α/2-stable with distri-bution sα/2(σ1, 1, 0) and s4 is symmetric α-stable random variable with distribution sα(σ2, 0, 0), σ1 = c −2/α α/2 , σ2 = c −1/α α , cα = ( ∫∞ 0 x−α sin xdx)−1 = [γ(1 −α) cos(πα/2)]−1. observe the rate of convergence (nh)1/α(log n)−1/α = (t )1/α(log n)−1/α. for α = 2, this rate is t 1/2(log n)−1/2. (b)consider the linear stochastic heat equation with multiplicative noise du(t,x) = θuxx (t,x)dt + u(t,x)dz(t,x) for 0 ≤ t ≤ t and x ∈ (0, 1) and θ > 0 with zero boundary conditions and nonzero initial value u(0) ∈ l2(0, 1). here a1 is the laplace operator on (0, 1) with zero boundary conditions that hasthe eigenfunctions hk(x) = √2/π sin(kx), k > 0 and the eigenvalues νk = −k2,ρk = 0, σk = 1,k > 0. uk(t) = ∫ 1 0 hk(x)u(t,x)dx, duk(t) = (θνk + ρk)uk(t)dt + σkuk(t)dzk(t).recall that ṽk,t := ln(uk,t/uk,0).the ccfe has the form θ̂k,t = ṽk,t − 1 k2 . (c) consider the following spde du(t,x) = [∆u(t,x) + θu(t,x)]dt + (1 − ∆)ru(t,x)dz(t,x). in this case a0 = ∆,a1 = i,m = (1 − ∆)r with the eigenvalues νk = 1,ρk = σk,µk = (1 + σk)r .it has a unique solution for any r ≤ 1/2.the ccfe has the form̂ θk,t = ṽk,t k2t 2(α−1)/α 2 − (1 −σk)2r k2t−((α−1) 2+1)/α2 − 1 σk(d) stable cox-ingersoll-ross model xiong and yang [44] studied existence and strong uniqueness of the following spde: duk(t) = (θνk + ρk)uk(t)dt + σk(uk(t)) 1/αdzk(t), k ≥ 1.the existence of the solution in the case of space-time white noise is shown by consideringthe weak limit of a sequence of sde systems which is obtained by replacing the laplacianoperator in the spde by its discrete version. the weak uniqueness follows from the uniqueness https://doi.org/10.28924/ada/ma.3.4 eur. j. math. anal. 10.28924/ada/ma.3.4 23 of solution to the martingale problem for the associated super-brownian motion. in the case of α-stable noise the existence and pathwise uniqueness of the solution is studied in xiong and yang [44]. concluding remark we considered levy process driving term in this paper. using fractionallevy process as the driving term, maximum quasi-likelihood estimation in fractional levy stochasticvolatility model was studied in bishwal [8]. recently, sub-fractional brownian (sub-fbm) motionwhich is a centered gaussian process with covariance function ch(s,t) = s 2h + t2h − 1 2 [ (s + t)2h + |s − t|2h ] , s,t > 0 for 0 < h < 1 introduced by bojdecki, gorostiza and talarczyk [13] has received some attentionrecently in finite dimensional models. the interesting feature of this process is that this processhas some of the main properties of fbm, but the increments of the process are nonstationary,more weakly correlated on non-overlapping time intervals than that of fbm, and its covariancedecays polynomially at a higher rate as the distance between the intervals tends to infinity. itwould be interesting to see extension of this paper to sub-fbm case. we generalize sub-fbm tosub-fractional levy process (sub-flp).sub-fractional levy process (sflp) is defined as sh,t = 1 γ(h + 1 2 ) ∫ r [(t − s)h−1/2+ − (−s) h−1/2 + ]dms, t ∈r where mt,t ∈ r is a levy process on r with e(m1) = 0, e(m21 ) < ∞ and without browniancomponent. sflp has the following properties:1) the covariance of the process is given by cov(sh,t,sh,s) = s 2h + t2h + e[l(1)2] 2γ(2h + 1) sin(πh) [|t|2h + |s|2h −|t − s|2h]. 2) sh is not a martingale. for a large class of levy processes, sh is neither a semimartingalenor a markov process. 3) sh is hölder continuous of any order β less than h − 12 . 4) sh hasnonstationary increments. 5) sh is symmetric. 6) sh is self similar. 7) sh has infinite totalvariation on compacts.it would be interesting to investigate qml estimation in spde driven by subfractional levyprocesses which incorporate both jumps and long memory apart from nonstationarity. references [1] d. applebaum, levy processes and stochastic calculus, second edition, cambridge 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sciences, national institute of technology karnataka, india-575 025 sgeorge@nitk.edu.in 4department of computer science, university of oklahoma, norman, 73019, ok, usa michael.i.argyros-1@ou.edu ∗correspondence: iargyros@cameron.edu abstract. developments are presented for the semi-local convergence of newton’s method to solvebanach space-valued nonlinear equations. by utilizing a new methodology, we provide a finer con-vergence analysis with no additional conditions than in earlier results. in particular, this is done byintroducing the center-lipschitz condition by which we construct a stricter domain than the originaldomain of the operator. then, the lipschitz constants in the new domain are at least as small asthe original constants leading to weaker sufficient convergence criteria, tighter error bounds on theerror distances involved, and a piece of better information on the location of the solution. thesebenefits are obtained under the same computational cost since in practice the computation of theoriginal constants requires the computation of the new constants as special cases. the same benefitsare obtained if the lipschitz conditions are replaced by hölder conditions or even more general ω−continuity conditions. this methodology can be applied to other methods using such as the secant,stirling’s newton-like, and other methods along the same lines. numerical examples indicate thatthe new results can be utilized to solve nonlinear equations, but not earlier ones. 1. introduction consider the problem of finding a solution x∗ ∈ ω of the equation f (x) = 0, (1.1) received: 26 jan 2023. key words and phrases. newton-kantorovich method; convergence; banach space.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 2 where f : ω −→ e2 is a continuously differentiable operator in the fréchet-sense, e1,e2 arebanach spaces and ω ⊂e1 is an open set.the solution x∗ in closed form is desirable. but this is possible only in special cases. so,most solution methods for (1.1) are iterative methods. the convergence regions for these methodsare small in general, so their applicability is reduced. the error bounds are also pessimistic (ingeneral).among the iterative methods, the most famous one is newton’s method (nm) defined for n = 0, 1, 2, . . . by xn+1 = xn −f ′(xn)−1f (xn) (1.2)kantorovich provided the semi-local convergence analysis of nm utilizing the contraction map-ping principle attributed to banach. in particular, he presented two different proofs using majorantfunctions or recurrence relations [15]. his so-called newton-kantorovich theorem is that no as-sumption on the solution is made and at the same time, the existence of the solution x∗ is established.numerous researchers used this theorem in applications and also as a theoretical tool [1–16]. butthe convergence criteria may not hold although nm may converge. motivated by these concernsand optimization considerations we present new results that not only extend the convergence regionbut also provide more precise error estimates and better knowledge of the location of the solution.the novelty of the article is that these benefits require no additional conditions. this is how theusage of nm is extended. the technique used can be applied to extend other iterative methodsalong the same lines. 2. convergence analysis let α > 0, λ ≥ 0 and x0 ∈ ω be such that ‖f ′(x0)−1‖ ≤ α, ‖f ′(x0)−1f (x0)‖ ≤ λ and f ′(x0) −1 ∈ l(e2,e1), the space of bounded linear operators from e2 to e1. by b(x,b),b[x,b] wedenote the open and closed balls in e1, respectively with center x ∈e1 and of radius b > 0.some lipschitz-type conditions are needed. definition 2.1. operator f ′ is center-lipschitz continuous about x0 on ω if there exists l0 > 0 such that for all u ∈ ω ‖f ′(u) −f ′(x0)‖≤ l0‖u −x0‖. (2.1) set ω0 = b(x0, 1 αl0 ) ∩ ω. (2.2) definition 2.2. operator f ′ is 1−restricted lipschitz continuous on ω0 if there exists l > 0 such that ‖f ′(u) −f ′(v)‖≤ l‖u −v‖ (2.3) for all u ∈ ω0, v = u −f ′(u)−1f (u) ∈ ω0. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 3 definition 2.3. operator f ′ is 2−restricted lipschitz continuous on ω0 if there exists l1 > 0 such that for all u,v ∈ ω0 ‖f ′(u) −f ′(v)‖≤ l1‖u −v‖. (2.4) definition 2.4. operator f ′ is lipschitz continuous on ω if there exists l2 > 0 such that for all u,v ∈ ω ‖f ′(u) −f ′(v)‖≤ l2‖u −v‖. (2.5) definition 2.5. assume: λαl0 < 1 (2.6) and ω1 = b(x1, 1 αl0 −‖x1 −x0‖) ⊂ ω (2.7) then, operator is 3− restricted lipschitz continuous on ω1 if there exists a constant k > 0 such that for all u ∈ ω1 ‖f ′(u) −f ′(v)‖≤ k‖u −v‖ (2.8) for v = u −f ′(u)−1f (u) ∈ ω1. remark 2.6. by the definition of sets ω0 and ω1, we get ω0 ⊆ ω, (2.9) and ω1 ⊆ ω0. (2.10) indeed, if y ∈ ω1, then we obtain ‖y −x1‖ ≤ 1 αl0 −‖x1 −x0‖⇒‖y −x1‖ + ‖x1 −x0‖≤ 1 αl0 ⇒ ‖y −x0‖≤ 1 αl0 ⇒ y ∈ ω0 ⇒ ω1 ⊆ ω0. it follows by these definitions, (2.9) and (2.10) that if the best constants are chosen in the definitions 2.1-2.5, then l ≤ l1 ≤ l2, (2.11) l0 ≤ l2, (2.12) and k ≤ l. (2.13) hence, parameter k can replace results on newton’s using the constants l,l1 and l2. notice also that l0 = l0(f ′, ω), l = l(f ′, ω0),l1 = l1(f ′, ω0), l2 = l2(f ′, ω) and k = k(f, ω0, ω1). examples, where (2.9)-(2.13) are strict can be found in the numerical section. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 4 notice that the computation of the constant l2 requires the computation of the other constants as special cases. hence, no additional effort is needed to compute them. moreover, they all depend on the initial data (x0,f, ω). it is also worth noticing that under (2.1) we obtain ‖f ′(u)−1‖≤ α 1 −αl0‖u −x0‖ . (2.14) this is a tighter estimate than using the stronger (2.5) to get ‖f ′(u)−1‖≤ α 1 −αl2‖u −x0‖ . (2.15) we assume from now on that l0 ≤ k. (2.16) but if k < l0 then, the following results hold with l0 replacing k. based on the above we present two extended theorems on newton’s method. an important role is played in the convergence of nm by the majorizing sequence {sn} definedby s0 = 0,sn+1 − sn = − p(sn) p′0(sn) = αk(sn − sn−1)2 1 −l0αsn , p(s) = k 2 s2 − s α + λ α , p0(s) = l0 2 s2 − s α λ α . theorem 2.7. (extended newton-kantorovich theorem [1, 2, 10, 12, 13, 15, 16]) under conditions (2.1), (2.6)-(2.8) further suppose b(x0,s∗) ⊂ ω, h = kαλ ≤ 1 2 . (2.17) then, newton’s method (1.2) initiated at x0 ∈ ω generates a sequence {xn} such that:{xn} ⊆ b(x0,s∗), limn−→∞xn = x∗ ∈ b[x0,s∗]. ‖xn+1 −xn‖≤ sn+1 − sn (2.18) ‖x∗ −xn‖≤ s∗ − sn, (2.19) where, limn−→∞ sn = s∗ = 1− √ 1−2h kα and s∗∗ = 1+ √ 1−2h kα . moreover, the following items hold for τ = s∗ s∗∗ s∗ − sn = { (s∗∗−s∗)τ2 n 1−τ2n , if s∗ < s∗∗ 1 2n s∗, if s∗ = s∗∗. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 5 furthermore, the element x∗ is the unique solution of equation f (x) = 0 in b[x0, s̄], where s̄ = 2 l0α − s∗ if l0αs∗ < 2. proof. simply replace l2 by k and use (2.14) instead of (2.15) in the proof of the version ofnewton-kantorovich theorem given in [10] (see also [3–9, 14–16]. � remark 2.8. (i)if k = l2, the result of theorem 2.7 reduces to one in the newton-kantorovich theorem where hk = l2αλ ≤ 1 2 , (2.20) t0 = 0,tn+1 − tn = − p̄(tn) p̄′(tn) = αl2(tn − tn−1)2 1 −l2αtn , p̄(s) = l2 2 s2 − s α + λ α , and limn−→∞ tn = t∗ = 1− √ 1−2kk l2α and t∗∗ = 1+ √ 1−2kk l2α , ¯̄s = 2 l2α − t∗, µ = t∗t∗∗ , t∗ − tn =  (t∗∗−t∗)µ2 n 1−µ2n , if t∗ < t∗∗ 1 2n t∗, if t∗ = t∗∗. then, in view of estimates (2.11)-(2.13) we have hk ≤ 1 2 ⇒ h ≤ 1 2 , (2.21) s∗ ≤ t∗, ¯̄s ≤ s̄, (2.22) 0 ≤ sn+1 − sn ≤ tn+1 − tn (2.23) and 0 ≤ s∗ − sn ≤ t∗ − tn. (2.24) estimates (2.21)-(2.24) justify the advantages (a) as stated in the introduction. (ii)a more careful look at the proof shows that tighter sequence {rn} defined by r0 = 0, r1 = λ, r2 = r1 + αl0(r1 − r0)2 2(1 −l0αr1) , rn+2 = rn+1 + kα(rn+1 − rn)2 2(1 −l0αrn+1) , also majorizes sequence {xn}. the sufficient convergence criterion for this sequence is given by ha = k̄αλ ≤ 1 2 , (2.25) https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 6 where k̄ = 1 8 (4l0 + √ kl0 + 8l 2 0 + √ l0k). this criterion was given by us in [4] for k = l− 2. notice that h ≤ 1 2 ⇒ ha ≤ 1 2 . (2.26) hence, if (2.25) and {rn} replace (2.17) and {sn} the conclusions of theorem 2.7 hold with these changes too. (iii)suppose that there exist a > 0,b > 0 such that ‖f ′(x0 + θ(x1 −x0)) −f ′(x0)‖≤ τa‖x1 −x0‖ (2.27) and ‖f ′(x1) −f ′(x0)‖≤ b‖x1 −x0‖ (2.28) for all τ ∈ [0, 1]. then, it was shown in [5] that sequence {qn} defined by q0 = 0,q1 = λ, q2 = q1 + αa(q1 −q0)2 2(1 −bαq1) , qn+2 = qn+1 + kα(qn+1 −qn)2 2(1 −l0αqn+1) is also majorizing for sequence {xn}. the convergence criterion for sequence {qn} is given by haa = λ 2c ≤ 1 2 , (2.29) where p1(s) = (ka + 2dl0(a− 2b))s2 + 4p(l0 + b)s − 4d, d = 2k k + √ k2 + 8l0k , and c =  1 l0+b , ka + 2dl0(a− 2b) = 0positive root of p1, ka + 2dl0(a− 2b) > 0smaller positive root of p1, ka + 2dl0(a− 2b) < 0. notice that b ≤ a ≤ l0. hence, {qn} is a tighter majorizing sequence than {rn}. criterion (2.29) was given by us in [4] for k = l2. therefore (2.29) and {qn} can also replace (2.17) and {sn} in theorem 2.7. (iv) it follows from the definition of sequence {sn} that if l0αsn < 1. (2.30) then, sequence {sn} is such that 0 ≤ sn ≤ sn+1 and limn−→∞ sn = s∗ ≤ 1l0α. hence, weaker than all conditions (2.30) can be used in theorem 2.7. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 7 3. examples we test the convergence criteria. example 3.1. defined the real function f on ω = b[x0, 1 −δ], x0 = 1, δ ∈ (0, 12) by f (s) = s3 −δ. then, the definitions are satisfied for λ = 1−δ 3 , α = 1 3 , l0 = 3(3 − δ),l2 = 6(2 − δ), l1 = 6(1 + 1 3−δ ),x1 = 2+δ 3 , l = 5(4−δ 3−δ) 3+δ 3(4−δ 3−δ) 2 , a = b = δ + 5, k = 5h 3+δ 3h2 , and h = δ+2 3 + 3−(1−δ)(3−δ) 3(1−δ) . denote by m1,m2,m3,m4 the set of values δ ∈ (0, 12) for which (2.20), (2.17), (2.25) and (2.29) are satisfied, respectively. then, by solving these inequalities for δ, we get m1 = ∅, m2 = (0.0751, 0.5), m3 = (0.1320, 0.5) and m4 = (0.3967, 0.5). notice in particular that the newton-kantorovich criterion (2.20) [1, 9–15] cannot assure convergence of nm since m1 = ∅. a second example is provided to show that our conditions can be used to solve equations incases where the ones in [1, 2, 10, 12, 13] cannot. example 3.2. consider e1 = e2 = c[0, 1] with the norm-max. set ω = b(x0, 3). define, hammerstein-type integral operator m on ω by m(z)(w) = z(w) −y(w) − ∫ 1 0 t (w,t)v3(t)dt, (3.1) w ∈ [0, 1], z ∈ c[0, 1], where y ∈ c[0, 1] is fixed and t is a green’s kernel defined by t (w,u) = { (1 −w)u, if u ≤ w w(1 −u), if w ≤ u. (3.2) then, the derivative m′ according to fréchet is defined by [m′(v)(z)](w) = z(w) − 3 ∫ 1 0 t (w,u)v2(t)z(t)dt, (3.3) w ∈ [0, 1], z ∈ c[0, 1]. let y(w) = x0(w) = 1. then, using (3.1)-(3.3), we obtain m′(x0)−1 ∈ l(e2,e1), ‖i − m′(x0)‖ < 38, ‖m ′(x0) −1‖ ≤ 8 5 := α, λ = 1 5 , l0 = 12 5 , l2 = 18 5 , and ω0 = b(1, 3) ∩ b(1, 5 12 ) = b(1, 5 12 ), so l1 = 32, and l0 < l2, l1 < l2. set k = l = l1. then, the old sufficient convergence criterion is not satisfied, since αλl2 = 15 8 5 18 5 = 144 125 > 1 2 holds. therefore, there is no guarantee that newton’s method (1.2) converges to x∗ under the conditions of the aforementioned references. but our condition hold, since dba = 1 5 8 5 3 2 = 24 50 < 1 2 . therefore, the conclusions of our theorem 2.7 follow. https://doi.org/10.28924/ada/ma.3.15 eur. j. math. anal. 10.28924/ada/ma.3.15 8 4. conclusion the technique of recurrent functions has been utilized to extend the sufficient conditions forconvergence of nm for solving nonlinear equations. the new results are finer than the earlierones. so, they can replace them. no additional conditions have been used. the technique is verygeneral rendering useful to extend the usage of other iterative methods. declarations the authors declare that there are no competing interests and that all authors contributedequally in conceptualization, methodology, formal analysis, and investigation. the original draftwas prepared by i. k. argyros and review and editing was done by s. regmi, s. george, and m.argyros. references [1] s. adly, h.v. ngai, v.v. nguyen, newton’s methods for solving generalized equations: kantorovich’s and smale’sapproaches, j. math. anal. appl. 439 (2016) 396-418. https://doi.org/10.1016/j.jmaa.2016.02.047.[2] s. adly, r. cibulka, h.v. ngai, newton’s method for solving inclusions using set-valued approximations, siam j.optim. 25 (2015) 159-184. https://doi.org/10.1137/130926730.[3] i.k. argyros, unified convergence criteria for iterative banach space valued methods with applications, mathematics,9 (2021) 1942. https://doi.org/10.3390/math9161942.[4] i.k. argyros, s. hilout, weaker conditions for the convergence of newton’s method, j. complex. 28 (2012) 364-387. https://doi.org/10.1016/j.jco.2011.12.003.[5] i.k. argyros, s. hilout, on an improved convergence analysis of newton’s method, appl. math. comp. 225 (2013)372-386; https://doi.org/10.1016/j.amc.2013.09.049.[6] i.k. argyros, a.a. magréñan, a contemporary study of iterative procedures, elsevier (academic press), new york,2018. https://doi.org/10.1016/c2015-0-04301-5.[7] i.k. argyros, s. george, mathematical modeling for the solution of equations and systems of equations with appli-cations, volume-iv, nova publisher, ny, 2021.[8] r. behl, p. maroju, e. martinez, s. singh, a study of the local convergence of a fifth order iterative procedure,indian j. pure appl. math. 51 (2020) 439-455. https://doi.org/10.1007/s13226-020-0409-5.[9] p.g. ciarlet, c. madare, on the newton-kantorovich theorem, anal. appl. 10 (2012) 249-269. https://doi.org/ 10.1142/s0219530512500121.[10] r. cibulka, a.l. dontchev, j. preininger, v. veliov, t. roubai, kantorovich-type theorems for generalized equations,j. convex anal. 25 (2018) 459-486. http://hdl.handle.net/20.500.12708/144705.[11] j.a. ezquerro, m.a. hernandez, newton’s procedure: an updated approach of kantorovich’s theory, cham switzer-land, (2018). https://www.booksandcranniesva.com/book/9783319559759.[12] l.v. kantorovich, g.p. akilov, functional analysis in normed spaces, the macmillan co, new york, (1964).[13] a.a. magréñan, j.m. gutiérrez, real dynamics for damped newton’s procedure applied to cubic polynomials, j.comp. appl. math. 275 (2015) 527–538; https://doi.org/10.1016/j.cam.2013.11.019.[14] f.a. potra, v. pták, nondiscrete induction and iterative processes, research notes in mathematics 103, pitman,boston (1984). https://archive.org/details/nondiscreteinduc0000potr.[15] p.d. proinov, new general convergence theory for iterative processes and its applications to newton-kantorovichtype theorems, j. complex. 26 (2010) 3-42. https://doi.org/10.1016/j.jco.2009.05.001. https://doi.org/10.28924/ada/ma.3.15 https://doi.org/10.1016/j.jmaa.2016.02.047 https://doi.org/10.1137/130926730 https://doi. org/10.3390/math9161942 https://doi.org/10.1016/j.jco.2011.12.003 https://doi.org/10.1016/j.amc.2013.09.049 https://doi.org/10.1016/c2015-0-04301-5 https://doi.org/10.1007/s13226-020-0409-5 https://doi.org/10.1142/s0219530512500121 https://doi.org/10.1142/s0219530512500121 http://hdl.handle.net/20.500.12708/144705 https://www.booksandcranniesva.com/book/9783319559759 https://doi.org/10.1016/j.cam.2013.11.019 https://archive.org/details/nondiscreteinduc0000potr https://doi.org/10.1016/j.jco.2009.05.001 eur. j. math. anal. 10.28924/ada/ma.3.15 9 [16] r. verma, new trends in fractional programming, nova science publisher, new york, usa, (2019). https:// novapublishers.com/shop/new-trends-in-fractional-programming/. https://doi.org/10.28924/ada/ma.3.15 https://novapublishers.com/shop/new-trends-in-fractional-programming/ https://novapublishers.com/shop/new-trends-in-fractional-programming/ 1. introduction 2. convergence analysis 3. examples 4. conclusion declarations references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 17doi: 10.28924/ada/ma.3.17 on certain properties of a degenerate sigmoid function thomas awinba akugre1,∗ , kwara nantomah2 , mohammed muniru iddrisu3 1department of mathematics, school of mathematical sciences, c. k. tedam university of technology and applied sciences, p. o. box 24, navrongo, upper-east region, ghana takugre.stu@cktutas.edu.gh 2department of mathematics, school of mathematical sciences, c. k. tedam university of technology and applied sciences, p. o. box 24, navrongo, upper-east region, ghana knantomah@cktutas.edu.gh 3department of mathematics, school of mathematical sciences, c. k. tedam university of technology and applied sciences, p. o. box 24, navrongo, upper-east region, ghana middrisu@cktutas.edu.gh ∗correspondence: takugre.stu@cktutas.edu.gh abstract. in this paper, we introduce a degenerate sigmoid function. by employing analytical tech-niques, we present some properties such as logarithmic concavity, monotonicity and inequalities ofthe new function. 1. introduction it is known that, what is currently referred to as the logistic equation or the s-shaped curve wasfirst introduced by verhulst (see [17]). it maps a very large input domain to a small range of outputof real numbers between 0 and 1. it is a one toone functioon and increases monotonically(see [8]). the sigmoid function, also known in the literature as the sigmoidal curve or standardlogistic function is defined as (see [13]), s (t) = et 1 + et = 1 1 + e−t , t ∈ (−∞,∞) , (1) = 1 2 + 1 2 tanh ( t 2 ) , t ∈ (−∞,∞) . (2) it has the following as its first and second derivatives received: 27 feb 2023. key words and phrases. degenerate sigmoid function; logarithmically concave; inequality.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.17 https://orcid.org/0009-0005-1387-377x https://orcid.org/0000-0003-0911-9537 https://orcid.org/0000-0001-7628-8168 eur. j. math. anal. 10.28924/ada/ma.3.17 2 s ′ (t) = et (1 + et) 2 = s (t) (1 −s (t)) , (3) s ′′ (t) = et ( 1 −et ) (1 + et) 3 = s (t) (1 −s (t)) (1 − 2s (t)) , (4) for all t ∈ (−∞,∞) .the sigmoid function is used in a wide range of scientific disciplines, including probability andstatistics, biology, demography, machine learning, population dynamics, ecology, and mathematicalpsychology(see [7], [16]). in the business sector, the sigmoid function has been utilized to analyzeperformance growth in manufacturing and service management (see [10]). at each neuron’s output,the function serves as an activation function in artificial neural networks (see [12], [18], [15]) andthe references therein.in addition, the function is used in medicine to research pharmacokinetic responses and mimictumor development (see [11]). in [5], the site index of unmanaged loblolly and slash pine plantationsin east texas is predicted using a generic variant of the sigmoid function. it is also used incomputer graphics and image processing to improve picture contrast (see [4], [9], [6]). it is clearfrom the above applications of the sigmoid function that, further research needs to be conductedon this very important function to unearth more of its properties and potential applications. recently, in [13], the author studied properties such as super multiplicativity, subadditivity,convexity and inequalities of the sigmoid function. in this paper, a degenerate sigmoid function is introduced and properties such as logarithmicconcavity, monotonicity and inequlities involving the function are provided. we start with thefollowing definitions and lemmas. 2. some definitions and lemmas definition 2.1. [1] a function m : (0,∞)×(0,∞) → (0,∞) is called a mean function if it satisfiesthe following. (1) m (r,t) = m (t,r) ,(2) m (t,t) = t,(3) r < m (r,t) < t, for r < t,(4) m (ηr,ηt) = ηm (r,t) , for η > 0. there are many well-known mean functions in the literature. amongst them are the following. (1) arithmetic mean: a (r,t) = r+t 2 ,(2) geometric mean: g (r,t) = √rt, https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 3 (3) harmonic mean: h (r,t) = 1 a(1r , 1 t ) = 2rt r+t ,(4) logarithmic mean: l (r,t) = r−t lnr−lnt , for r 6= t and l (t,t) = t,(5) identric mean: i (r,t) = 1 e ( rr tt ) 1 r−t , for r 6= t and i (t,t) = t. definition 2.2. [1] let g : i ⊆ (0,∞) → (0,∞) be a continuous function and u and v be any twomean functions. then, g is said to be uv−convex (uv−concave) if g (u (r,t)) ≤ (≥) v (g (r) ,g (t)) , for all r,t ∈ i. lemma 2.3. [1] let f : i ⊆ (0,∞) → (0,∞) be a differentiable function. then (1) f is ag-convex(or concave) if and only if f ′(t) f (t) is increasing(or decreasing) for all t ∈ i. (2) f is ah-convex( or concave) if and only if f ′(t) f (t) 2 is increasing(or decreasing) for all t ∈ i. lemma 2.4. [2] let f : i ⊆ (b,∞) → (−∞,∞) with b ≥ 0. if the function defined by g (t) = f (t)−1 t is increasing on (b,∞) , then the function h (t) = f ( t2 ) satisfies the grumbaum-type inequality 1 + h ( z2 ) ≥ h ( r2 ) + h ( t2 ) , (5) where r,t ≥ b and z2 = r2 + t2. if g is decreasing, then the inequality (5) is reversed. 3. main results definition 3.1. the degenerate sigmoid function is defined for λ ∈ (0,∞) and t ∈ (−∞,∞) as sλ (t) = (1 + λt) 1 λ 1 + (1 + λt) 1 λ (6) = 1 1 + (1 + λt) −1 λ (7) = 1 2 + 1 2 tanhλ ( t 2 ) . (8) it is clear that, taking the limit of sλ (t) as λ → 0, then sλ (t) → s (t) .the first derivative of the degenerate sigmoid function is given as s ′ λ (t) = (1 + λt) 1 λ −1[ 1 + (1 + λt) 1 λ ]2 > 0, (9) for all t ∈ (−∞,∞) and λ ∈ (0,∞) . the degenerate sigmoid function satisfies the following identities. https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 4 sλ (t) + sλ (−t) = 1, (10) sλ (t) sλ (−t) = (1 + λt) s ′ λ (t) , (11) s ′ λ (t) = s ′ λ (−t) , (12) lim t→∞ sλ (t) = 1, (13) lim t→0 sλ (t) = 1 2 , (14) lim t→0 s ′ λ (t) = 1 4 , (15) lim t→∞ s ′ λ (t) = 0. (16) theorem 3.2. the function sλ (t) is ag-concave on (0,∞). in other words, for all r,t,λ ∈ (0,∞) , the inequality sλ ( r + t 2 ) ≥ [sλ (r) sλ (t)] 1 2 (17) is satisfied. proof. we have s ′ λ (t) sλ (t) =  (1 + λt) 1λ−1[ 1 + (1 + λt) 1 λ ]2 (1 + (1 + λt) 1λ (1 + λt) 1 λ ) = 1 (1 + λt) + (1 + λt) 1 λ +1 and ( s ′ λ (t) sλ (t) )′ = − λ + (1 + λ) (1 + λt) 1 λ[ (1 + λt) + (1 + λt) 1 λ +1 ]2 < 0, (18) which imlplies that s′λ(t) sλ(t) is decreasing on (0,∞). hence, by lemma 2.3(1), we obtain the desiredresult (17). � theorem 3.3. the function sλ (t) is ah-concave on (0,∞). in other words, for all r,t,λ ∈ (0,∞) , the inequality sλ ( r + t 2 ) ≥ 2sλ (r) sλ (t) sλ (r) + sλ (t) (19) is valid. https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 5 proof. now we have s ′ λ (t) sλ (t) 2 =  (1 + λt) 1λ−1[ 1 + (1 + λt) 1 λ ]2   [ 1 + (1 + λt) 1 λ ]2 (1 + λt) 2 λ  = 1 (1 + λt) (1 + λt) 1 λ = 1 (1 + λt) 1 λ +1 and ( s ′ λ (t) sλ (t) 2 )′ = − (1 + λ) (1 + λt) 1 λ (1 + λt) 2 λ +2 < 0. by lemma 2.3(2), we conclude that sλ (t) is ah-concave on (0,∞). this implies inequality(19). � theorem 3.4. the function sλ (t), for r,t,λ ∈ (0,∞) and z2 = r2+t2, satisfies the grunbaum-type inequality 1 + sλ ( z2 ) ≥ sλ ( r2 ) + sλ ( t2 ) . (20) proof. let h (t) be defined for t,λ ∈ (0,∞) as h (t) = sλ(t)−1 t . this implies h (t) = (1+λt) 1 λ 1+(1+λt) 1 λ − 1 t = − 1 t + t (1 + λt) 1 λ . differentiating h (t), we have h ′ (t) = 1 + (1 + λt) 1 λ + t (1 + λt) 1 λ −1[ t + t (1 + λt) 1 λ ]2 > 0, which implies that h (t) is increasing. by applying lemma 2.4, we obtain the desired result (20). � theorem 3.5. for λ ∈ (0,∞) , the function sλ (t) satisfies the inequalities s2λ (r + t) ≥ sλ (r) sλ (t) , r,t ∈ [0,∞) (21) and s2λ (r + t) ≤ sλ (r) sλ (t) , r,t ∈ (−∞, 0] . (22) equality holds if r = t = 0. https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 6 proof. let r,t ∈ [0,∞) and λ ∈ (0,∞). recall that sλ (t) is increasing. thus we have sλ (r + t) ≥ sλ (r) > 0, (23) sλ (r + t) ≥ sλ (t) > 0, (24) since r + t ≥ r and r + t ≥ t. now by multiplying (23)and (24), we obtain the desired result (21).next, let r,t ∈ (−∞, 0] and λ ∈ (0,∞), we have 0 < sλ (r + t) ≤ sλ (r) , (25) 0 < sλ (r + t) ≤ sλ (t) , (26) since r + t ≤ r and r + t ≤ t. by multiplying the inequalities (25) and (26), we have the desiredresult. � theorem 3.6. the function sλ (t) , for λ ∈ (0,∞) , satisfies the inequalities s2λ (rt) ≤ sλ (r) sλ (t) , r,t ∈ [0, 1] (27) and s2λ (rt) ≥ sλ (r) sλ (t) , r,t ∈ [1,∞) . (28) equality holds if r = t = 1. proof. let r,t ∈ [0, 1] and λ ∈ (0,∞). recall that sλ (t) is increasing. thus we have 0 < sλ (rt) ≤ sλ (r) , (29) 0 < sλ (rt) ≤ sλ (t) , (30) since rt ≤ r and rt ≤ t. now by multiplying (29)and (30), we obtain the result (27).next, let r,t ∈ [1,∞, ) and λ ∈ (0,∞), we have sλ (rt) ≥ sλ (r) > 0, (31) sλ (rt) ≥ sλ (t) > 0, (32) since rt ≥ r and rt ≥ t. by multiplying the inequalities (31) and (32), the desired result isobtained (28). � theorem 3.7. for r,t ∈ (−∞,∞) and λ ∈ (0,∞), the function sλ (t) is logarithmically concave. in other words, the inequality sλ ( r a + t b ) ≥ [sλ (r)] 1 a [sλ (t)] 1 b (33) is satisfied. where a > 1 and 1 a + 1 b = 1. https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 7 proof. let q (t) = ln sλ (t) . then, q ′ (t) = s ′ λ (t) sλ (t) = (1+λt) 1 λ −1[ 1+(1+λt) 1 λ ]2 (1+λt) 1 λ 1+(1+λt) 1 λ =  (1 + λt) 1λ (1 + λt) [ 1 + (1 + λt) 1 λ ]2 (1 + (1 + λt) 1λ (1 + λt) 1 λ ) = 1 (1 + λt) + (1 + λt) 1 λ +1 . taking the second derivative of q (t) , we have q ′′ (t) = − λ + (1 + λ) (1 + λt) 1 λ[ (1 + λt) + (1 + λt) 1 λ +1 ]2 < 0, and this completes the proof. � corollary 3.8. for λ ∈ (0,∞) and t ∈ (−∞,∞) , the inequalities s ′′ λ (t) sλ (t) ≤ [ s ′ λ (t) ]2 (34) and sλ (1 + u) sλ (1 −u) ≤ [ (1 + λ) 1 λ 1 + (1 + λ) 1 λ ]2 (35) are valid. proof. since sλ (t) is logarithmically concave, then [ln (sλ (t))]′′ ≤ 0, for all t ∈ (−∞,∞) and λ ∈ (0,∞) . this implies that, [ln (sλ (t))] ′′ = [ s ′ λ (t) sλ (t) ]′ = s ′′ λ (t) sλ (t) −s ′ λ (t) s ′ λ (t) [sλ (t)] 2 = s ′′ λ (t) sλ (t) − [ s ′ λ (t) ]2 [sλ (t)] 2 ≤ 0. hence, s′′λ (t) sλ (t) − [s′λ (t)]2 ≤ 0, which yields equation (34). https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 8 next, let a = b = 2,t = 1 + u and r = 1 −u in equation (33). we have sλ ( 1 + u 2 + 1 −u 2 ) ≥ [sλ (1 + u)] 1 2 [sλ (1 −u)] 1 2 sλ (1) ≥ ([sλ (1 + u)] [sλ (1 −u)]) 1 2[ (1 + λ) 1 λ 1 + (1 + λ) 1 λ ]2 ≥ sλ (1 + u) sλ (1 −u) , resulting in equation (35). this concludes the proof. � theorem 3.9. for t,λ ∈ (0,∞) , the function sλ (t) satisfies the inequality 1 < sλ (t + 1) sλ (t) < 2 (1 + λ) 1 λ 1 + (1 + λ) 1 λ . (36) proof. recall from equation (18), that( s ′ λ (t) sλ (t) )′ = − λ + (1 + λ) (1 + λt) 1 λ[ (1 + λt) + (1 + λt) 1 λ +1 ]2 < 0, for all t,λ ∈ (0,∞) . this implies, the function s′λ(t) sλ(t) is decreasing on the given interval.now, let p (t) = sλ (t + 1) sλ (t) = ( [1 + λ (t + 1)] 1 λ 1 + [1 + λ (t + 1)] 1 λ )( 1 + (1 + λt) 1 λ (1 + λt) 1 λ ) = [1 + λ (t + 1)] 1 λ + (1 + λt) 1 λ [1 + λ (t + 1)] 1 λ (1 + λt) 1 λ + (1 + λt) 1 λ [1 + λ (t + 1)] 1 λ and ω (t) = ln p (t) = ln sλ (t + 1) − ln sλ (t) . then, ω ′ (t) = s ′ λ (t + 1) sλ (t + 1) − s ′ λ (t) sλ (t) < 0, since s′λ(t) sλ(t) is decreasing. this implies ω (t) and consequently p (t) are decreasing. hence, forall t,λ ∈ (0,∞) , we have 1 = lim t→∞ p (t) < p (t) < lim t→0 p (t) = 2 (1 + λ) 1 λ 1 + (1 + λ) 1 λ , which yields the desired result (36). � https://doi.org/10.28924/ada/ma.3.17 eur. j. math. anal. 10.28924/ada/ma.3.17 9 4. conclusion we have introduced a degenerate sigmoid function. properties such as concavity, monotonicity 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https://doi.org/10.1007/bf02309004 https://doi.org/10.1007/bf02309004 https://doi.org/10.5772/intechopen.80416 https://doi.org/10.5772/intechopen.80416 1. introduction 2. some definitions and lemmas 3. main results 4. conclusion 5. conflicts of interest references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 16doi: 10.28924/ada/ma.3.16 best proximity points for generalized geraghty quasi-contraction type mappings in metric spaces j. c. umudu1,∗, j. o. olaleru2, h. olaoluwa2, a. a. mogbademu2 1department of mathematics, faculty of natural sciences, university of jos, nigeria umuduj@unijos.edu.ng 2department of mathematics, faculty of science, university of lagos, nigeria jolaleru@unilag.edu.ng, holaoluwa@unilag.edu.ng, amogbademu@unilag.edu.ng ∗correspondence: umuduj@unijos.edu.ng abstract. in this paper, we introduce a new concept of α-φ-geraghty proximal quasi-contractiontype mappings and establish best proximity point theorems for those mappings in proximal t-orbitallycomplete metric spaces. this generalizes and complements the proofs of some known fixed and bestproximity point results. 1. introduction let a and b be two nonempty subsets of a metric space (x,d). a best proximity point of anon-self mapping t : a → b, is the point x ∈ a, satisfying d(x,tx) = d(a,b). numerousresults on best proximity point theory were studied by several authors ( [1], [3], [4], [5]) imposingsufficient conditions that would assure the existence and uniqueness of such points. these resultsare generalizations of the contraction principle and other contractive mappings ( [2], [6], [8], [16],[21], [22], [24]) in the case of self-mappings, which reduces to a fixed point if the mapping underconsideration is a self-mapping. the notion of best proximity point was introduced in [14], the classof proximal quasi contraction mappings was introduced in [11] and thereafter, several known resultswere derived ( [10], [12], [13]). best proximity pair theorems analyse the conditions under which theoptimization problem, namely minx∈ad(x,tx) has a solution and is known to have applicationsin game theory. for additional information on best proximity point, see [7], [9], [10], [11], [12], [13],[14], [15], [17], [18], [20], [23]. definition 1.1 [4]. let t : x → x be a map on metric space. for each x ∈ x and for any positiveinteger n, ot (x,n) = {x,tx,...,tnx} and ot (x,∞) = {x,tx,...,tnx, ...}. received: 8 feb 2023. key words and phrases. best proximity; quasi-contraction; metric space.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 2 the set ot (x,∞) is called the orbit of t at x and the metric space x is called t-orbitally completeif every cauchy sequence in ot (x,∞) is convergent in x. quasi contraction mapping is known in literature as one of the most generalized contractive map-pings and is defined as follows. definition 1.2 [6]. a mapping t : x → x of a metric space x into itself is said to be a quasi-contraction if and only if there exists a number k, 0 ≤ k < 1, such that d(tx,ty) ≤ k max{d(x,y); d(x,tx); d(y,ty); d(x,ty); d(y,tx)} holds for every x,y ∈ x. consider the class f of functions β : [0,∞) → [0, 1) satisfying the condition: lim n→∞ β(tn) = 1 implies lim n→∞ tn = 0. recently, using these class of functions, umudu et al. [22] introduced a new class of quasi-contraction type mappings called generalized α-φ-geraghty quasi-contraction type mappings andproved the existence of its unique fixed point as follows. definition 1.3 [22]. let (x,d) be a metric space and α : x ×x →r+. a mapping t : x → x iscalled a generalized α-geraghty quasi-contraction type mapping if there exists β ∈ f such thatfor all x,y ∈ x, α(x,y)(d(tx,ty)) ≤ β(mt (x,y))(mt (x,y)), (1) where mt (x,y) = max{d(x,y),d(x,tx),d(y,ty),d(x,ty),d(y,tx)}. let φ denote the class of the functions φ : [0,∞) → [0,∞) which satisfies the following conditions: (i) φ is nondecreasing;(ii) φ is continuous;(iii) φ(t) = 0 ⇐⇒ t = 0. definition 1.4 [22]. let (x,d) be a metric space and α : x×x →r+. a self mapping t : x → xis called a generalized α-φ-geraghty quasi-contraction type mapping if there exists β ∈ f suchthat for all x,y ∈ x, α(x,y)φ(d(tx,ty)) ≤ β(φ(mt (x,y)))φ(mt (x,y)), (2) where mt (x,y) = max{d(x,y),d(x,tx),d(y,ty),d(x,ty),d(y,tx)}, and φ ∈ φ. if φ(t) = t, inequality (2) reduces to inequality (1). the generalized α-φ-geraghty quasi-contraction type self mapping is a generalization of other quasi-contraction type self mappingsin literature. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 3 the following mappings introduced by popescu [19] and used by umudu et al. [22] to establish theexistence of a fixed point will also be needed in this paper. definition 1.5 [19]. let t : x → x be a self-mapping and α : x ×x → r+ be a function. then t is said to be α-orbital admissible if α(x,tx) ≥ 1 implies α(tx,t2x) ≥ 1. definition 1.6 [19]. let t : x → x be a self-mapping and α : x × x → r+ be a function.then t is said to be triangular α-orbital admissible if t is α-orbital admissible, α(x,y) ≥ 1 and α(y,ty) ≥ 1 imply α(x,ty) ≥ 1. the main result obtained in [22] is the following. theorem 1.7. let (x,d) be a t orbitally complete metric space, α : x ×x → r+ be a function,and let t : x → x be a self-mapping. suppose that the following conditions are satisfied: (i) t is a generalized α-φ-geraghty quasi-contraction type mapping;(ii) t is triangular α-orbital admissible mapping;(iii) there exists x1 ∈ x such that α(x1,tx1) ≥ 1; then t has a fixed point x∗ ∈ x and {tnx1} converges to x∗. in this paper, we extend the concept of generalized α-φ-geraghty quasi-contraction typemapping to generalized α-φ-geraghty proximal quasi-contraction type mapping in the case ofnon-self mappings. more precisely, we study the existence and uniqueness of best proximitypoints for generalized α-φ-geraghty proximal quasi-contraction for non-self mappings. 2. preliminaries we start this section with the following definitions.let a and b be non-empty subsets of a metric space (x,d). we denote by a0 and b0 the followingsets: d(a,b) = inf{d(a,b) : a ∈ a, b ∈ b}. a0 = {x ∈ a : d(x,y) = d(a,b) for some y ∈ b}. b0 = {y ∈ b : d(x,y) = d(a,b) for some x ∈ a}. definition 2.1 [14]. an element x ∈ a is said to be a best proximity point of the non-self-mapping t : a → b if it satisfies the condition that d(x,tx) = d(a,b).we denote the set of all best proximity points of t by pt (a), that is, pt (a) := {x ∈ a : d(x,tx) = d(a,b)}. the following were introduced by [11]. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 4 definition 2.2 [11]. a non-self mapping t : a → b is said to be a proximal quasi-contraction ifand only if there exists a number q, 0 ≤ q < 1, such that{ d(u,tx) = d(a,b) d(v,ty) = d(a,b) =⇒ d(u,v) ≤ q max{d(x,y); d(x,u); d(y,v); d(x,v); d(y,u)}, where x,y,u,v ∈ a. if t is a self mapping on a, then definition 2.2 reduces to definition 1.2. lemma 2.3 [11]. let t : a → b be a non-self mapping. suppose that the following conditionshold: (i) a0 6= ∅;(ii) t (a0) ⊆ b0.then, for all a ∈ a0, there exists a sequence {xn}⊂ a0 such that{ x0 = a, d(xn+1,txn) = d(a,b), ∀n ∈n.any sequence {xn} ⊂ a0 satisfying the equation in lemma 2.3 is called a proximal picardsequence associated to a ∈ a0 and we denote by pp(a) the set of all proximal picard sequencesassociated to a. suppose a ∈ a0 and {xn} ∈ pp(a). for all (i, j) ∈ n2, the following sets are definedby: ot (xi, j) := {xl : i ≤ l ≤ j + i} and ot (xi,∞) := {xl : l ≥ i}. definition 2.4 [11] a0 is said to be proximal t-orbitally complete if and only if every cauchysequence {xn}∈pp(a) for some a ∈ a0, converges to an element in a0. if t is a self mapping on a, then the preceding definition reduces to the condition that a is t-orbitally complete. the concepts of α-orbital proximal admissible mapping and triangular α-orbital proximaladmissible mapping are hereby introduced as follows. definition 2.5 let t : a → b be a non-self mapping and α : a×a → [0,∞) be a function. themapping t is said to be α-orbital proximal admissible if  α(x,u) ≥ 1 d(u,tx) = d(a,b) d(v,tu) = d(a,b) =⇒ α(u,v) ≥ 1, for all x,u,v ∈ a. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 5 definition 2.6 let t : a → b be a non-self mapping and α : a × a → [0,∞) be a function.the mapping t is said to be triangular α-orbital proximal admissible if it is α-orbital proximaladmissible and  α(x,y) ≥ 1 α(y,u) ≥ 1 d(u,ty) = d(a,b) =⇒ α(x,u) ≥ 1, for all x,y,u ∈ a. remark 2.7. if t is a self mapping, that is, if a = b, α-orbital proximal admissible mappingreduces to α-orbital admissible mapping while triangular α-orbital proximal admissible mappingreduces to triangular α-orbital admissible mapping defined in [19] . example 2.8. let x be the euclidean plane r2 and consider the two subsets: a = {(0, 0), (0, 1), (0, 2), (0, 3)} b = {(1, 0), (2, 1), (2, 2), (1, 3)} define a mapping t : a → b such that t (0, 0) = (1, 0), t (0, 1) = (2, 2), t (0, 2) = (2, 1) and t (0, 3) = (1, 3).also define a mapping α : a×a → [0,∞) such that α(x,y) =  1, if x = y ∈{(0, 0), (0, 3)} 0 elsewhere. for all x,y ∈ a. one can see that d(a,b) = 1. let u,v,x ∈ a. one can check that α(x,u) ≥ 1 d(u,tx) = 1 d(v,tu) = 1 =⇒ x = u = v ∈{(0, 0), (0, 3)} =⇒ α(u,v) = 1. hence, t is α-orbital proximal admissible. let u,x,y ∈ a. one can check that  α(x,u) ≥ 1 α(y,u) ≥ 1 d(u,ty) = 1 =⇒ x = y = u ∈{(0, 0), (0, 3)} =⇒ α(x,u) = 1. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 6 thus, t is also triangular α-orbital proximal admissible. we introduce the following new classes of non-self mappings. definition 2.9 let a and b be two nonempty subsets of a metric space (x,d) and α : a×a →r+be a function. a non-self mapping t : a → b is called a generalized α-φ-geraghty proximalquasi-contraction type mapping if there exists β ∈ f such that for all x,y,u,v ∈ a,{ d(u,tx) = d(a,b) d(v,ty) = d(a,b) =⇒ α(x,y)φ(d(u,v)) ≤ β(φ(mt (x,y)))φ(mt (x,y)), (3) where mt (x,y) = max{d(x,y),d(x,u),d(y,v),d(x,v),d(y,u)}, for all x,y,u,v ∈ a and φ ∈ φ. if φ(t) = t, then definition 2.9 reduces to the following. definition 2.10 let a and b be two nonempty subsets of a metric space (x,d) and α : a×a →r+be a function. a non-self mapping t : a → b is called an α-geraghty proximal quasi-contractiontype mapping if there exists β ∈ f such that for all x,y,u,v ∈ a,{ d(u,tx) = d(a,b) d(v,ty) = d(a,b) =⇒ α(x,y)d(u,v) ≤ β(mt (x,y))(mt (x,y)), (4) for all x,y,u,v ∈ a. where mt (x,y) = max{d(x,y),d(x,u),d(y,v),d(x,v),d(y,u)} for all x,y,u,v ∈ a. 3. main results now we state and prove our main results. theorem 3.1. let a and b be two nonempty subsets of a metric space such that a0 isproximal t-orbitally complete, where t : a → b is a non-self mapping, α : a × a → r+ is afunction and the following conditions are satisfied: (i) t is a generalized α-φ-geraghty proximal quasi-contraction type mapping;(ii) t (a0) ⊆ b0 and t is a triangular α-orbital proximal admissible mapping;(iii) there exists x0,x1 ∈ a0 such that d(x1,tx0) = d(a,b) and α(x0,x1) ≥ 1.then there exists an element x∗ ∈ a0 such that d(x∗,tx∗) = d(a,b). moreover, if α(x,y) ≥ 1 for all x,y ∈ pt (a), then x∗ is the unique best proximity point of t . proof.let x0,x1 ∈ a0 be such that d(x1,tx0) = d(a,b) and α(x0,x1) ≥ 1. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 7 t (a0) ⊆ b0 and there exists x2 ∈ a0 such that d(x2,tx1) = d(a,b). now, we have α(x0,x1) ≥ 1 d(x1,tx0) = d(a,b), d(x2,tx1) = d(a,b). since t is α-orbital proximal admissible, α(x1,x2) ≥ 1. thus, we have d(x2,tx1) = d(a,b) and α(x1,x2) ≥ 1. by induction, we can construct a sequence {xi}⊆ a0 such that d(xi+1,txi) = d(a,b) and α(xi,xi+1) ≥ 1, f or all i ∈n. (5) for all i ≥ 0  α(xi,xi+1) ≥ 1 α(xi+1,xi+2) ≥ 1 d(xi+2,txi−1) = d(a,b), =⇒ α(xi,xi+2) ≥ 1, since t is triangular α-orbital proximal admissible. thus by induction, α(xi,xj) ≥ 1 for all i, jsuch that 0 ≤ i < j.therefore for any i ∈n, we have α(xi−1,xj−1) ≥ 1 d(xi,txi−1) = d(a,b), d(xj,txj−1) = d(a,b) for all i, j such that 1 ≤ i < j.clearly, if xi+1 = xi for some i ∈ n from inequality (5), xi will be a best proximity point, sohenceforth, in this proof, we assume d(xi,xi+1) > 0, ∀ i ∈n. from inequality (3), we have φ(d(xi,xj)) ≤ α(xi−1,xj−1)φ(d(xi,xj)) ≤ β(φ(mt (xi−1,xj−1)))φ(mt (xi−1,xj−1)) (6) 1 ≤ i < j where φ(mt (xi−1,xj−1)) ≤ φ(max{d(xi−1,xj−1),d(xi−1,xi),d(xj−1,xj), d(xi−1,xj),d(xj−1,xi)}) ≤ φ(δ[ot (xi−1,n)]), f or i ≤ j ≤ n + i. https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 8 note that the case φ(mt (xi−1,xj−1)) = φ(d(xi,xj)) is impossible. indeed, by inequality (6), φ(d(xi,xj)) ≤ β(φ(mt (xi−1,xj−1)))φ(mt (xi−1,xj−1)) ≤ β(φ(d(xi,xj)))φ(d(xi,xj)) < φ(d(xi,xj)), is a contradiction. thus, we conclude that φ(d(xi,xj)) < φ(d(xi−1,xj−1)) for all 0 < i < j and sothe sequence {φ(d(xi,xj))} is positive and decreasing. consequently, there exists r ≥ 0 such that lim i,j→∞ φ(d(xi,xj)) = r. we claim that r = 0. suppose, on the contrary, that r > 0. then we have φ(d(xi,xj)) φ(d(xi−1,xj−1)) ≤ β(φ(mt (xi−1,xj−1))) ≤ 1 f or each i, j ∈n such that i < j. then, since β ∈ f , lim i,j→∞ β(φ(mt (xi−1,xj−1))) = 1, implying that lim i,j→∞ φ(mt (xi−1,xj−1)) = 0, (7) and so by inequality (6) lim i,j→∞ φ(d(xi,xj)) = 0, which is a contradiction. now, by the continuity property of φ, φ ( lim i,j→∞ (d(xi,xj)) ) = φ(0). (8) but φ(t) = 0 if and only if t = 0 and so (8) gives lim i,j→∞ (d(xi,xj)) = 0. therefore, {xn} is a cauchy sequence in a0 and since a0 is proximal t-orbitally complete, thereexists x∗ ∈ a0 such that lim i→∞ xi = x ∗. also, since t (a0) ⊆ b0, then there exists y ∈ a0 such that d(y,tx∗) = d(xi,txi−1) = d(a,b) ∀n ∈n, ∀i ≥ 0. t being a generalized α-φ-geraghty proximal quasi-contraction type mapping gives φ(d(y,xi)) ≤ α(x∗,xi−1)φ(d(y,xi)) ≤ β(φ(mt (x∗,xi−1)φ(mt (x∗,xi−1)) https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 9 provided that α(x∗,xi−1) ≥ 1 where φ(mt (x ∗,xi−1)) = φ(max{d(x∗,xi−1),d(x∗,xi),d(xi−1,xi),d(x∗,y),d(xi−1,y)}). but taking the limit, φ(d(y,x∗)) ≤ lim i→∞ β(φ(mt (x ∗,xi−1)))φ(d(x ∗,y)), which gives, 1 ≤ lim i→∞ β(φ(mt (x ∗,xi−1))) = β(φ(d(y,x ∗))) = 1 implying φ(d(y,x∗)) = 0 and d(y,x∗) = 0 i.e y = x∗. we have d(x∗,tx∗) = d(y,tx∗) = d(a,b) and x∗ ∈ a0 is a bestproximity point of t .for uniqueness, suppose the best proximity point of t is not unique. let x∗, y∗ be two bestproximity points of t with x∗ 6= y∗. then, α(x∗,y∗) ≥ 1 d(x∗,tx∗) = d(a,b) d(y∗,ty∗) = d(a,b)  since t is a generalized α-φ-geraghty proximal quasi-contraction type mapping, φ(d(x∗,y∗)) ≤ α(x∗,y∗)φ(d(x∗,y∗)) ≤ β(mt (x∗,y∗))φ(mt (x∗,y∗)) < φ(mt (x ∗,y∗)) where mt (x ∗,y∗) = max{d(x∗,y∗),d(x∗,x∗),d(y∗,y∗),d(x∗,y∗),d(y∗,x∗)} = d(x∗,y∗). this gives d(x∗,y∗) < d(x∗,y∗), which is a contradiction. therefore x∗ = y∗, and the bestproximity point of t is unique. corollary 3.2. let a and b be two nonempty subsets of a metric space such that a0 isproximal t-orbitally complete, where t : a → b is a non-self mapping, α : a × a → r+ is afunction and the following conditions are satisfied: (i) t is a generalized α-geraghty proximal quasi-contraction type mapping;(ii) t (a0) ⊆ b0 and t is a triangular α-orbital proximal admissible mapping;(iii) there exists x0,x1 ∈ a0 such that d(x1,tx0) = d(a,b) and α(x0,x1) ≥ 1.then there exists an element x∗ ∈ a0 such that d(x∗,tx∗) = d(a,b). moreover, if α(x,y) ≥ 1 for all x,y ∈ pt (a), then x∗ is the unique best proximity point of t . https://doi.org/10.28924/ada/ma.3.16 eur. j. math. anal. 10.28924/ada/ma.3.16 10 4. conclusion in this paper, we introduced the notion of generalized α-φ-geraghty proximal quasi-contractiontype mappings which, for a self mapping, reduces to that in umudu et al. 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in uniform spaces, int.j. anal. 2017 (2017), 6173468. https://doi.org/10.1155/2017/6173468.[19] o. popescu, some new fixed point theorems for α-geraghty contraction type maps in metric spaces, fixed pointtheory appl. 2014 (2014), 190. https://doi.org/10.1186/1687-1812-2014-190.[20] v. sankar raj, a best proximity point theorem for weakly contractive non-self-mappings, nonlinear anal.: theorymeth. appl. 74 (2011), 4804-4808. https://doi.org/10.1016/j.na.2011.04.052.[21] b.e. rhoades, a comparison of various definitions of contractive maps, trans. amer. math. soc. 226 (1977), 257-290.[22] j.c. umudu, j.o. olaleru, a.a. mogbademu, fixed point results for geraghty quasi-contraction type mappingsin dislocated quasi-metric spaces, fixed point theory appl. 2020 (2020), 16. https://doi.org/10.1186/ s13663-020-00683-z.[23] j.c. umudu, j.o. olaleru, a.a. mogbademu, best proximity point results for geraghty p-proximal cyclic quasi-contraction in uniform spaces, divulgaciones mat. 21 (2020), 21-31.[24] j. umudu, a. mogbademu, j. olaleru, fixed point results for geraghty contractive type operators in uniform spaces,caspian j. math. sci. 11 (2022), 191-202. https://doi.org/10.22080/cjms.2021.3052. https://doi.org/10.28924/ada/ma.3.16 https://doi.org/10.1186/1687-1812-2013-180 https://doi.org/10.1155/2017/6173468 https://doi.org/10.1186/1687-1812-2014-190 https://doi.org/10.1016/j.na.2011.04.052 https://doi.org/10.1186/s13663-020-00683-z https://doi.org/10.1186/s13663-020-00683-z https://doi.org/10.22080/cjms.2021.3052 1. introduction 2. preliminaries 3. main results 4. conclusion competing interests: authors' contributions: references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 11doi: 10.28924/ada/ma.3.11 woven k −g−fusion frames in hilbert c∗−modules fakhr-dine nhari1, mohamed rossafi2,∗ 1laboratory analysis, geometry and applications department of mathematics, faculty of sciences, university of ibn tofail, p. o. box 133 kenitra, morocco nharidoc@gmail.com 2lasma laboratory, department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, p. o. box 1796 fez atlas, morocco rossafimohamed@gmail.com ∗correspondence: rossafimohamed@gmail.com abstract. in this paper, we introduced the notion of woven k − g−fusion frames in hilbert c∗−modules. we present necessary and sufficient conditions for these woven and also constructthem by linear bounded operator. finally we study perturbation of weaving k−g−fusion frames. 1. introduction basis is one of the most important concepts in vector spaces study. however, frames generaliseorthonormal bases and were introduced by duffin and schaefer [3] in 1952 to analyse some deepproblems in nonharmonic fourier series by abstracting the fundamental notion of gabor [5] for signalprocessing. in 2000, frank-larson [4] introduced the concept of frames in hilbet c∗−modulesas a generalization of frames in hilbert spaces. the basic idea was to consider modules over c∗−algebras of linear spaces and to allow the inner product to take values in the c∗−algebras [6].many generalizations of the concept of frame have been defined in hilbert c∗-modules [7,9,11–16].throughout this paper, h is considered to be a countably generated hilbert c∗−module. let {hj}j∈j are the collection of hilbert c∗−module and {wj}j∈j is a collection of closed orthogonallycomplemented submodules of h, where j be finite or countable index set. end∗a(h,hj) is a setof all adjointable operator from h to hj . in particular end∗a(h) denote the set of all adjointableoperators on h. pwj denote the orthogonal projection onto the closed submodule orthogonally received: 31 jul 2022.2010 mathematics subject classification. primary 41a58; secondary 42c15. key words and phrases. fusion frames; k − g−fusion frames; woven k − g−fusion frames; c∗-algebra; hilbert c∗-modules. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 2 complemented wj of h. define the module l2({hj}j∈j) = {{fj}j∈j : fj ∈ hj,‖ ∑ j∈j 〈fj, fj〉‖ < ∞} with a−valued inner product 〈f ,g〉 = ∑j∈j〈fj,gj〉, where f = {fj}j∈j and g = {gj}j∈j, clearly l2({hj}j∈j) is a hilbert a−module. definition 1.1. [8] let a be a unital c∗-algebra and h be a left a-module, such that the linearstructures of a and h are compatible. h is a pre-hilbert a-module if h is equipped with an a-valued inner product 〈., .〉 : h×h →a, such that is sesquilinear, positive definite and respectsthe module action. in the other words, (i) 〈f , f 〉≥ 0 for all f ∈ h and 〈f , f 〉 = 0 if and only if f = 0.(ii) 〈af + g,h〉 = a〈f ,h〉 + 〈g,h〉 for all a ∈a and f ,g,h ∈ h.(iii) 〈f ,g〉 = 〈g,f 〉∗ for all f ,g ∈ h. for f ∈ h, we define ||f || = ||〈f , f 〉||12 . if h is complete with ||.||, it is called a hilbert a-moduleor a hilbert c∗-module over a. for every a in a c∗-algebra a, we have |a| = (a∗a) 12 and the a-valued norm on h is defined by |f | = 〈f , f 〉12 for f ∈ h. lemma 1.2. [10] let {wj}j∈j be a sequence of orthogonally complemented closed submodules of h and t ∈ end∗a(h) invertible, if t ∗twj ⊂ wj for each j ∈ j, then {twj}j∈j is a sequence of orthogonally complemented closed submodules and pwjt ∗ = pwjt ∗ptwj . lemma 1.3. [2]. let h and k two hilbert a-modules and t ∈ end∗a(h,k). then the following statements are equivalent: (i) t is surjective.(ii) t∗ is bounded below with respect to norm, i.e., there is m > 0 such that ‖t∗x‖ ≥ m‖x‖ for all x ∈ k.(iii) t∗ is bounded below with respect to the inner product, i.e., there is m′ > 0 such that 〈t∗x,t∗x〉≥ m′〈x,x〉 for all x ∈ k. lemma 1.4. [1]. let u and h two hilbert a-modules and t ∈ end∗a(u,h). then:(i) if t is injective and t has closed range, then the adjointable map t∗t is invertible and ‖(t∗t )−1‖−1 ≤ t∗t ≤‖t‖2. (ii) if t is surjective, then the adjointable map tt∗ is invertible and ‖(tt∗)−1‖−1 ≤ tt∗ ≤‖t‖2. definition 1.5. [10] let {wi}i∈i be a sequence of closed orthogonally complemented submodulesof h, {vi}i∈i be a familly of positive weights in a, i.e., each vi is a positive invertible element from https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 3 the center of the c∗−algebra a and λi ∈ end∗a(h,hi ) for all i ∈ i. we say that λ = {wi, λi,vi}i∈iis a g−fusion frame for h if and only if there exists two constants 0 < a ≤ b < ∞ such that a〈x,x〉≤ ∑ i∈i v2i 〈λipwix, λipwix〉≤ b〈x,x〉, ∀x ∈ h. (1.1) the constants a and b are called the lower and upper bounds of g−fusion frame, respectively. if a = b then λ is called tight g-fusion frame and if a = b = 1 then we say λ is a parseval g−fusionframe. if λ satisfies the inequality∑ i∈i v2i 〈λipwix, λipwix〉≤ b〈x,x〉, ∀x ∈ h. then it is called a g−fusion bessel sequence with bound b in h. definition 1.6. [10]let λ = {wj, λj,vj}j∈j be a g−fusion bessel sequence for h. then the operator tλ : l2({hj}j∈j) → h defined by tλ({fj}j∈j) = ∑ j∈j vjpwj λ ∗ j fj, ∀{fj}j∈j ∈ l 2({hj}j∈j). is called synthesis operator. we say the adjoint uλ of the synthesis operator the analysis operatorand it is defined by uλ : h→ l2({hj}j∈j) such that uλ(f ) = {vjλjpwj (f )}j∈j, ∀f ∈ h. the operator sλ : h → h defined by sλf = tλuλf = ∑ j∈j v2j pwj λ ∗ j λjpwj (f ), ∀f ∈ h. is called g−fusion frame operator. it can be easily verify that 〈sλf , f 〉 = ∑ j∈j v2j 〈λjpwj (f ), λjpwj (f )〉, ∀f ∈ h. (1.2) furthermore, if λ is a g−fusion frame with bounds a and b, then a〈f , f 〉≤ 〈sλf , f 〉≤ b〈f , f 〉, ∀f ∈ h. it easy to see that the operator sλ is bounded, self-adjoint, positive, now we proof the inversibilityof sλ. let f ∈ h we have ||uλ(f )|| = ||{vjλjpwj (f )}j∈i|| = || ∑ j∈j v2j 〈λjpwj (f ), λjpwj (f )〉|| 1 2 . since λ is g−fusion frame then √ a||〈f , f 〉|| 1 2 ≤ ||uλf ||.then √ a||f || ≤ ||uλf ||. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 4 frome lemma 1.3, tλ is surjective and by lemma 1.4, tλuλ = sλ is invertible. we now, aih ≤ sλ ≤ bih and this gives b−1ih ≤ s−1λ ≤ a−1ih. 2. woven k −g−fusion frames in hilbert c∗−modules throughout this paper, [m] = {1, 2, ...,m} for each m > 1, {wij}j∈j,i∈[m] is a collection of closedorthogonally complemented submodules of h, {vij}j∈j,i∈[m] is a family of weights, k ∈ end∗a(h)and {λij}j∈j,i∈[m] ∈ end∗a(h,hij) where hij are hilbert a−modules. definition 2.1. a family of g−fusion frames {wij, λij,vij}j∈j,i∈[m] for h is said to be k−g−fusionwoven if there exist universal positive constants 0 < a ≤ b such that for each partition {σi}i∈[m]of j, the family {wij, λij,vij}j∈σi,i∈[m] is a k −g−fusion frame for h with bounds a and b. in next theorem, we provide a necessary and sufficient condition for weaving k−g−fusion frames. theorem 2.2. assume that {wj, λj,vj}j∈j and {vj,θj,µj}j∈j are two k − g−fusion frames for h where λj ∈ end∗a(h,hj) and θj ∈ end ∗ a(h,hj) for any j ∈ j, the following assertions are equivalent. (1) {wj, λj,vj}j∈j and {vj,θj,µj}j∈j are k −g−fusion woven.(2) there exists α > 0 such that for each σ ⊂ j there exists a bounded linear operator ψσ : l σ 2 ({hj}j∈j) → h, ψσ{xj}j∈j = ∑ j∈σ vjpwj λ ∗ j xj + ∑ j∈σc µjpvjθ ∗ j xj, such that αkk∗ ≤ ψσψ∗σ, where lσ2 ({hj}j∈j) = {{xj}j∈j = {fj}j∈σ ∪{gj}j∈σc : fj ∈ hj,gj ∈ hj,‖ ∑ j∈j 〈xj,xj〉‖ < ∞}. proof. (1) =⇒ (2): suppose that a is an universal lower frame bound for {wj, λj,vj}j∈j and {vj,θj,µj}j∈j. choose α = a and ψσ = tσ for every σ ⊂ j, where tσ is the synthesis operator of {wj, λj,vj}j∈σ ∪{vj,θj,µj}j∈σc . then, for any {xj}j∈j ∈ lσ2 ({hj}j∈j) we have ψσ{xj}j∈j = tσ{xj}j∈j = ∑ j∈σ vjpwj λ ∗ j xj + ∑ j∈σc µjpvjθ ∗ j xj, and also, for each f ∈ h, a〈k∗f ,k∗f 〉≤ 〈t∗σf ,t ∗ σf 〉 = 〈ψ ∗ σf ,ψ ∗ σf 〉. thus, αkk∗ ≤ ψσψ∗σ. (2) =⇒ (1): let σ ⊂ j and f ∈ h, so it is easy to check that ψ∗σf = {vjλjpwjf}j∈σ ∪{µjθjpvjf}j∈σc. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 5 therefore, α〈k∗f ,k∗f 〉 = 〈αkk∗f , f 〉 ≤ 〈ψσψ∗σf , f 〉 = 〈ψ∗σf ,ψ ∗ σf 〉 = ∑ j∈σ v2j 〈λjpwjf , λjpwjf 〉 + ∑ j∈σc µ2j 〈θjpvjf ,θjpvjf 〉. this gives that α is an universal lower frame bound of {wj, λj,vj}j∈j and {vj,θj,µj}j∈j. � in next results, we construct a k −g−fusion woven by using a bounded linear operator. theorem 2.3. let {wij, λij,vij}j∈j,i∈[m] be a k−g−fusion woven for h with common frame bounds a,b and assume that u ∈ end∗a(h) has closed range so that r(k ∗) ⊂ r(u) and ku = uk. then {uwij, λijpwiju ∗,vij}j∈j,i∈[m] is also k −g−fusion woven for r(u). proof. by the open mapping theorem, uwij is closed for any j ∈ j and i ∈ [m]. using lemme(refk-g-fusion ), we can write for each f ∈r(u), a〈k∗f ,k∗f 〉 = a〈(u+)∗u∗k∗f , (u+)∗u∗k∗f 〉 ≤ a‖u+‖2〈k∗u∗f ,k∗u∗f 〉 ≤ ‖u+‖2 ∑ i∈[m] ∑ j∈j v2ij〈λijpwiju ∗f , λijpwiju ∗f 〉 = ‖u+‖2 ∑ i∈[m] ∑ j∈j v2ij〈λijpwiju ∗puwijf , λijpwiju ∗puwijf 〉. the upper bound is obvious. � theorem 2.4. let k have closed range, {wij, λij,vij}j∈j,i∈[m] be a k−g−fusion woven for h with the universal bounds a,b and u ∈ end∗a(h) has closed range so that r(u ∗) ⊂ r(k). then {uwij, λijpwiju ∗,vij}j∈j,i∈[m] is a k − g−fusion woven for h if and only if there exists a δ > 0 such that for every f ∈ h, 〈u∗f ,u∗f 〉≥ δ〈k∗f ,k∗f 〉. proof. let f ∈ h and {uwij, λijpwiju∗,vij}j∈j,i∈[m] is a k − g−fusion woven for h with lowerbound c, we get c〈k∗f ,k∗f 〉≤ ∑ i∈[m] ∑ j∈j v2ij〈λijpwiju ∗puwijf , λijpwiju ∗puwijf 〉 = ∑ i∈[m] ∑ j∈j v2ij〈λijpwiju ∗f , λijpwiju ∗f 〉 ≤ b〈u∗f ,u∗f 〉. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 6 therefore, 〈u∗f ,u∗f 〉≥ √c b 〈k∗f ,k∗f 〉. for the opposite implication, we can write for all f ∈ h, 〈u∗f ,u∗f 〉 = 〈(k+)∗k∗u∗f , (k+)∗k∗u∗f 〉≤ ‖k+‖2〈k∗u∗f ,k∗u∗f 〉. hence, we have aδ‖k+‖−2〈k∗f ,k∗f 〉≤ a‖k+‖−2〈u∗f ,u∗f 〉 ≤ a〈k∗u∗f ,k∗u∗f 〉 ≤ ∑ i∈[m] ∑ j∈j v2ij〈λijpwiju ∗f , λijpwiju ∗f 〉 = ∑ i∈[m] ∑ j∈j v2ij〈λijpwiju ∗puwijf , λijpwiju ∗puwijf 〉 ≤ b‖u‖2〈f , f 〉. so, {uwij, λijpwiju∗,vij}j∈j,i∈[m] is a k − g−fusion woven for h with frame bounds aδ‖k+‖−2and b‖u‖2. � theorem 2.5. let {wij, λij,vij}j∈j,i∈[m] be a k − g−fusion woven for h with common frame bounds a and b. suppose that 0 ≤ c ≤ |w(i) j |2 ≤ d < ∞ for any i ∈ [m] and j ∈ j, then {wij,w (i) j λij,vij}j∈j,i∈[m] is a k −g−fusion woven for h with frame bounds ac and bd. proof. for any partition {σi}i∈[m] of j and f ∈ h, we get ac〈k∗f ,k∗f 〉 = min i∈[m] |w(i) j |2a〈k∗f ,k∗f 〉≤ ∑ i∈[m] ∑ j∈σi v2ij〈w (i) j λijpwijf ,w (i) j λijpwijf 〉 ≤ max i∈[m] |w(i) j |2b〈f , f 〉 = bd〈f , f 〉. � theorem 2.6. let i ⊂ j be arbitrary and {wij, λij,vij}j∈i,i∈[m] be a k − g−fusion woven for h. then {wij, λij,vij}j∈j,i∈[m] is a k −g−fusion woven. proof. assume that σi ⊂ j, so σi ∩i⊂ i and a is the lower bound of {wij, λij,vij}j∈σi∩i,i∈[m], thenfor every f ∈ h we have a〈k∗f ,k∗f 〉≤ ∑ i∈[m] ∑ j∈σi∩i v2ij〈λijpwijf , λijpwijf 〉 ≤ ∑ i∈[m] ∑ j∈σi v2ij〈λijpwijf , λijpwijf 〉. this implies the statement. � next theorem is shows that even if one subspace is deleted, it dose not still remain a k−g−fusionwoven. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 7 theorem 2.7. let k has closed range, i⊂ j and {wij, λij,vij}j∈j,i∈[m] be a k −g−fusion woven for h with the bounds a,b. if c = ∑ i∈[m] ∑ j∈i v2ij‖λijpwij‖ 2 < a‖k+‖2, then {wij, λij,vij}j∈j−i,i∈[m] is a k −g−fusion woven for r(k). proof. the upper bound is obvious. suppose that σi i∈[m] ⊂ j− i and f ∈r(k), so we get∑ i∈[m] ∑ j∈σi v2ij〈λijpwijf , λijpwijf 〉 = ∑ i∈[m] ∑ j∈σi∪i v2ij〈λijpwijf , λijpwijf 〉− ∑ i∈[m] ∑ j∈i v2ij〈λijpwijf , λijpwijf 〉 ≥ a〈k∗f ,k∗f 〉− ∑ i∈[m] ∑ j∈i v2ij‖λijpwij‖ 2〈f , f 〉 ≥ (a−c‖k+‖2)〈k∗f ,k∗f 〉. � theorem 2.8. let {wij, λij,vij}j∈j,i∈[m] be a k − g−fusion woven for h with bounds a,b. for each i ∈ [m],j ∈ j and a index set iij , suppose that {f (k) ij }k∈iij ∈ λij(wij) is a parseval frame for hij such that for every finite subset kij ⊂ iij , the set {f kij }k∈iij−kij is a frame with the lower bound cij . let w̃ij = span{λ∗ijf (k) ij }k∈iij−kij for any i ∈ [m] and j ∈ j, then {w̃ij, λij,vij}j∈j,i∈[m] is a k −g−fusion woven for h with the bounds (mini∈[m],j∈jcij)a and b. proof. obviously, b is the upper bound of {w̃ij, λij,vij}j∈j,i∈[m]. assume that f ∈ h and {σi}i∈[m] ∈ j, so ∑ i∈[m] ∑ j∈σi v2ij〈λijpw̃ijf , λijpw̃ijf 〉 = ∑ i∈[m] ∑ j∈σi v2ij ∑ k∈iij 〈λijpw̃ijf , f (k) ij 〉〈f (k) ij , λijpw̃ij f 〉 ≥ ∑ i∈[m] ∑ j∈σi v2ij ∑ k∈iij−kij 〈λijpw̃ijf , f (k) ij 〉〈f (k) ij , λijpw̃ij f 〉 ≥ ∑ i∈[m] ∑ j∈σi v2ijcij〈λijpwijf , λijpwijf 〉 ≥ ( min i∈[m],j∈j cij) ∑ i∈[m] ∑ j∈σi v2ij〈λijpwijf , λijpwijf 〉 ≥ ( min i∈[m],j∈j cij)a〈k∗f ,k∗f 〉. � theorem 2.9. let {wij, λij,vij}j∈j is a k−g−fusion frame for h for each i ∈ [m]. suppose that for a partition collection of disjoint finite sets {δi}i∈[m] of j and for any � > 0 there exists a partition {σi}i∈[m] of the set j−∪i∈[m]δi such that {wij, λij,vij}j∈(σi∪δi ),i∈[m] has a lower k − g−fusion frame bound less than �. then {wij, λij,vij}j∈j,i∈[m] is not a woven. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 8 proof. we can write j = ∪j∈njj , where jj are disjoint index sets. assume that δ1j = ∅ for all i ∈ [m] and � = 1. then, there exists a partition σi1i∈[m] of j such that {wij, λij,vij}j∈(σi1∪δi1),i∈[m]has a lower bound (also, optimal lower bound) less than 1. thus, there is a f1 ∈ h such that∑ i∈[m] ∑ j∈(σi1∪δi1) v2ij〈λijpwijf1, λijpwijf1〉 < 〈k ∗f1,k ∗f1〉. since ∑ i∈[m] ∑ j∈j v2ij〈λijpwijf1, λijpwijf1〉 < ∞, so, there is a k1 ∈n such that∑ i∈[m] ∑ j∈k1 v2ij〈λijpwijf1, λijpwijf1〉 < 〈k ∗f1,k ∗f1〉, where, k1 = ∪i≥k1+1jj . continuing this way, for � = 1n and a partition {δni}i∈[m] of j1 ∪ ...∪jkn−1such that δni = δ(n−1)i ∪ (σ(n−1)i ∩ (j1 ∪ ...∪jkn−1)) for all i ∈ [m], there exists a partition {σni}i∈[m] of j − (j1 ∪ ... ∪ jkn−1) such that {wij, λij,vij}j∈(σni∪δni ),i∈[m] has a lower bound less than 1n . therefore, there is a fn ∈ h and kn ∈n such that kn > kn−1 and∑ i∈[m] ∑ j∈kn v2ij〈λijpwijfn, λijpwijfn〉 < 1 n 〈k∗fn,k∗f1〉, where, kn = ∪i≥kn+1jj . choose a partition {αi}i∈[m] of j, where αi = ∪j∈n{δji} = δ(n+1)i ∪ (αi ∩ j− (j1 ∪ ... ∪ jn)). assume that {wij, λij,vij}j∈αi,i∈[m] is a k − g−fusion frame for h with theoptimal lower bound a. then, by the archimedean property, there exits a r ∈n such that r > 2 a .now, there exists a fr ∈ h such that∑ i∈[m] ∑ j∈αi v2ij〈λijpwijfr, λijpwijfr〉 = ∑ i∈[m] ∑ j∈δ(r+1)i v2ij〈λijpwijfr, λijpwijfr〉 + ∑ i∈[m] ∑ j∈αi∩j−(j1∪...∪jr ) v2ij〈λijpwijfr, λijpwijfr〉 ≤ ∑ i∈[m] ∑ j∈(σri∪δri ) v2ij〈λijpwijfr, λijpwijfr〉 + ∑ i∈[m] ∑ j∈∪k≥r+1jk v2ij〈λijpwijfr, λijpwijfr〉 < 1 r 〈k∗fr,k∗fr〉 + 1 r 〈k∗fr,k∗fr〉 < a〈k∗fr,k∗fr〉 and this is a contradiction with the lower bound of a. � https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 9 corollary 2.10. let {wij, λij,vij}j∈j,i∈[m] be a k − g−fusion woven for h. then there exists a collection of disjoint finite subsets {δi}i∈[m] of j and a > 0 such that for each partition {σi}i∈[m] of the set j−∪i∈[m]δi , some the family {wij, λij,vij}j∈(σi∪δi ),i∈[m] is a k −g−fusion frame for h with the lower frame bound a. theorem 2.11. let {wij, λij,vij}j∈j be a k−g−fusion frame for h with bounds ai and bi for each i ∈ [m]. suppose that there exists n > 0 such that for all i,k ∈ [m] with i 6= k, i⊂ j and f ∈ h,∑ j∈i 〈(vijλijpwij −vkjλkjpwkj )f , (vijλijpwij −vkjλkjpwkj )f 〉≤ n min{ ∑ j∈i v2ij〈λijpwijf , λijpwijf 〉,∑ j∈i v2kj〈λkjpwkjf , λkjpwkjf 〉}. then the family {wij, λij,vij}j∈j,i∈[m] is woven with universal bounds a (m− 1)(n + 1) + 1 and b, where a = ∑ i∈[m] ai and b = ∑ i∈[m] bi . proof. let {σi}i∈[m] be a partition of j and f ∈ h. therefore,∑ i∈[m] ai〈k∗f ,k∗f 〉 ∑ i∈[m] ∑ j∈j v2ij〈λijpwijf , λijpwijf 〉 = ∑ i∈[m] ∑ k∈[m] ∑ j∈σk v2ij〈λijpwijf , λijpwijf 〉 ≤ ∑ i∈[m] (∑ j∈σi v2ij〈λijpwijf , λijpwijf 〉 + ∑ k∈[m],k 6=i ∑ j∈σk {v2kj〈λkjpwkjf , λkjpwkjf 〉 + 〈(vijλijpwij −vkjλkjpwkj )f , (vijλijpwij −vkjλkjpwkj )f 〉} ) ≤ ∑ i∈[m] (∑ j∈σi v2ij〈λijpwijf , λijpwijf 〉 + ∑ k∈[m],k 6=i ∑ j∈σk (n + 1)v2kj〈λkjpwkjf , λkjpwkjf 〉 ) = {(m− 1)(n + 1) + 1} ∑ i∈[m] (∑ j∈σi v2ij〈λijpwijf , λijpwijf 〉 ) . thus, we get a (m− 1)(n + 1) + 1 〈k∗f ,k∗f 〉≤ ∑ i∈[m] (∑ j∈σi v2ij〈λijpwijf , λijpwijf 〉 ) ≤ b〈f , f 〉. � in next theorem we study a paley-wiener type perturbation for weaving k −g−fusion frames. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 10 theorem 2.12. let {wj, λj,wj}j∈j and {vj,θj,vj}j∈j be two k−g−fusion frames for h with frame bounds a1,b1 and a2,b2, respectively. suppose that there exist non-negative scalers µ and 0 ≤ λ < 1 2 such that ( 1 2 −λ)a1 > µ and for each f ∈ h,∑ j∈j 〈(wjλjpwj −vjθjpvj )f , (wjλjpwj −vjθjpvj )f 〉≤ λ ∑ j∈j 〈wjλjpwjf ,wjλjpwjf 〉 + µ〈k ∗f ,k∗f 〉. then, {wj, λj,wj}j∈j and {vj,θj,vj}j∈j are k−g−fusion woven for h with universal frame bounds ( 1 2 −λ)a1 −µ and b1 + b2. proof. the upper frame bound is clear. for the lower frame bound, assume that σ ⊂ j and we get,by the arithmetic-quadratic mean, for any f ∈ h ∑ j∈σ w2j 〈λjpwjf , λjpwjf 〉 + ∑ j∈σc v2j 〈θjpvjf ,θjpvjf 〉 = ∑ j∈σ w2j 〈λjpwjf , λjpwjf 〉 + ∑ j∈σc 〈wjλjpwjf − (wjλjpwj −vjθjpvj )f ,wjλjpwjf − (wjλjpwj −vjθjpvj )f 〉 ≥ ∑ j∈σ w2j 〈λjpwjf , λjpwjf 〉 + 1 2 ∑ j∈σc w2j 〈λjpwjf , λjpwjf 〉 − ∑ j∈σc 〈(wjλjpwj −vjθjpvj )f , (wjλjpwj −vjθjpvj )f 〉 = 1 2 ∑ j∈j w2j 〈λjpwjf , λjpwjf 〉 + 1 2 ∑ j∈σ w2j 〈λjpwjf , λjpwjf 〉 − ∑ j∈σc 〈(wjλjpwj −vjθjpvj )f , (wjλjpwj −vjθjpvj )f 〉 ≥ 1 2 ∑ j∈j w2j 〈λjpwjf , λjpwjf 〉− ∑ j∈σc 〈(wjλjpwj −vjθjpvj )f , (wjλjpwj −vjθjpvj )f 〉 ≥ 1 2 ∑ j∈j w2j 〈λjpwjf , λjpwjf 〉−λ ∑ j∈j w2j 〈λjpwjf , λjpwjf 〉−µ〈k ∗f ,k∗f 〉 ≥ ( ( 1 2 −λ )a1 −µ ) 〈k∗f ,k∗f 〉. this completes the proof. � declarations availablity of data and materialsnot applicable. https://doi.org/10.28924/ada/ma.3.11 eur. j. math. anal. 10.28924/ada/ma.3.11 11 human and animal rightswe would like to mention that this article does not contain any studies with animals and does notinvolve any studies over human being. competing intereston behalf of all authors, the corresponding author states that there is no conflict of interest. fundingsauthors declare that there is no funding available for this article. authors’ contributionsthe authors equally conceived of the study, participated in its design and coordination, drafted themanuscript, participated in the sequence alignment, and read and approved the final manuscript. references [1] a. alijani, m. dehghan, ∗-frames in hilbert c∗modules, u.p.b. sci. bull., ser. a, 73 (2011), 89-106.[2] lj. arambašić, on frames for countably generated hilbert c∗-modules, proc. amer. math. soc. 135 (2007) 469-478. https://doi.org/10.1090/s0002-9939-06-08498-x.[3] r.j. duffin, a.c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952), 341–366. https://doi.org/10.1090/s0002-9947-1952-0047179-6.[4] m. frank, d.r. larson, a-module frame concept for hilbert c∗-modules, 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https://doi.org/10.2307/2372552 https://doi.org/10.1142/s0219691308002458 https://doi.org/10.28924/2291-8639-19-2021-836 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.2478/aupcsm-2018-0002 https://doi.org/10.52547/ijmsi.17.1.1 https://doi.org/10.52547/ijmsi.17.1.1 https://doi.org/10.1007/s11785-022-01229-4 1. introduction 2. woven k-g-fusion frames in hilbert c-modules declarations references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 34-44doi: 10.28924/ada/ma.1.34 katugampola fractional calculus with generalized k−wright function ahmad y. a. salamooni1,∗ , d. d. pawar2 1department of mathematics, faculty of education zabid, hodeidah university, al-hodeidah, yemen ayousss83@gmail.com 2school of mathematical sciences, swami ramanand teerth marathwada university, nanded-431606, india dypawar@yahoo.com ∗correspondence: ayousss83@gmail.com abstract. in this article, we present some properties of the katugampola fractional integrals andderivatives. also, we study the fractional calculus properties involving katugampola fractional inte-grals and derivatives of generalized k−wright function nφkm(z). 1. introduction and preliminaries in recent years, researchers have introduced new fractional integral and differential operatorswhich are generalizations of the famous definitions of riemann-liouville, caputo, hadamard, hilfer,etc. they have made a qualitative contribution to fractional differential equations. for more details,see [1, 5-7, 9-14] and references therein. definition 1.1. [9] let ω = [a,b], the katugampola fractional integrals ρiγ0+ϕ and ρiγ−ϕ of order γ ∈c(r(γ) > 0) are defined for ρ > 0, a = 0 and b = ∞ as (ρi γ 0+ϕ)(s) = ρ1−γ γ(γ) ∫ s 0 τρ−1ϕ(τ) (sρ −τρ)1−γ dτ (s > 0), (1.1) and (ρi γ −ϕ)(s) = ρ1−γ γ(γ) ∫ ∞ s τρ−1ϕ(τ) (τρ − sρ)1−γ dτ (s > 0), (1.2) the corresponding katugampola fractional derivatives ρdγ0+ϕ and ρdγ−ϕ are defined with (n = 1 + [r(γ)] ) as (ρd γ 0+ϕ)(s) := ( s1−ρ d ds )1+[r(γ)]( ρ i 1−γ+[r(γ)] 0+ ϕ ) (s) received: 30 aug 2021. key words and phrases. katugampola fractional integral and derivative; k−gamma function; k−wright function.34 https://adac.ee https://doi.org/10.28924/ada/ma.1.34 https://orcid.org/0000-0001-8227-3093 https://orcid.org/0000-0001-8986-5243 eur. j. math. anal. 1 (2021) 35 = ργ−[r(γ)] γ(1 −γ + [r(γ)]) ( s1−ρ d ds )1+[r(γ)] ∫ s 0 τρ−1ϕ(τ) (sρ −τρ)γ−[r(γ)] dτ (s > 0), (1.3) and (ρd γ −ϕ)(s) := ( − s1−ρ d ds )1+[r(γ)]( ρ i 1−γ+[r(γ)] − ϕ ) (s) = ργ−[r(γ)] γ(1 −γ + [r(γ)]) ( − s1−ρ d ds )1+[r(γ)] ∫ ∞ s τρ−1ϕ(τ) (τρ − sρ)γ−[r(γ)] dτ (s > 0). (1.4) definition 1.2. [2] the generalized k−gamma function γk(y) is defined by γk(y) = lim n→∞ n!kn(nk) y k −1 (y)n,k (k > 0; y ∈c\kz−), (1.5) where (y)n,k is the k−pochhammer symbol given as (y)n,k :=  γk (y+nk) γk (y) (k ∈r; y ∈c\{0}) y(y + k)(y + 2k)...(y + (n− 1)k) (n ∈n+; y ∈c) (1.6) and for r(y) > 0, the k−gamma function γk(y) is defined by the integral γk(y) = ∫ ∞ 0 xy−1e− xk k dx. (1.7) this gives a relation with euler’s gamma function as γk(y) = k y k −1γ( y k ). (1.8) also, in [8], we have γ(1 −y)γ(y) = π sin(yπ) . (1.9) definition 1.3. [14] the beta function b(υ,ω) is defined as b(υ,ω) = ∫ 1 0 zυ−1(1 −z)ω−1dz, r(υ) > 0, r(ω) > 0, = γ(υ)γ(ω) γ(υ + ω) (1.10) furthermore, we have∫ ∞ x̂ (z − x̂)υ−1(z − ŷ)ω−1dz = (x̂ − ŷ)υ+ω−1b(υ, 1 −υ −ω), x̂ > ŷ, 0 < r(υ) < 1 −r(ω). (1.11) recently, the generalized k−wright function introduced by (gehlot and prajapati [3]) is definedas follows: eur. j. math. anal. 1 (2021) 36 definition 1.4. for k ∈ r+; z ∈ c; pi,qj ∈ c, αi,βj ∈ r (αi,βj 6= 0; i = 1, 2, ...,n; j = 1, 2, ...,m) and (pi + αir), (qj + βjr) ∈ c \ kz−, the generalized k−wright function nφkm isdefined by nφ k m(z) = nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣z] = ∞∑ r=0 ∏n i=1 γk(pi + αir)∏m j=1 γk(qj + βjr) zr r! , (1.12) with the convergence conditions described as ∆ = m∑ j=1 (βj k ) − n∑ i=1 (αi k ) ; µ = n∏ i=1 ∣∣αi k ∣∣−αik m∏ j=1 ∣∣βj k ∣∣βjk ; ν = m∑ j=1 (qj k ) − n∑ i=1 (pi k ) + n−m 2 . lemma 1.1. [3] for k ∈ r+; z ∈ c; pi,qj ∈ c, αi,βj ∈ r (αi,βj 6= 0; i = 1, 2, ...,n; j = 1, 2, ...,m) and (pi + αir), (qj + βjr) ∈c\kz− (1) if ∆ > −1, then series (1.12) is absolutely convergent for all z ∈c and generalized k−wrightfunction nφkm(z) is an entire function of z. (2) if ∆ = −1, then series (1.12) is absolutely convergent for all |z| < µ and of |z| = µ,r(µ) > 1 2 . 2. properties of katugampola fractional integral and derivative in this section, we investigate some properties of the katugampola fractional integrals andderivatives (1.1), (1.2) and (1.3), (1.4) for the power function ϕ(s) = sα−1 and the exponentialfunction e−λ sρ. lemma 2.1. let ρ > 0,r(γ) = 0 and n = 1 + [r(γ)] (1) if r(α) > 0, then (ρi γ 0+τ α−1)(s) = ρ−γγ(1 + α−1 ρ ) γ(1 + α−1 ρ + γ) sργ+(α−1) (r(γ) ≥ 0; r(α) > 0) (2.1) (ρd γ 0+τ α−1)(s) = ργ−nγ(1 + α−1 ρ ) γ(1 + α−1 ρ −γ) s(α−1)−ργ (r(γ) = 0; r(α) > 0). (2.2) (2) if α ∈c, then (ρi γ −τ α−1)(s) = ρ−γγ( 1−α ρ −γ) γ( 1−α ρ ) sργ+(α−1) (r(γ) ≥ 0; r(γ + α) < 1) (2.3) (ρd γ −τ α−1)(s) = ργ−nγ( 1−α ρ + γ) γ( 1−α ρ ) s(α−1)−ργ (r(γ) = 0; r(γ + α− [r(γ)]) < 1). (2.4) (3) if r(λ) > 0, then (ρi γ −e −λτρ)(s) = (λρ)−γe−λ s ρ (r(γ) ≥ 0) (2.5) eur. j. math. anal. 1 (2021) 37 (ρd γ −e −λτρ)(s) = (λρ)γe−λ s ρ (r(γ) = 0). (2.6) proof. to prove this lemma, let the substitution x = τρ sρ in parts (1) and (2). (1) firstly, by the equation (1.1) and the given substitution, we have (ρi γ 0+τ α−1)(s) = ρ−γsργ+α−1 γ(γ) ∫ 1 0 x α−1 ρ (1 −x)1−γ dx = ρ−γsργ+α−1 γ(γ) b ( γ, 1 + α− 1 ρ ) . now, using equation (1.10), we obtain the result (2.1).secondly, by the equation (1.3), the given substitution and by using the result (2.1), we have (ρd γ 0+τ α−1)(s) = ( s1−ρ d ds )n( ρ i n−γ 0+ τ α−1)(s) = ργ−nγ(1 + α−1 ρ ) γ(1 + α−1 ρ + n−γ) ( s1−ρ d ds )n sρ(n−γ)+α−1 = ργ−nγ(1 + α−1 ρ ) γ(1 + α−1 ρ −γ) s(α−1)−ργ. (2) firstly, by the equation (1.2) and the given substitution, we have (ρi γ −τ α−1)(s) = ρ−γsργ+α−1 γ(γ) ∫ ∞ 1 x α−1 ρ (x − 1)γ−1dx. now, using the equation (1.11) with x̂ = 1 and ŷ = 0, we obtain (ρi γ −τ α−1)(s) = ρ−γsργ+α−1 γ(γ) b ( γ, 1 −γ − (1 + α− 1 ρ ) ) . by using equation (1.10), we obtain the result (2.3).secondly, by the equation (1.4), the given substitution and by using the result (2.3), we have (ρd γ −τ α−1)(s) = ( − s1−ρ d ds )n( ρ i n−γ − τ α−1)(s) = (−1)nργ−nγ( 1−α ρ + γ −n) γ( 1−α ρ ) ( s1−ρ d ds )n sρ(n−γ)+α−1 = (−1)nργ−n γ( 1−α ρ ) γ( 1−α ρ + γ −n)γ(1 − [ 1−α ρ + γ −n]) γ(1 − [γ − α−1 ρ ]) . (2.7) also, by using (1.9), we have γ( 1 −α ρ + γ −n)γ(1 − [ 1 −α ρ + γ −n]) = π sin([ 1−α ρ + γ −n]π) = (−1)nπ sin([γ − α−1 ρ ]π) (2.8) and 1 γ(1 − [γ − α−1 ρ ]) = γ(γ − α−1 ρ ) γ(γ − α−1 ρ )γ(1 − [γ − α−1 ρ ]) = γ(γ − α−1 ρ ) π sin([γ − α− 1 ρ ]π) (2.9) substituting relations (2.8) and (2.9) in (2.7), we obtain (2.4). eur. j. math. anal. 1 (2021) 38 (3) for this part, let the substitution x = τρ − sρ.firstly, by the equation (1.2) and the given substitution in this part, we have (ρi γ −e −λτρ)(s) = ρ−γ γ(γ) e−λ s ρ ∫ ∞ 0 e−λ xxγ−1dx, then by use the substitution ϑ = λ x, we obtain (ρi γ −e −λτρ)(s) = ρ−γ γ(γ) e−λ s ρ λ−γ ∫ ∞ 0 e−ϑϑγ−1dϑ, since ∫∞ 0 e−ϑϑγ−1dϑ = γ(γ) [8], then the result is satisfied.secondly, by the equation (1.4) and by using the result (2.5), we have (ρd γ −e −λτρ)(s) = ( − s1−ρ d ds )n( ρ i n−γ − e −λτρ)(s) = (−1)n ( s1−ρ d ds )n( (λρ)γ−ne−λ s ρ) = (−1)n s(1−ρ)n (λρ)γ−n ( dn dsn e−λ s ρ) = (λρ)γe−λ s ρ . � remark 2.1. (a) in lemma 2.1, if the power function is ϕ(s) = (sρ ρ )α−1 , then (1) if r(α) > 0, then( ρi γ 0+ (τρ ρ )α−1) (s) = γ(α) γ(α + γ) (sρ ρ )α+γ−1 (r(γ) ≥ 0; r(α) > 0) ( ρd γ 0+ (τρ ρ )α−1) (s) = γ(α) γ(α−γ) (sρ ρ )α−γ−1 (r(γ) = 0; r(α) > 0). (2) if α ∈c, then( ρi γ − (τρ ρ )α−1) (s) = γ(1 −γ −α) γ(1 −α) (sρ ρ )α+γ−1 (r(γ) ≥ 0; r(γ + α) < 1) ( ρd γ − (τρ ρ )α−1) (s) = γ(1 + γ −α) γ(1 −α) (sρ ρ )α−γ−1 (r(γ) = 0; r(γ + α− [r(γ)]) < 1). (b) if r(α) > r(γ) > 0, then (ρi γ −τ −α)(s) = ρ−γγ(α ρ −γ) γ(α ρ ) sργ−α. (2.10) eur. j. math. anal. 1 (2021) 39 3. katugampola fractional integration for generalized k−wright function in this section, we establish the katugampola fractional integration for generalized k−wrightfunction (1.12). theorem 3.1. let γ, α ∈ c such that r(γ) > 0, r(α) > 0; λ ∈ c, ρ > 0, ν > 0, then for ∆ > −1, the katugampola fractional integration ρiγ0+ for generalized k−wright function nφkm(z)is given as( ρi γ 0+ ( τ α k −1 nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ νk ]))(s) = ( k ρ )γ s α k +ργ−1 n+1φ k m+1 [ ( pi,αi ) 1,n , ( 1 ρ (α + (ρ− 1)k), ν ρ )( qj,βj ) 1,m , ( 1 ρ (α + (ρ(γ + 1) − 1)k), ν ρ )∣∣∣∣∣ λ s νk ] . (3.1) proof. according to lemma 1.1, a generalized k−wright function in both sides of the equation (3.1)exists for s > 0. we consider that m ≡ ( ρi γ 0+ ( τ α k −1 nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ νk ]))(s). using (1.12), we can write the above equation as m ≡ ( ρi γ 0+ ( τ α k −1 ∞∑ r=0 ∏n i=1 γk(pi + αir)∏m j=1 γk(qj + βjr) (λ τ ν k )r r! )) (s). now, using the integration of the series term by term, we obtain m ≡ ∞∑ r=0 ∏n i=1 γk(pi + αir)∏m j=1 γk(qj + βjr) (λ)r r! ( ρi γ 0+ ( τ α k + νr k −1 )) (s). applying (2.1), the above equation is reduced to m ≡ ∞∑ r=0 ∏n i=1 γk(pi + αir)∏m j=1 γk(qj + βjr) (λ)r r! ρ−γγ(1 + α k + νr k −1 ρ ) γ(1 + α k + νr k −1 ρ + γ) s α+νr k +ργ−1. using (1.8), we obtain m ≡ ( k ρ )γ s α k +ργ−1 n+1φ k m+1 [ ( pi,αi ) 1,n , ( 1 ρ (α + (ρ− 1)k), ν ρ )( qj,βj ) 1,m , ( 1 ρ (α + (ρ(γ + 1) − 1)k), ν ρ )∣∣∣∣∣ λ s νk ] . � theorem 3.2. let γ, α ∈ c such that r(γ) > 0, r(α) > 0; λ ∈ c, ρ > 0, ν > 0, then for ∆ > −1, the katugampola fractional integration ρiγ− for generalized k−wright function nφkm(z) isgiven as ( ρi γ − ( τ− α k nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ−νk ]))(s) eur. j. math. anal. 1 (2021) 40 = ( k ρ )γ sργ− α k n+1φ k m+1 [( pi,αi ) 1,n , ( α ρ −kγ, ν ρ )( qj,βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s−νk ] . (3.2) proof. according to lemma 1.1, a generalized k−wright function in both sides of the equation (3.2)exists for s > 0. we consider that n ≡ ( ρi γ − ( τ− α k nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ−νk ]))(s). using (1.12), we can write the above equation as n ≡ ∞∑ r=0 ∏n i=1 γk(pi + αir)∏m j=1 γk(qj + βjr) (λ)r r! ( ρi γ − ( τ− α+νr k )) (s). applying (2.10), the above equation is reduced to n ≡ ∞∑ r=0 ∏n i=1 γk(pi + αir)∏m j=1 γk(qj + βjr) (λ)r r! ρ−γγ( α+νr k ρ −γ) γ( α+νr k ρ ) sργ− α+νr k . using (1.8), we obtain n ≡ ( k ρ )γ sργ− α k n+1φ k m+1 [( pi,αi ) 1,n , ( α ρ −kγ, ν ρ )( qj,βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s−νk ] . � 4. katugampola fractional differentiation for generalized k−wright function this section deals with the katugampola fractional differentiation for generalized k−wrightfunction (1.12). theorem 4.1. let γ, α ∈ c such that r(γ) > 0, r(α) > 0; λ ∈ c, ρ > 0, ν > 0, thenfor ∆ > −1, the katugampola fractional differentiation ρdγ0+ for generalized k−wright function nφ k m(z) is given as( ρd γ 0+ ( τ α k −1 nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ νk ]))(s) = ( k ρ )−γ s α k −ργ−1 n+1φ k m+1 [ ( pi,αi ) 1,n , ( 1 ρ (α + (ρ− 1)k), ν ρ )( qj,βj ) 1,m , ( 1 ρ (α + (ρ(1 −γ) − 1)k), ν ρ )∣∣∣∣∣ λ s νk ] . (4.1) proof. according to lemma 1.1, a generalized k−wright function in both sides of the equation (4.1)exists for s > 0. let n = 1 + [r(γ)]. then, we consider that p ≡ ( ρd γ 0+ ( τ α k −1 nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ νk ]))(s). eur. j. math. anal. 1 (2021) 41 using (1.3), we have p ≡ ( s1−ρ d ds )n( ρi n−γ 0+ ( τ α k −1 nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ νk ]))(s). using theorem 3.1, we obtain p ≡ ( s1−ρ d ds )n( ( k ρ )n−γ s α k +ρ(n−γ)−1 n+1φ k m+1 [ ( pi,αi ) 1,n , ( 1 ρ (α + (ρ− 1)k), ν ρ )( qj,βj ) 1,m , ( 1 ρ (α + (ρ(n−γ + 1) − 1)k), ν ρ )∣∣∣∣∣ λ s νk ]) . using (1.12), we can write the above equation as p ≡ ( k ρ )n−γ ∞∑ r=0 ∏n i=1 γk(pi + αir)γk( 1 ρ (α + (ρ− 1)k) + ν ρ r)∏m j=1 γk(qj + βjr)γk( 1 ρ (α + (ρ(n−γ + 1) − 1)k) + ν ρ r) (λ)r r! ( s1−ρ d ds )n( s α k + ν k +ρ(n−γ)−1). also, the above equation can be written as p ≡ kn−γ ργ ∞∑ r=0 ∏n i=1 γk(pi + αir)γk( 1 ρ (α + (ρ− 1)k) + ν ρ r)∏m j=1 γk(qj + βjr)γk( 1 ρ (α + (ρ(n−γ + 1) − 1)k) + ν ρ r) (λ)r r! × γ( 1 ρ (α k + νr k + (n−γ)ρ + ρ− 1) γ( 1 ρ (α k + νr k −γρ + ρ− 1) s α k + ν k −ργ−1. using (1.8), we obtain p ≡ ( k ρ )−γ s α k −ργ−1 n+1φ k m+1 [ ( pi,αi ) 1,n , ( 1 ρ (α + (ρ− 1)k), ν ρ )( qj,βj ) 1,m , ( 1 ρ (α + (ρ(1 −γ) − 1)k), ν ρ )∣∣∣∣∣ λ s νk ] . � theorem 4.2. let γ, α ∈c such that r(γ) > 0, r(α) > 1+[r(γ)]−r(γ); λ ∈c, ρ > 0, ν > 0, then for ∆ > −1, the katugampola fractional differentiation ρdγ− for generalized k−wrightfunction nφkm(z) is given as( ρd γ − ( τ− α k nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ−νk ]))(s) = ( k ρ )−γ s−ργ− α k n+1φ k m+1 [( pi,αi ) 1,n , ( α ρ + kγ, ν ρ )( qj,βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s−νk ] (4.2) proof. according to lemma 1.1, a generalized k−wright function in both sides of the equation (4.2)exists for s > 0. let n = 1 + [r(γ)]. then, we consider that q ≡ ( ρd γ − ( τ− α k nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ−νk ]))(s). using (1.4), we have q ≡ ( − s1−ρ d ds )n( ρi n−γ − ( τ− α k nφ k m [ (pi,αi )1,n (qj,βj)1,m ∣∣∣ λ τ−νk ]))(s). eur. j. math. anal. 1 (2021) 42 using theorem 3.2, we obtain q ≡ ( − s1−ρ d ds )n ( k ρ )n−γ sρ(n−γ)− α k n+1φ k m+1 [( pi,αi ) 1,n , ( α ρ −k(n−γ), ν ρ )( qj,βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s−νk ] . using (1.12), we can write the above equation as q ≡ (−1)n( k ρ )n−γ ∞∑ r=0 ∏n i=1 γk(pi + αir)γk( α ρ − (n−γ)k + ν ρ r)∏m j=1 γk(qj + βjr)γk( α ρ + ν ρ r) (λ)r r! ( s1−ρ d ds )n( sρ(n−γ)− α k −ν k ) . on simplifying the above equation, we obtain q ≡ (−1)nkn−γργ ∞∑ r=0 ∏n i=1 γk(pi + αir)γk( α ρ − (n−γ)k + ν ρ r)∏m j=1 γk(qj + βjr)γk( α ρ + ν ρ r) (λ)r r! × γ(1 + (n−γ) − α ρk − ν ρk r) γ(1 −γ − α ρk − ν ρk r) ( s−ργ− α k −ν k ) . using (1.8), we obtain q ≡ (−1)nργ ∞∑ r=0 ∏n i=1 γk(pi + αir)∏m j=1 γk(qj + βjr)γ( α ρk + ν ρk r) (λ)r r! × γ(γ −n + α ρk + ν ρk r)γ(1 − (γ −n + α ρk + ν ρk r)) γ(1 − (γ + α ρk + ν ρk r)) ( s−ργ− α k −ν k ) . (4.3) using (1.9), we have γ(γ −n + α ρk + ν ρk r)γ(1 − (γ −n + α ρk + ν ρk r)) = π sin[(γ + α ρk + ν ρk r)π −nπ] = π sin[(γ + α ρk + ν ρk r)π] cos(nπ) = (−1)nπ sin[(γ + α ρk + ν ρk r)π] (4.4) and 1 γ(1 − (γ + α ρk + ν ρk r)) = γ(γ + α ρk + ν ρk r) sin[(γ + α ρk + ν ρk r)π] π . (4.5) substituting (4.4) and (4.5) in (4.3) and finally by using (1.8), we obtain q ≡ ( k ρ )−γ s−ργ− α k n+1φ k m+1 [( pi,αi ) 1,n , ( α ρ + kγ, ν ρ )( qj,βj ) 1,m , ( α ρ , ν ρ ) ∣∣∣∣∣ λ s−νk ] . � eur. j. math. anal. 1 (2021) 43 5. concluding remarks • if ρ = 1, thentheorems 3.1, 3.2, 4.1 and 4.2, are reduced to theorems 2, 3, 4 and 5 respectively(see [4]). • some general properties of the katugampola fractional integrals and derivatives for thepower function ϕ(s) = sα−1 and the exponential function e−λ sρ are investigated. • the katugampola fractional integration ρiγ0+ and ρiγ− for generalized k−wright function nφ k m(z) are established. • the katugampola fractional differentiation ρdγ0+ and ρdγ− for generalized k−wright func-tion nφkm(z) are established. acknowledgment the authors are would like to thank the reviewers for their important remarks and suggestions. references [1] r. almeida, a.b. malinowska, t. odzijewicz, fractional differential equations with dependence on the 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and stability results for hilfer-katugampola-type fractional implicit dif-ferential equations with nonlocal conditions, j. nonlinear sci. appl. 14 (3) (2021), 124-138. http://dx.doi.org/ 10.22436/jnsa.014.03.02.[14] a.y.a. salamooni, d.d. pawar, existence and uniqueness of nonlocal boundary conditions for hilfer-hadamard-type fractional differential equations, adv. differ. equations, 2021 (2021), 198. https://doi.org/10.1186/ s13662-021-03358-0.[15] s.g. samko, a.a. kilbas, o.i. marichev, fractional integrals and derivatives: theory and applications, gordon andbreach, new york (1993). http://dx.doi.org/10.22436/jnsa.014.03.02 http://dx.doi.org/10.22436/jnsa.014.03.02 https://doi.org/10.1186/s13662-021-03358-0 https://doi.org/10.1186/s13662-021-03358-0 1. introduction and preliminaries 2. properties of katugampola fractional integral and derivative 3. katugampola fractional integration for generalized k-wright function 4. katugampola fractional differentiation for generalized k-wright function 5. concluding remarks acknowledgment references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 22doi: 10.28924/ada/ma.3.22 local stability analysis of onchocerciasis transmission dynamics with nonlinear incidence functions in two interacting populations k. m. adeyemo department of mathematics, hallmark university ijebu-itele, ogun state, nigeria ∗correspondence: mikyade2019@gmail.com abstract. a deterministic compartmental model for the transmission dynamics of onchocerciasis withnonlinear incidence functions in two interacting populations is studied. the model is qualitatively an-alyzed to investigate its local asymptotic behavior with respect to disease-free and endemic equilibria.it is shown, using routh-hurwitz criteria, that the disease-free equilibrium is locally asymptoticallystable when the associated basic reproduction number is less than the unity. when the basic repro-duction number is greater than the unity, we prove the existence of a locally asymptotically stableendemic equilibrium. 1. introduction onchocerciasis is one of the neglected tropical diseases caused by the parasite onchocercavolvulus, a filarial nematode [2]. the disease is transmitted from one person to another by repeatedbites of black flies. the disease is endemic in sub-saharan africa. many researchers have workedon many ways to reduce the spread of the disease. for instance, remme et al. [10] used skin snipsurvey in west africa to investigate the impact of controlling black flies by larviciding. plaisieret al. [9] used micro simulation model to determine the period required for combining annualivermectin treatment and vector control in the onchocerciasis control programme in west africa.alley et al. [1] used a computer simulation model to study prevention of onchocerciasis by usingmacrofilaricide which kills the adult worms. asha hassan & nyimvua shaban [3] investigated theeffects of four control strategies on the spread of the disease.in this paper, we consider onchocerciasis transmission dynamics with nonlinear incidence functions.the human population is sub-divided into four compartments and the vector population is sub-divided into three compartments. we show local asymptotic behaviour in disease-free and endemicequilibria. the rest of the paper is organized as follows: the description of the model and theorems received: 27 apr 2023. key words and phrases. basic reproduction number; diseases free equilibrium; onchocerciasis epidemic model; non-linear incidence function. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 2 on positivity of solutions are given in section 2 while section 3 is devoted to the proof local stabilitytheorems. 2. model description two interacting populations are considered; the humans and the black-flies populations. thehuman population is partitioned into four compartments: the susceptible human compartment; sh„ the exposed compartment; eh, the infectious human compartment; ih and the recoveredhuman compartment; rh. the black-fly population is partitioned into three compartments:susceptible vector; sv , the exposed vector compartment; ev and the infective vector compart-ment. the total human and vector populations at any given time, t, are respectively given by; n = sh(t) + eh(t) + ih(t) + rh(t) and v = sv (t) + ev (t) + iv (t). we assume that thetransmission of onchocerciaisis in susceptible hosts is only through contact with infectious vector.we also assume that susceptible vector becomes infectious as a result of contact with infectioushosts during blood meal. the population under study is assumed to be large enough to bemodelled deterministically. the following system of non-linear ordinary differential equations,with non-negative initial conditions, describes the dynamics of onchocerciaisis epidemics. dsh(t,xi ) dt = ψh(xi ) − ∑l i=0 δλh(xi )sh(t,xi )iv (t) 1+νh(xi )iv (t) −µh(xi )sh + w(xi )rh(t,xi )) deh(t,xi ) dt = ∑l i=0 δλh(xi )sh(t,xi )iv (t) 1+νh(xi )iv (t) − (αh(xi ) + µh(xi ))eh(t,xi ) dih(t,xi ) dt = ∑l i=0 αh(xi )eh − (r(xi ) + γh(xi ) + µh(xi ))ih(t,xi ) drh(t,xi ) dt = ∑l i=0 r(xi )ih − (µh(xi ) + w(xi ))rh(t,xi ) dsv dt = ψv − δλv (xi )sv (t)ih(xi,t) 1+νvih(xi,t) −µvsv (t) dev dt = δλv (xi )sv (t)ih(xi,t) 1+νvih(xi,t) − (αv + µv )ev (t) div dt = αvev (t) − (µv + γv )iv (t)  (2.1) subject to the following initial conditions: sh(0,xi ) = s0h(xi ),eh(0,xi ) = e0h(xi ), ih(0,xi ) = i0h(xi ),rh(0,xi ) = r0h(xi ) sm(0) = s0m,em(0) = e0m, im(0) = i0m (2.2) https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 3 symbols definitionss sh(t,xi ) number of susceptible humans at time t and discrete age xi eh(t,xi ) number of exposed humans at time t and discrete age xi ih(t,xi ) number of infectious humans at time t and discrete age xi rh(t,ai) number of recovered humans at time t and discrete age xi sv (t) number of susceptible black-flies at time t ev (t) number of exposed black-flies at time t iv (t) number of infectious black-flies at time t ψh(xi ) recruitment term of the susceptible humans at discrete age xi ψv recruitment term of the susceptible vectors δ biting rate of the vector λh(xi ) probability that a bite by an infectious vector results in transmission of disease to human at discrete age xi λv probability that a bite results in transmission of parasite to a susceptible vector µh(xi ) per capita death rate of humans at discrete age xi µv per capita death rate of vector γh(xi ) disease-induced death rate of humans at discrete age xi γv disease-induced death rate of vectors αh(xi ) per capita rate of progression of humans from the exposed state to the infectious state at discrete age xi αv per capita rate of progression of vectors from the exposed state to the infectious state r(xi ) per capita recovery rate for humans from the infectious state to the recovered state due to treatment at discrete age xi ω(xi ) per capita transition rate of recovered humans to the susceptible state at discrete age xi νh(xi ) humans disease-inhibiting factor at discrete age xi νv vectors disease-inhibiting factor model assumptionsthe formulation of the compartmental model is based on the following assumptions: 1. that all humans are born susceptible. that is, humans are liable to contract the disease.2. that the susceptible humans, when infected, becomes exposed humans who are not yetinfectious.3. that the exposed humans progress to become infectious only.4. that the infectious humans may either die naturally or as a result of the disease, and ifnot, they become recovered humans due to treatment.5. that the recovered humans become susceptible again.6. all black-flies are born susceptible.7. that the susceptible black-flies, when infected, becomes exposed black-flies who are notyet infectious.8. that the exposed black-flies progress to become infectious only.9. that the infectious black-flies remain infectious for life. that is, there is no recovered classfor black-fly population. 2.1. existence and positivity of solutions. in this section, we analyse the general properties ofthe system (2.1) with positive initial conditions. it describes the population dynamics both in humanand black-fly populations. the system is biologically relevant in the set given by ω = (sh(t,xi ),eh(t,xi ), ih(t,xi ),rh(t,xi )) ∈r4+ : nh ≤ l∑ i=0 ψh(xi ) µh(xi ) , (sv (t),ev (t), iv (t)) ∈r3+ : nv ≤ ψv µv https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 4 here, the following results are provided which guarantee that the model governed by system (2.1)is mathematically well-posed in a feasible region ω defined by: ω = ωh × ωv ⊂r4 ×r3 theorem 1:there exists a domain ω in which the solution set sh(t,xi ),eh(t,xi ), ih(t,xi ),rh(t,xi ),sv (t),ev (t), iv (t)is contained and bounded. proofif the total human population size is given by nh = sh(t,xi ) + eh(t,xi ) + ih(t,xi ) + rh(t,xi ), andthe total size of black-fly population is nv = sv (t) + ev (t) + iv (t). from model (2.1), we havethat dnh(t,xi ) dt ≤ ψh(xi ) − l∑ i=0 µh(xi )nh(t,xi ) (2.3) and dnv dt ≤ ψv −µvnv (2.4)it follows from (2.3) and (2.4) that nh(t,xi ) ≤ ψh(xi ) µh(xi ) [1 −e1−µh(xi )t]+nh(0,xi )e −µh(xi )t] and nv ≤ ψv µv [1 −e−µvt] + nv (0)e−µvt taking the lim sup as t → ∞ gives nh ≤ ψh(xi )µh(xi ) and nv ≤ ψvµv . this shows that all solu-tions of the humans population only are confined in the solution set ωh and all solutions of theblack-fly population are confined in ωv . it also suffices to say that ω is positively invariant as nh(t,xi ) ≤ ∑l i=0 ψh(xi ) µh(xi ) whenever nh(0,xi ) ≤ ψh(xi )µh(xi ) and nv (t) ≤ ψvµv if nv (0) ≤ ψvµv , therefore thesolution set for the model (2.1) exists and is given by ω = ωh × ωv ⊂r4+ ×r3+ 2it remains to show that the solutions of system (2.1) are nonnegative in ω for any time t > 0 sincethe variables represent human and black-fly populations. theorem 2:the solutions, sh(t,xi ), eh(t,xi ), ih(t,xi ), rh(t,xi ), sv (t), ev (t), iv (t), of model (2.1) with non-negative initial conditions in ω, remain nonnegative in ω for all t > 0. proof: given that the initial conditions, s0h(xi ), e0h(xi ), i0h(xi ), r0h(xi ), s0v,e0v,i0v , are non-negative and from (2.1), dsh(t,xi ) dt + l∑ i=0 [ bλh(xi )iv (t) 1 + νh(xi )iv (t) + µh(xi ) ] sh(t,xi ) ≥ 0 so that d dt [ l∑ i=0 sh(t,xi )exp (∫ t 0 bλh(xi )iv (η) 1 + νh(xi )iv (η) dη + µh(xi )t )] ≥ 0, (2.5) https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 5 integrating (2.5), we have l∑ i=0 sh(t,xi ) ≥ l∑ i=0 s0h(xi )exp [ − (∫ t 0 bλh(xi )iv (η) 1 + νh(xi )iv (η) dη + µh(xi )t )] ≥ 0, which implies that for all t > 0 and for all a ∈r+, we have sh(t,xi ) ≥ l∑ i=0 s0h(xi )exp [ − (∫ t 0 bλh(xi )iv (η) 1 + νh(xi )iv (η) dη + µh(xi )t )] ≥ 0. hence, sh(t,xi ) > 0 for any arbitrary xi . also, we have deh(t,xi ) dt + l∑ i=0 ((αh(xi ) + µh(xi )))eh(t,xi ) ≥ 0 so that d dt [ l∑ i=0 eh(t, (xi ))exp(αh(xi ) + µh(xi )t) ] ≥ 0 (2.6) integrating (2.6), we have for all t > 0 and for all a ∈ mathbbr+, that eh(t,a) ≥ l∑ i=0 e0h(xi )exp [−(αh(xi ) + µh(xi ))t] hence, eh(t,xi ) > 0 for any arbitrary xi also we have dih(t,xi ) dt ≥− l∑ i=0 (r(xi ) + γh(xi ) + µh(xi ))ih(t) so that d dt [ih(t)exp(r(xi ) + γh(xi ) + µh(xi ))t] ≥ 0 (2.7) similarly, (2.7) becomes ih(t,a) ≥ l∑ i=0 i0hexp [−(r(xi ) + γh(xi ) + µh(xi ))t] > 0for all t > 0 for all a ∈r+ hence, ih(t,xi ) > 0 for any arbitrary xi . also from (2.1), we have drh(t,xi ) dt + l∑ i=0 (µh(xi ) + w(xi ))rh(t,xi ) ≥ 0 and we have d dt [ l∑ i=0 rh(t,xi )exp((µh(xi ) + w(xi ))t ] ≥ 0 (2.8) integrating (2.8), we have, for all t > 0 and a ∈r, that rh(t,a) ≥ l∑ i=0 r0h(xi )exp(−(µh(xi ) + w(xi ))t) > 0 https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 6 hence, rh(t,xi ) > 0 for any arbitrary xi . in a similar manner, we have dsv dt + [ l∑ i=0 bλvih(t)) 1 + νvih(t) + µv ] sv (t) ≥ 0 so that d dt [ sv (t)exp (∫ t 0 bλvih(η)) 1 + νvih(η) d(η) + µvt )] ≥ 0 (2.9) integrating (2.9), we have sv (t) ≥ s0vexp [ − (∫ t 0 bλvih(η)) 1 + νvih(η) d(η) + µvt )] > 0 ∀ t > 0 also we have dev dt ≥−(αv + µv )ev (t) which on integration gives ev (t) ≥ ev (0)exp [−(αv + µv )t] > 0 ∀ t > 0 (2.10) and finally, we have div dt + (µv + γv )iv (t) so that d dt [iv (t)exp(µv + γv )t] ≥ 0 (2.11) and we have iv (t) ≥ iv (0)exp [−(µv + γv )t] > 0, ∀ t > 0 this completes the proof 2 3. existence and stability of the equilibrium points 3.1. disease-free equilibrium. the disease-free equilibrium (dfe) points are steady state solu-tions that depict the absence of infection in both the human host and black-fly vector populations,i.e, onchocerciasis does not exist in the population. thus, the disease-free equilibrium point, e0, forthe model (2.1) implies that s∗(xi )h 6= 0, e∗h(xi ) = i∗h = 0(xi ) = r∗h(xi ) = 0, s∗v 6= 0, ev = iv = 0and putting these into (2.1), we have s∗(xi )h = ψh(xi )µh(xi ) and s∗v = ψvµv . consequently we obtain e0as e0 = ( ψh(xi ) µh(xi ) , 0, 0, 0, ψv µv , 0, 0 ) (3.1) a key notion in the analysis of infectious disease models is the basic reproduction number r0 , anepidemiological threshold that determines whether disease dies out or persists in the population.thebasic reproduction number r0 of the system (2.1) is computed using the next generation matrixmethod and is given by r0 = √ rhrv https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 7 where rh = ∑li=0 δαhλh(xi )ψh(xi )µh(xi )(αh(xi )+µh(xi ))(r(xi )+γh(xi )+µh(xi )) and rv = δαvλv ψvµv (αv +µv )(γv +µv ) . the basicreproduction number r0, determines whether onchocerciasis dies out or persists in the population.therefore, rh describes the number of humans that one infectious black-fly infects over its expectedinfectious period in a completely susceptible humans population, while rv is the number of blac-flies infected by one infectious human during the period of infectiousness in a completely susceptibleblack-fly population. 3.2. local stability of the disease-free equilibrium point e0. using the basic reproduction num-ber obtained for the model (2.1), we analyse the stability of the equilibrium point in the followingresult. theorem 3:the disease-free equilibrium point, e0, is locally asymptotically stable if r0 < 1, and unstable if r0 > 1. proof: the jacobian matrix of the system (2.1) evaluated at the disease-free equilibrium point e0,is obtained as m(e0) =  m11 0 0 m14 0 0 m17 0 m22 0 0 0 0 m27 0 m32 m33 0 0 0 0 0 0 m43 m44 0 0 0 0 0 m53 0 m55 0 0 0 0 m63 0 0 m66 0 0 0 0 0 0 m76 m77  where m11 = −µh(xi ), m14 = w(a1), m17 = −∑li=0 δλh(xi )ψh(xi )µh(xi ) , m22 = −(αh(xi ) + µh(xi )), m27 = ∑l i=0 δλh(xi )ψh(xi ) µh(xi ) , m32 = αh(xi ), m33 = −(r(xi ) + γh(xi ) + µh(xi )), m43 = r(xi ), m44 = −(µh(xi ) + w(xi )), m53 = −δλv ψvµv , m55 = −µv , m63 = δλv ψvµv , m66 = −(αv + µv ), m76 = αv , m77 = −(µv + γv ) we need to show that all the eigenvalues of m(e0) are negative. as the firstand fifth columns form the two negative eigenvalues, h(xi ) and −v , the other five eigenvalues canbe obtained from the sub-matrix, m1(e0), formed by excluding the first and fifth rows and columnsof m(e0). hence m1(e0) =  m′11 0 0 0 m ′ 15 αh(xi ) m ′ 22 0 0 0 0 r(xi ) m ′ 33 0 0 0 0 δλv ψv µv 0 −(αv + µv ) 0 0 0 0 αv −(µv + λv )  in the same way, the third column of m1(e0) contains only the diagonal term which forms anegative eigenvalue, (µh(xi ) + w(xi )). the remaining four eigenvalues are obtained from the https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 8 sub-matrix m2(e0) given by m2(e0) =  m′′11 0 0 m ′′ 14 αh(xi ) m ′ 22 0 0 0 δλv ψv µv −(αv + µv ) 0 0 0 αv −(µv + λv )  thus, the eigenvalues of the matrix m2(e0) are the roots of the characteristic equation of the form (ξ + αh(xi ))(ξ + r(xi ) + γh(xi ) + µh(xi ))(ξ + µv + γ)− l∑ i=0 δ2αh(xi )λh(xi )ψh(xi )vλv ψv µh(xi )µv = 0 (3.2) if we let y1 = αh(xi ) + µh(xi ), y2 = r(xi ) + γh(xi ) + µh(xi ), y3 = αv + µv , and y4 = µv + γv , then(3.2) becomes x4ξ 4 + x3ξ 3 + x2ξ 2 + x1ξ + x0 = 0, (3.3) where x4 = 1 x3 = y1 + y2 + y3 + y4 x2 = (y1 + y2)(y2 + y4) + y1y2 + y3y4 x1 = (y1 + y2)y3y4 + (y3 + y4)y1y2 x0 = y1y2y3y4 − ∑l i=0 δ2αh(xi )λh(xi )ψh(xi )vλv ψv µh(xi )µv  (3.4) expressing x0 in terms of reproduction number r0, we have x0 = y1y2y3y4(1 −r20) (3.5) we can see from (3.4) that x1 > 0, x2 > 0, x3 > 0, x4 > 0, since all yis are positive. moreover,if r0 < 1, it follows from (3.5) that x0 > 0. thus, using the routh-hurwitz criterion, we have h1 = x3 > 0 h2 = ∣∣∣∣∣ x3 x4x1 x2 ∣∣∣∣∣ = y1(y2 + y3 + y4)(y1 + y2 + y3 + y4) + (y2 + y3)(y2 + y4)(y3 + y4) > 0similarly we have h3 > 0 and h4 > 0 where h3 = ∣∣∣∣∣∣∣∣ x3 x4 0 x1 x2 x3 0 x0 x1 ∣∣∣∣∣∣∣∣and h4 = ∣∣∣∣∣∣∣∣∣∣∣ x3 x4 0 0 x1 x2 x3 x4 0 x0 x1 x2 0 0 0 x0 ∣∣∣∣∣∣∣∣∣∣∣ theref ore,alltheeigenvaluesof thejacobianmatrixm(e0) have negative real parts when r0 < 1 and the disease-free equilibrium point is locally asymptotically stable. however, when r0 > 1,we see that x0 < 0 and there is one eigenvalue with positive real part and therefore the disease-free equilibrium point is unstable 2 https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 9 3.3. endemic equilibrium point ee. we shall show that the formulated model (2.1) has an endemicequilibrium point, ee. the endemic equilibrium point is a positive steady state solution where thedisease persists in the population. theorem 4: the model (2.1) has a unique endemic equilibrium ee whenever r0 > 1. proof: let ee = (s′′h(xi ),e′′h (xi ), i′′h (xi ),r′′h(xi ),s′′v ,e′′v , i′′v ) be a nontrivial equilibrium of the model(2.1). that is, all components of ee are positive. then the onchocerciasis model (2.1) at steady-statebecomes ψh(xi ) − l∑ i=0 ( δλh(xi )s ′′ h(xi )iv 1 + νh(xi )i ′′ v −µh(xi )s′h(xi ) + ω(xi )r ′′ h(xi ) ) = 0 (3.6) l∑ i=0 ( δλh(xi )s ′′ h(xi )iv 1 + νh(xi )i ′′ v − (αh(xi ) + µh(xi ))e′′h (xi ) ) = 0 (3.7) l∑ i=0 (αh(xi )e ′′ h (xi ) − (r(xi ) + µh(xi ) + γh(xi ))i ′′ h (xi )) = 0 (3.8) l∑ i=0 r(xi )i ′′ h (xi ) − (µh(xi ) + ω(xi ))r ′′ h(xi ) = 0 (3.9) ψv − δλvs ′′ vih(xi ) 1 + νv (xi )i ′′ h (xi ) −µvs′′v = 0 (3.10) δλvs ′′ vih(xi ) 1 + νv (xi )i ′′ h (xi ) − (αv + µv )e′′v = 0 (3.11) αve ′′ v − (µv + γv )i ′′ v = 0 (3.12)from the last three equations, we have i′′v = αve ′′ v µv + γv (3.13) e′′v = δλvs ′′ vih(xi ) 1 + νv (xi )i ′′ h (xi )(αv + µv ) (3.14) and s′′v = ψv δλvs′′vih(xi ) 1+νv (xi )i ′′ h (xi ) + µv (3.15) substituting (3.14) and (3.15) into (3.13) yields i′′v = rvµvi′′h (xi ) µv + (δλv + µvνv )i ′′ h (xi ) (3.16) from (3.8) and (3.9), we have e′′h (xi ) = l∑ i=0 (r(xi ) + µh(xi ) + γh(xi ))ih(xi ) αh(xi ) (3.17) and r′′h(xi ) = l∑ i=0 r(xi )i ′′(xi ) µh(xi ) + ω(xi ) (3.18) https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 10 if we put (3.16) and (3,17) in (3.7) in terms of r0, we have s′′h(xi ) = ∑l i=0 ψh(xi )[µv + (δλv + µvνv + νh(xi )µvrv )i ′′ h (xi )] µh(xi )µvr20 (3.19) finally, using (3.16), (3.18) and (3.19) in (3.7), we have i′′h (xi ) = l∑ i=0 µh(xi )µv ψh(xi )(µh(xi ) + ω(xi )) (r20 − 1) ρ (3.20) where ρ = l∑ i=0 (µh(xi )+ω(xi ))[δλh(xi )µvrv +ψh(xiµh(xi )(δλv +µvνv +νh(xi )µv )rm)]− l∑ i=0 µh(xi )µvω(xi )r(xi )r20. (3.21)if in (3.20), ω(xi ) = 0 then ρ > 0. from this, one sees that model (2.1) has no positive solutionwhen r0 < 1. however, with ω(xi ) = 0, a unique endemic equilibrium exists when r0 > 1. thiscompletes the proof. 2 remark 1: it is important to have a remark that positive solution exists for the model (2.1) in acase where ρ < 0 and r0 < 1. this implies that the disease-free equilibrium co-exists with theendemic equilibrium state when r0 is slightly less than unity resulting into a phenomenon ofsubcritical (backward) bifurcation. references [1] w.s. alley, b.a.b. boatin, n.j.d.n. nagelkerke, macrofilaricides and onchocerciasis control, mathematical modellingof the prospects for elimination. bmc public health. 1 (2001) 12.[2] u. amazigo, m. noma, j. bump, b. bentin, b. liese, l. yameogo, h. zouré, and a. seketeli, onchocerciasis diseaseand mortality in sub saharan africa, chapter 15, world bank, washington, dc, 2006.[3] a. hassan, n. shaban, onchocerciasis dynamics: modelling the effects of treatment, education and vector control,j. biol. dyn. 14 (2020) 245-268.[4] e.m. poolman, a.p. galvani, modeling targeted ivermectin treatment for controlling river blindness, amer. j. trop.med. hygiene, 75 (2006) 921–927.[5] j.p. mopecha, h.r. thieme, competitive dynamics in a model for onchocerciasis with cross-immunity, can. appl.math. quart. 11 (2003) 339–376.[6] m.g. basanez, m. boussinesq, population biology of human onchocerciasis, phil. trans. r. soc. lond. b: biol. sci.354 (1999) 809–826.[7] m.g. basanez, j. ricardez-esquinca, models for the population biology and control of human onchocerciasis, trendsparasitol. 17 (2001) 430–438.[8] j.d. murray, mathematical biology i., an introduction. 3rd ed. heidelberg: springer-verlag berlin, 2002.[9] a.p. plaisier, e.s. alley, g.j. van oortmarssen, b.a. boatin, j.d.f habbema, required duration of combined annualivermectin treatment and vector control program in west africa, bull. world health organ. 75 (1997) 237-245.[10] j. remme, g. de sole, g.j. van oortmarssen, the predicted and observed decline in onchocerciasis infection during14 years of successful control of black flies in west africa, bull. world health organ. 68 (1990) 331–339. https://doi.org/10.28924/ada/ma.3.22 eur. j. math. anal. 10.28924/ada/ma.3.22 11 [11] s.i. omade, a.t. omotunde, a.s. gbenga, mathematical modeling of river blindness disease with demography usingeuler method, math. theory model. 5 (2015), 75–85.[12] world health organization, african programme for onchocerciasis control: meeting of national onchocerciasis taskforces, september 2012, weekly epidemiol. record 87(49–50) (2012), pp. 494–502. https://doi.org/10.28924/ada/ma.3.22 1. introduction 2. model description 2.1. existence and positivity of solutions 3. existence and stability of the equilibrium points 3.1. disease-free equilibrium 3.2. local stability of the disease-free equilibrium point e0 3.3. endemic equilibrium point ee references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 12doi: 10.28924/ada/ma.3.12 coordination of classical and dynamic inequalities complying on time scales muhammad jibril shahab sahir department of mathematics and statistics, the university of lahore, lahore, pakistan correspondence: jibrielshahab@gmail.com abstract. in this research article, we present extensions of some classical inequalities such asschweitzer, pólya–szegö, kantorovich and greub–rheinboldt inequalities of fractional calculus ontime scales. to investigate generalizations of such types of classical inequalities, we use the timescales riemann–liouville type fractional integrals. we explore dynamic inequalities on delta calcu-lus and their symmetric nabla versions. a time scale is an arbitrary nonempty closed subset of thereal numbers. the theory of time scales is applied to combine results in one comprehensive form.the calculus of time scales unifies and extends continuous versions and their discrete and quantumanalogues. by using the calculus of time scales, results are presented in more general form. thishybrid theory is also widely applied on dynamic inequalities. 1. introduction the calculus of time scales was initiated by stefan hilger [11]. the three most popular examplesof calculus on time scales are differential calculus, difference calculus, and quantum calculus, i.e.,when t = r, t = n and t = qn0 = {qt : t ∈n0} where q > 1. the time scales calculus is studiedas delta calculus, nabla calculus and diamond-α calculus. during the last two decades, manyresearchers investigated several dynamic inequalities [1–4, 7, 16–18]. the basic work on dynamicinequalities is done by ravi agarwal, george anastassiou, martin bohner, allan peterson, donalo’regan, samir saker and many other authors.there have been recent achievements of the theory and applications of dynamic inequalities ontime scales. from the theoretical point of view, the study provides a harmonious reconciliation andextension of commonly known differential, difference and quantum equations. moreover, it is animportant tool in many computational, biological, economical and numerical applications.in this paper, it is assumed that all considerable integrals exist and are finite and t is a timescale, a,b ∈t with a < b and an interval [a,b]t means the intersection of a real interval with thegiven time scale. received: 22 aug 2021. key words and phrases. time scales; fractional riemann–liouville integral; schweitzer, pólya–szegö, kantorovichand greub–rheinboldt inequalities. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 2 2. preliminaries we need here basic concepts of delta calculus. the results of delta calculus are adopted frommonographs [7, 8].for t ∈t, the forward jump operator σ : t→t is defined by σ(t) := inf{s ∈t : s > t}. the mapping µ : t → r+0 = [0, +∞) such that µ(t) := σ(t) − t is called the forward graininessfunction. the backward jump operator ρ : t→t is defined by ρ(t) := sup{s ∈t : s < t}. the mapping ν : t→r+0 = [0, +∞) such that ν(t) := t −ρ(t) is called the backward graininessfunction. if σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. also, if t < supt and σ(t) = t, then t is called right-dense, and if t > inf t and ρ(t) = t, then t is called left-dense. if t has a left-scattered maximum m, then tk = t−{m},otherwise tk = t.for a function f : t→r, the delta derivative f ∆ is defined as follows:let t ∈ tk . if there exists f ∆(t) ∈ r such that for all � > 0, there is a neighborhood u of t,such that |f (σ(t)) − f (s) − f ∆(t)(σ(t) − s)| ≤ �|σ(t) − s|, for all s ∈ u, then f is said to be delta differentiable at t, and f ∆(t) is called the delta derivativeof f at t.a function f : t→r is said to be right-dense continuous (rd-continuous), if it is continuous ateach right-dense point and there exists a finite left-sided limit at every left-dense point. the setof all rd-continuous functions is denoted by crd(t,r).the next definition is given in [7, 8]. definition 2.1. a function f : t → r is called a delta antiderivative of f : t → r, provided that f ∆(t) = f (t) holds for all t ∈tk . then the delta integral of f is defined by∫ b a f (t)∆t = f (b) −f (a). the following results of nabla calculus are taken from [6–8].if t has a right-scattered minimum m, then tk = t−{m}, otherwise tk = t. a function f : tk → r is called nabla differentiable at t ∈ tk , with nabla derivative f∇(t), if there exists f∇(t) ∈r such that given any � > 0, there is a neighborhood v of t, such that |f (ρ(t)) − f (s) − f∇(t)(ρ(t) − s)| ≤ �|ρ(t) − s|, for all s ∈ v . https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 3 a function f : t→r is said to be left-dense continuous (ld-continuous), provided it is continuousat all left-dense points in t and its right-sided limits exist (finite) at all right-dense points in t.the set of all ld-continuous functions is denoted by cld(t,r).the next definition is given in [6–8]. definition 2.2. a function g : t→r is called a nabla antiderivative of g : t→r, provided that g∇(t) = g(t) holds for all t ∈tk . then the nabla integral of g is defined by∫ b a g(t)∇t = g(b) −g(a). the following definition is taken from [2, 4]. definition 2.3. for α ≥ 1, the time scale ∆-riemann–liouville type fractional integral for a function f ∈ crd is defined by iαa f (t) = ∫ t a hα−1(t,σ(τ))f (τ)∆τ, (1) which is an integral on [a,t)t, see [9] and hα : t × t → r, α ≥ 0 are the coordinate wiserd-continuous functions, such that h0(t,s) = 1, hα+1(t,s) = ∫ t s hα(τ,s)∆τ, ∀s,t ∈t. (2) notice that i1af (t) = ∫ t a f (τ)∆τ, which is absolutely continuous in t ∈ [a,b]t, see [9]. the following definition is taken from [3, 4]. definition 2.4. for α ≥ 1, the time scale ∇-riemann–liouville type fractional integral for a function f ∈ cld is defined by jαa f (t) = ∫ t a ĥα−1(t,ρ(τ))f (τ)∇τ, (3) which is an integral on (a,t]t, see [9] and ĥα : t × t → r, α ≥ 0 are the coordinate wiseld-continuous functions, such that ĥ0(t,s) = 1, ĥα+1(t,s) = ∫ t s ĥα(τ,s)∇τ, ∀s,t ∈t. (4) notice that j 1a f (t) = ∫ t a f (τ)∇τ, which is absolutely continuous in t ∈ [a,b]t, see [9]. https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 4 we will generalize the following classical inequalities [13] by using the calculus of time scales.first we consider the inequality given by schweitzer [19] such that( 1 p p∑ k=1 xk )( 1 p p∑ k=1 1 xk ) ≤ (m + m)2 4mm , (5) where 0 < m ≤ xk ≤ m for k = 1, . . . ,p.in the same paper, schweitzer has also shown that if functions y 7→ f (y) and y 7→ 1 f (y) areintegrable on [a,b] and 0 < m ≤ f (y) ≤ m on [a,b], then∫ b a f (y)dy ∫ b a 1 f (y) dy ≤ (m + m)2 4mm (b−a)2. (6) pólya and szegö [15] proved that( p∑ k=1 x2k )( p∑ k=1 y2k ) ( p∑ k=1 xkyk )2 ≤  √ mn mn + √ mn mn 2 2 , (7) where 0 < m ≤ xk ≤ m and 0 < n ≤ yk ≤ n for k = 1, . . . ,p.kantorovich [12] proved that( p∑ k=1 xky 2 k )( p∑ k=1 1 xk y2k ) ≤ 1 4 (√ m m + √ m m )2 ( p∑ k=1 y2k )2 , (8) where 0 < m ≤ xk ≤ m and yk ∈ r for k = 1, . . . ,p, and he pointed out that inequality (8) is aparticular case of (7).greub and rheinboldt [10] proved that( p∑ k=1 x2kz 2 k )( p∑ k=1 y2k z 2 k ) ≤ (mn + mn) 2 4mnmn ( p∑ k=1 xkykz 2 k )2 , (9) where 0 < m ≤ xk ≤ m, 0 < n ≤ yk ≤ n and zk ∈r for k = 1, . . . ,p with p∑ k=1 z2k < ∞. 3. main results in order to present our main results, first we give a simple proof for an extension of pólya–szegö’sinequality by using the time scale ∆-riemann–liouville type fractional integral. theorem 3.1. let w,f,g ∈ crd ([a,b]t,r−{0}) be ∆-integrable functions. assume that there exist four positive ∆-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]t,∀x ∈ [a,b]t). https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 5 let α,β ≥ 1 and hα−1(., .),hβ−1(., .) > 0. then we have the following inequality iαa ((f1f2)(x)|w(x)|)i β a ((g1g2)(x)|w(x)|)iαa ( |w(x)||f (x)|2 ) iβa ( |w(x)||g(x)|2 ){ iαa (f1(x)|(wf )(x)|)i β a (g1(x)|(wg)(x)|) + iαa (f2(x)|(wf )(x)|)i β a (g2(x)|(wg)(x)|) }2 ≤ 14. (10) proof. using the given conditions, for y,z ∈ [a,x]t, ∀x ∈ [a,b]t, we have( f2(y) g1(z) − |f (y)| |g(z)| ) ≥ 0, and ( |f (y)| |g(z)| − f1(y) g2(z) ) ≥ 0, which imply that ( f1(y) g2(z) + f2(y) g1(z) ) |f (y)| |g(z)| ≥ |f (y)|2 |g(z)|2 + f1(y)f2(y) g1(z)g2(z) . multiplying both sides by g1(z)g2(z)|g(z)|2, we have f1(y)g1(z)|f (y)g(z)| + f2(y)g2(z)|f (y)g(z)| ≥ g1(z)g2(z)|f (y)|2 + f1(y)f2(y)|g(z)|2. (11) multiplying both sides of (11) by hα−1(x,σ(y))|w(y)|hβ−1(x,σ(z))|w(z)| and double integratingover y and z from a to x, respectively, we have iαa (f1(x)|w(x)f (x)|)i β a (g1(x)|w(x)g(x)|) + i α a (f2(x)|w(x)f (x)|)i β a (g2(x)|w(x)g(x)|) ≥iαa ( |w(x)||f (x)|2 ) iβa (g1(x)g2(x)|w(x)|) + i α a (f1(x)f2(x)|w(x)|)i β a ( |w(x)||g(x)|2 ) . (12) applying the am-gm inequality √ζη ≤ ζ+η 2 , ζ ≥ 0, η ≥ 0, the inequality (12) takes the form iαa (f1(x)|w(x)f (x)|)i β a (g1(x)|w(x)g(x)|) + i α a (f2(x)|w(x)f (x)|)i β a (g2(x)|w(x)g(x)|) ≥ 2 √ iαa (|w(x)||f (x)|2)i β a (g1(x)g2(x)|w(x)|)iαa (f1(x)f2(x)|w(x)|)i β a (|w(x)||g(x)|2). (13) inequality (13) directly yields inequality (10). the proof of theorem 3.1 is completed. � now, we give an extension of pólya–szegö’s inequality by using the time scale ∇-riemann–liouville type fractional integral. theorem 3.2. let w,f,g ∈ cld ([a,b]t,r−{0}) be ∇-integrable functions. assume that there exist four positive ∇-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]t,∀x ∈ [a,b]t). let α,β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. then we have the following inequality jαa ((f1f2)(x)|w(x)|)j β a ((g1g2)(x)|w(x)|)jαa ( |w(x)||f (x)|2 ) jβa ( |w(x)||g(x)|2 ){ jαa (f1(x)|(wf )(x)|)j β a (g1(x)|(wg)(x)|) + jαa (f2(x)|(wf )(x)|)j β a (g2(x)|(wg)(x)|) }2 ≤ 14. (14) https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 6 proof. similar to the proof of theorem 3.1. � corollary 3.3. let w,f,g ∈ crd ([a,b]t,r−{0}) be ∆-integrable functions such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a,x]t, ∀x ∈ [a,b]t. let α,β ≥ 1 and hα−1(., .),hβ−1(., .) > 0. then we have the following inequality iαa (|w(x)|)i β a (|w(x)|)iαa ( |w(x)||f (x)|2 ) iβa ( |w(x)||g(x)|2 ){ iαa (|(wf )(x)|)i β a (|(wg)(x)|) }2 ≤ 14 (√ mn mn + √ mn mn )2 . (15) proof. putting f1 = m, f2 = m, g1 = n and g2 = n in theorem 3.1, we get the inequality (15). � corollary 3.4. let w,f,g ∈ cld ([a,b]t,r−{0}) be ∇-integrable functions such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a,x]t, ∀x ∈ [a,b]t. let α,β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. then we have the following inequality jαa (|w(x)|)j β a (|w(x)|)jαa ( |w(x)||f (x)|2 ) jβa ( |w(x)||g(x)|2 ){ jαa (|(wf )(x)|)j β a (|(wg)(x)|) }2 ≤ 14 (√ mn mn + √ mn mn )2 . (16) proof. similar to the proof of corollary 3.3. � remark 3.1. let t = r, α,β > 0, a = 0, x > 0, w ≡ 1, f > 0 and g > 0. then (10) reduces to iα0 ((f1f2)(x))i β 0 ((g1g2)(x))i α 0 ( f 2(x) ) iβ0 ( g2(x) ){ iα0 ((f1f )(x))i β 0 ((g1g)(x)) + i α 0 ((f2f )(x))i β 0 ((g2g)(x)) }2 ≤ 14, (17) as given in [14]. theorem 3.5. let w,f,g ∈ crd ([a,b]t,r−{0}) be ∆-integrable functions. assume that there exist four positive ∆-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]t,∀x ∈ [a,b]t). let α,β ≥ 1 and hα−1(., .),hβ−1(., .) > 0. then we have the following inequality iαa ( |w(x)||f (x)|2 ) iβa ( |w(x)||g(x)|2 ) ≤iαa ( f2(x) g1(x) |(wf g)(x)| ) iβa ( g2(x) f1(x) |(wf g)(x)| ) . (18) proof. using the given condition, for y ∈ [a,x]t, ∀x ∈ [a,b]t, we have |f (y)|2 ≤ f2(y) g1(y) |f (y)g(y)|. multiplying both sides of the last inequality by hα−1(x,σ(y))|w(y)| and integrating over y from a to x, we have∫ x a hα−1(x,σ(y))|w(y)||f (y)|2∆y ≤ ∫ x a hα−1(x,σ(y)) f2(y) g1(y) |w(y)||f (y)g(y)|∆y. (19) https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 7 the inequality (19) takes the form iαa ( |w(x)||f (x)|2 ) ≤iαa ( f2(x) g1(x) |w(x)||f (x)g(x)| ) . (20) similarly, we have that iβa ( |w(x)||g(x)|2 ) ≤iβa ( g2(x) f1(x) |w(x)||f (x)g(x)| ) . (21) multiplying (20) and (21), we get the desired inequality (18). � theorem 3.6. let w,f,g ∈ cld ([a,b]t,r−{0}) be ∇-integrable functions. assume that there exist four positive ∇-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]t,∀x ∈ [a,b]t). let α,β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. then we have the following inequality jαa ( |w(x)||f (x)|2 ) jβa ( |w(x)||g(x)|2 ) ≤jαa ( f2(x) g1(x) |(wf g)(x)| ) jβa ( g2(x) f1(x) |(wf g)(x)| ) .(22) proof. similar to the proof of theorem 3.5. � corollary 3.7. let w,f,g ∈ crd ([a,b]t,r−{0}) be ∆-integrable functions. assume that there exist four positive constants m, m, n and n such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a,x]t, ∀x ∈ [a,b]t. let α,β ≥ 1 and hα−1(., .),hβ−1(., .) > 0. then we have the following inequality iαa ( |w(x)||f (x)|2 ) iβa ( |w(x)||g(x)|2 ) iαa (|(wf g)(x)|)i β a (|(wf g)(x)|) ≤ mn mn . (23) proof. putting f1 = m, f2 = m, g1 = n and g2 = n in theorem 3.5, we get the desired inequality. � corollary 3.8. let w,f,g ∈ cld ([a,b]t,r−{0}) be ∇-integrable functions. assume that there exist four positive constants m, m, n and n such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a,x]t, ∀x ∈ [a,b]t. let α,β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. then we have the following inequality jαa ( |w(x)||f (x)|2 ) jβa ( |w(x)||g(x)|2 ) jαa (|(wf g)(x)|)j β a (|(wf g)(x)|) ≤ mn mn . (24) proof. similar to the proof of corollary 3.7. � remark 3.2. let t = r, α > 0, α = β, a = 0, x > 0, w ≡ 1, f > 0 and g > 0. then (23) reducesto iα0 ( f 2(x) ) iα0 ( g2(x) ){ iα0 (f (x)g(x)) }2 ≤ mnmn . (25)inequality (25) may be found in [5]. https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 8 theorem 3.9. let w,f,g ∈ crd ([a,b]t,r−{0}) be ∆-integrable functions. assume that there exist four positive ∆-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]t,∀x ∈ [a,b]t). let α ≥ 1 and hα−1(., .) > 0. then we have the following inequality iαa ( g1(x)g2(x)|w(x)||f (x)|2 ) iαa ( f1(x)f2(x)|w(x)||g(x)|2 ) {iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)} 2 ≤ 1 4 . (26) proof. using the given conditions, for y ∈ [a,x]t, ∀x ∈ [a,b]t, we have( f2(y) g1(y) − |f (y)| |g(y)| ) ≥ 0, and ( |f (y)| |g(y)| − f1(y) g2(y) ) ≥ 0. multiplying the last two inequalities, we have( f2(y) g1(y) − |f (y)| |g(y)| )( |f (y)| |g(y)| − f1(y) g2(y) ) ≥ 0, which implies ( f1(y) g2(y) + f2(y) g1(y) ) |f (y)| |g(y)| ≥ |f (y)|2 |g(y)|2 + f1(y)f2(y) g1(y)g2(y) . multiplying both sides by g1(y)g2(y)|g(y)|2, we have f1(y)g1(y)|f (y)g(y)| + f2(y)g2(y)|f (y)g(y)| ≥ g1(y)g2(y)|f (y)|2 + f1(y)f2(y)|g(y)|2. (27) multiplying both sides of (27) by hα−1(x,σ(y))|w(y)| and integrating over y from a to x, we have iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|) ≥iαa ( g1(x)g2(x)|w(x)||f (x)|2 ) + iαa ( f1(x)f2(x)|w(x)||g(x)|2 ) . (28) applying the am-gm inequality, we get iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|) ≥ 2 √ iαa (g1(x)g2(x)|w(x)||f (x)|2)iαa (f1(x)f2(x)|w(x)||g(x)|2). (29) analogously, we have that iαa ( g1(x)g2(x)|w(x)||f (x)|2 ) iαa ( f1(x)f2(x)|w(x)||g(x)|2 ) ≤ 1 4 {iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)} 2 . (30) this directly yields the desired inequality (26). � https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 9 theorem 3.10. let w,f,g ∈ cld ([a,b]t,r−{0}) be ∇-integrable functions. assume that there exist four positive ∇-integrable functions f1, f2, g1 and g2 such that: 0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]t,∀x ∈ [a,b]t). let α ≥ 1 and ĥα−1(., .) > 0. then we have the following inequality jαa ( g1(x)g2(x)|w(x)||f (x)|2 ) jαa ( f1(x)f2(x)|w(x)||g(x)|2 ) {jαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)} 2 ≤ 1 4 . (31) proof. similar to the proof of theorem 3.9. � remark 3.3. let t = r, α > 0, a = 0, x > 0, w ≡ 1, f > 0 and g > 0. then (26) reduces to iα0 ( g1(x)g2(x)f 2(x) ) iα0 ( f1(x)f2(x)g 2(x) ){ iα0 ((f1(x)g1(x) + f2(x)g2(x)) f (x)g(x)) }2 ≤ 14. (32) inequality (32) may be found in [14]. corollary 3.11. let w,f,g ∈ crd ([a,b]t,r−{0}) be ∆-integrable functions. assume that there exist four positive constants m, m, n and n such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a,x]t, ∀x ∈ [a,b]t. let α ≥ 1 and hα−1(., .) > 0. then we have the following inequality iαa ( |w(x)||f (x)|2 ) iαa ( |w(x)||g(x)|2 ) {iαa (|w(x)||f (x)g(x)|)} 2 ≤ 1 4 (√ mn mn + √ mn mn )2 . (33) proof. putting f1 = m, f2 = m, g1 = n and g2 = n in theorem 3.9, we get the desired inequality(33). � corollary 3.12. let w,f,g ∈ cld ([a,b]t,r−{0}) be ∇-integrable functions. assume that there exist four positive constants m, m, n and n such that 0 < m ≤ |f (y)| ≤ m < ∞ and 0 < n ≤ |g(y)| ≤ n < ∞ on the set [a,x]t, ∀x ∈ [a,b]t. let α ≥ 1 and ĥα−1(., .) > 0. then we have the following inequality jαa ( |w(x)||f (x)|2 ) jαa ( |w(x)||g(x)|2 ) {jαa (|w(x)||f (x)g(x)|)} 2 ≤ 1 4 (√ mn mn + √ mn mn )2 . (34) proof. similar to the proof of corollary 3.11. � remark 3.4. we have the following: (i) let α = 1, t = z, a = 1, x = b = p + 1, xk > 0, w(k) = wk = 1xk , f (k) = xk for k = 1, . . . ,p and n = g = n = 1. then inequality (33) reduces to inequality (5).(ii) let α = 1, t = r, x = b, 0 < m ≤ f (y) ≤ m < ∞, w(y) = 1 f (y) on [a,b] and n = g = n = 1. then inequality (33) reduces to inequality (6).(iii) let α = 1, t = z, a = 1, x = b = p + 1, w ≡ 1, f (k) = xk > 0 and g(k) = yk > 0 for k = 1, . . . ,p. then inequality (33) reduces to inequality (7). https://doi.org/10.28924/ada/ma.3.12 eur. j. math. anal. 10.28924/ada/ma.3.12 10 (iv) let α = 1, t = z, a = 1, x = b = p + 1, xk > 0, yk ∈r, w(k) = wk = 1xk y2k , f (k) = xk for k = 1, . . . ,p and n = g = n = 1. then inequality (33) reduces to inequality (8).(v) let α = 1, t = z, a = 1, x = b = p + 1, zk ∈ r, w(k) = wk = z2k , f (k) = xk > 0 and g(k) = yk > 0 for k = 1, . . . ,p. then inequality (33) reduces to inequality (9). 4. conclusion the subject of dynamic inequalities on time scales has become a crucial field of pure and appliedmathematics. many researchers developed interesting results concerning fractional calculus on timescales. due to utility of dynamic inequalities in many branches of mathematics, this field is given aprominent importance. this field has a wide scope. recently, interesting results have obtained byusing specht’s ratio and kantorovich’s ratio on time scales as given in [18]. by using these ratios,we can explore further results.dynamic inequalities may be extended by applying other techniques such as diamond-α inte-gral, which is defined as a linear operator of delta and nabla integrals on time scales. quantumcalculus, α,β-symmetric quantum calculus, functional generalization, fractional derivatives and n-tuple diamond-alpha integral are some other developed techniques and we will continue them toinvestigate other dynamic inequalities in future research. references [1] r.p. agarwal, d. o’regan, s.h. saker, dynamic inequalities on time scales, springer, cham, switzerland, 2014. https://doi.org/10.1007/978-3-319-11002-8.[2] g.a. anastassiou, principles of delta fractional calculus on time scales and inequalities, math. comp. model. 52(2010) 556–566. https://doi.org/10.1016/j.mcm.2010.03.055.[3] g.a. anastassiou, foundations of nabla fractional calculus on time scales and inequalities, comp. math. appl. 59(2010) 3750–3762. https://doi.org/10.1016/j.camwa.2010.03.072.[4] g.a. anastassiou, integral operator inequalities on time scales, int. j. diff. equ. 7 (2012) 111–137.[5] a. anber, z. dahmani, new integral results using pólya–szegö inequality, acta comment. univ. tart. math. 17(2013) 171–178. https://doi.org/10.12697/acutm.2013.17.15.[6] d. anderson, j. bullock, l. erbe, a. peterson, h. tran, nabla dynamic equations on time scales, pan-amer. math.j. 13 (2003) 1–47.[7] m. bohner, a. peterson, dynamic equations on time scales, birkhäuser boston, inc., boston, ma, 2001.[8] m. bohner, a. peterson, advances in dynamic equations on time scales, birkhäuser boston, boston, ma, 2003. https://doi.org/10.1007/978-0-8176-8230-9.[9] m. bohner, h. luo, singular second-order multipoint dynamic boundary value problems with mixed derivatives, adv.diff. equ. (2006) 1–15. https://doi.org/10.1155/ade/2006/54989.[10] w. greub, w. rheinboldt, on a generalization of an inequality of l.v. kantorovich, proc. amer. math. soc. 10 (1959)407–415. https://doi.org/10.1090/s0002-9939-1959-0105028-3.[11] s. hilger, ein maβkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten, ph.d. thesis, universität würzburg,1988. https://doi.org/10.28924/ada/ma.3.12 https://doi.org/10.1007/978-3-319-11002-8 https://doi.org/10.1016/j.mcm.2010.03.055 https://doi.org/10.1016/j.camwa.2010.03.072 https://doi.org/10.12697/acutm.2013.17.15 https://doi.org/10.1007/978-0-8176-8230-9 https://doi.org/10.1155/ade/2006/54989 https://doi.org/10.1090/s0002-9939-1959-0105028-3 eur. j. math. anal. 10.28924/ada/ma.3.12 11 [12] l.v. kantorovich, functional analysis and applied mathematics (russian), uspehi mat. nauk (n.s.). 3 (1948) 89–185(in particular, pp. 142–144) [also translated from russian into english by c.d. benster, nat. bur. standards rep.no. 1509. 1952, 202 pp. (in particular, pp. 106–109)].[13] d.s. mitrinović, analytic inequalities, springer-verlag, berlin, 1970. https://doi.org/10.1007/ 978-3-642-99970-3.[14] s.k. ntouyas, p. agarwal, j. tariboon, on pólya–szegö and chebyshev types inequalities involving the riemann–liouville fractional integral operators, j. math. ineq. 10 (2016) 491–504. https://doi.org/10.7153/jmi-10-38.[15] g. pólya, g. szegö, aufgaben und lehrsätze aus der analysis, berlin. 1 (1925) 213–214. https://doi.org/10. 1007/978-3-662-38381-0.[16] m.j.s. sahir, formation of versions of some dynamic inequalities unified on time scale calculus, ural math. j. 4(2018) 88–98. https://doi.org/10.15826/umj.2018.2.010.[17] m.j.s. sahir, symmetry of classical and extended dynamic inequalities unified on time scale calculus, turk. j. ineq.2 (2018) 11–22.[18] m.j.s. sahir, parity of classical and dynamic inequalities magnified on time scales, bull. int. math. virtual inst. 10(2020) 369–380.[19] p. schweitzer, an inequality concerning the arithmetic mean (hungarian), math. phys. lapok. 23 (1914) 257–261. https://doi.org/10.28924/ada/ma.3.12 https://doi.org/10.1007/978-3-642-99970-3 https://doi.org/10.1007/978-3-642-99970-3 https://doi.org/10.7153/jmi-10-38 https://doi.org/10.1007/978-3-662-38381-0 https://doi.org/10.1007/978-3-662-38381-0 https://doi.org/10.15826/umj.2018.2.010 1. introduction 2. preliminaries 3. main results 4. conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 18doi: 10.28924/ada/ma.3.18 a modified algorithms for new krasnoselskii’s type for strongly monotone and lipschitz mappings furmose mendy, john t mendy∗ university of the gambia, gambia furmosemendy111@gmail.com, jt.mendy@yahoo.com ∗correspondence: jt.mendy@yahoo.com abstract. let e be a 2 uniformly smooth and convex real banach space and let a mapping a : e → e∗be lipschitz and strongly monotone such that a−1(0) 6= ∅. for an arbitrary ({x1},{y1}) ∈ e, wedefine the sequences {xn} and {yn} by{ yn = xn −θnj−1(axn), n ≥ 1 xn+1 = yn −λnj−1(ayn), n ≥ 1where λn and θn are positive real number and j is the duality mapping of e. letting (λn,θn) ∈ (0, 1), then xn and yn converges strongly to ρ∗, a unique solution of the equation ax = 0. we also appliedour algorithm in convex minimization and also proved the convergence of it in lp,`p or wm,p. at theend we proposed the algorithm of it in lp(ω) and its inverse lq(ω). 1. introduction definition 1.1. a map a : e → e∗ is called monotone if for each x,y ∈ e, the following inequalityholds: 〈ax −ay,x −y〉≥ 0 a is called strongly monotone if there exists k ∈ (0, 1) such that for each x,y ∈ e, the followinginequality holds: 〈ax −ay,x −y〉≥ k‖x −y‖2 a map a : e → e is called accretive if for each x,y ∈ e, there exists j(x − y) ∈ j(x − y) suchthat 〈ax −ay,j(x −y)〉≥ 0 a is called strongly accretive if there exists k ∈ (0, 1) such that for each x,y ∈ e, there exists j(x −y) ∈ j(x −y) such that 〈ax −ay,j(x −y)〉≥ k‖x −y‖2 received: 23 apr 2023. key words and phrases. krasnoselskii-type algorithm; monotone operators; lipschitz mappings.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.18 https://orcid.org/0000-0002-3774-0761 eur. j. math. anal. 10.28924/ada/ma.3.18 2 a map a : e → e∗ is called lipschitzian, if for each constant l > 0 and for all x,y ∈ e, thefollowing inequality holds: i): 〈ax −ay〉≤ l‖x −y‖ ii): l2(d2 − 1) < k2 many physical problems in applications can be modeled in the following form: find x ∈ h suchthat 0 ∈ ax (1.1) where a is a monotone operator on a real hilbert space h. typical examples where monotone oper-ators occur and satisfy the inclusion 0 ∈ ax include the equilibrium state of evolution equations andcritical points of some functionals and convex optimization, linear programing, monotone inclusionsand elliptic differential equations defined on hilbert spaces (see e.g., browder [2], mustafa [19],stephen [26], sina [24], mendy et al, [17] and chidume [3]). for precisely, the classical convexoptimization problem: let h : h →r∪{+∞} be a proper convex function. the sub-differential of h at x ∈ h; is defined by ∂h : h → 2h ∂(x) = {x∗ ∈ h : h(y) −h(x) ≥〈y −x,x∗〉,∀y ∈ h}. (1.2) clearly, ∂h : h → 2his monotone operator on h, and 0 ∈ ∂(x0) if and only if x0 is a minimizer of h. in the case of setting ∂(x) ≡ a ; solving the inclusion 0 ∈ ax is solving for a minimizer of h.there have been fruitful works on approximating zero point of a in hilbert spaces (see e.g.,takahashi and ueda [31], song and chen [25], and cho et al. [9]). the proximal point algorithm (ppa) is recognized as a powerful and successful algorithm in finding a numerical solution ofmonotone operators equation 0 ∈ ax which was introduced by martinet [13] and studied furtherby rockafellar [22] and a host of other authors. that is, given xk ∈ h; xn+1 = jλnxn. (1.3) where jλn = (i + λna)−1 is the resolvent of operator a. since rockafellar [22] only obtained theweak convergence of the algorithm 1.3 as λn →∞ ; so he proposed two open questions for obtainingthe strong convergence of the proximal point algorithm: (1) does the proximal point algorithmalways converge weakly? (2) can the proximal point algorithm be modified to guarantee strongconvergence? in studying the strong convergence, many authors have modified the proximal pointalgorithm (ppa) to guarantee strong convergence under different settings, see e.g., takahashi [29],reich [20], lehdili and moudafi [12], chidume et al. [6], and the references therein.let e be a real normed space, e∗ its topological dual space. the map j : e → 2e∗ defined by jx : { x∗ ∈ e∗ : 〈x,x∗〉 = ‖x‖.‖x∗‖ = ‖x‖2 = ‖x∗‖2 } . https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 3 is called the normalized duality map on e. where 〈,〉 denotes the generalized duality pairingbetween e and e∗.in a hilbert space, the normalized duality map is the identity map. hence, in hilbert spaces,monotonicity and accretivity coincide. for an accretive-type operator a,solutions of the equation ax = 0, in many cases, represent the equilibrium state of somedynamical system (see, for example, [29], page 116). to approximate a solution of ax = 0, assumingexistence, where a : e → e is of accretive type, browder [2] defined an operator t : e → e by t := i − a, where i is the identity map on e. he called such an operator pseudo-contractive.it is trivial to observe that zeros of a correspond to fixed points of t . for lipschitz stronglypseudo-contractive maps, chidume [6] proved the following theorem. theorem 1.1. (chidume, [7]. let e = lp, 2 ≤ p < 8, and k ⊂ e be nonempty closed convex and bounded. let t : k → k be a strongly pseudo-contractive and lipschitz map. for arbitrary x0 ∈ k, let a sequence {xn} be defined iteratively by xn+1 = (1 −λn)xn + λntxn,n ≥ 0, where {λn}⊂ (0, 1) satisfies the following conditions: ,(i) ∞∑ n=1 λn = ∞ , (ii) ∞∑ n=1 λ2n ≤∞. then {xn} converges strongly to the unique fixed point of t . by setting t := i − a in theorem 1.1, the following theorem for approximating a solution of ax = 0 where a is a strongly accretive and bounded operator can be proved.unfortunately, the success achieved in using geometric properties developed from the mid-1980sto early 1990s in approximating zeros of accretive-type mappings has not carried over to approx-imating zeros of monotone-type operators in general banach spaces. part of the problem is thatsince a maps e to e∗, for xn ∈ e,axn is in e∗. consequently, a recursion formula containing xnand axn may not be well defined. attempts have been made to overcome this difficulty by introduc-ing the inverse of the normalized duality mapping in the recursion formulas for approximating zerosof monotone-type mappings.examples chidume [4], [5], moudafi [18], reich [21], takahashi [30],zegeye [37], djitte [17], mendy [ [15], [10]]motivated by approximating zeros of monotone mappings, chidume et al. [8] proposed akrasnoselskii-type scheme and proved a strong convergence theorem in lp, 2 ≤ p < ∞. in fact,they obtained the following result. theorem 1.2. (chidume et al. [8]). let x = lp, 2 ≤ p < ∞, and a : x → x∗ be a lipschitz map. assume that there exists a constant k ∈ (0, 1) such that a satisfies the condition 〈ax −ay,x −y〉≥ k‖x −y‖ p p−1 (1.4) and that a−1(0) 6= ∅. for arbitrary x1 ∈ x, define the sequence {xn} iteratively by xn+1 = j −1(jxn −λnaxn) n ≥ 0 https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 4 where λn ∈ (0,δp) and δp is some positive constant. then the sequence {xn} converges strongly to the unique solution of the equation ax = 0. in [8], the authors posed the following open problem. if e = lp, 2 ≤ p < ∞, attempts to obtainstrong convergence of the krasnoselskii-type sequence defined for x0 ∈ e by xn+1 = j −1(jxn −λnaxn) n ≥ 0 to a solution of the equation ax = 0, where a is strongly monotone and lipschitz, have notyielded any positive result.following the works of chidume et al [8], and motivation of finding the zeros of the monotonetype mapping, several strong convergence results have been established by various authors (seee.g [17], [10], [15], [23], [16]).following this great work, in 2023, mendy [16] constructed the following two-step proximalalgorithm for the zero point of monotone mapping and proof a strong convergency of the sequences {xn} and {yn} to a unique point x∗ ∈ a−1(0).{ yn+1 = j −1(jxn −λnaxn), n ≥ 0 xn+1 = j −1(jyn+1 −λn+1ayn+1), n ≥ 0 (1.5) in this paper, we study the two step size of the new krasnoselskii-type algorithm introduced bysene et al. [23] and prove a strong convergence theorem to approximate the unique zero of alipschitz and strongly monotone mapping 2−uniformly smooth and convex real banach space for p ≥ 2. this class of banach spaces contains all lp-spaces, 2 ≤ p < ∞ and sobolev space. thenwe apply our results to the convex minimization problem. finally, our method of proof generalizedand extended various authors in this way of work. 2. preliminaries let e be a normed linear space. e is said to be smooth if lim t→0 ‖x + ty‖−‖x‖ t (2.1) exist for each x,y ∈ se (here se := {x ∈ e : ||x|| = 1} is the unit sphere of e). e is said to beuniformly smooth if it is smooth and the limit is attained uniformly for each x,y ∈ se, and e isfréchet differentiable if it is smooth and the limit is attained uniformly for y ∈ se.let e be a real normed linear space of dimension ≥ 2. the modulus of smoothness of e , ρe, isdefined by: ρe(τ) := sup { ‖x + y‖ + ‖x −y‖ 2 − 1 : ‖x‖ = 1,‖y‖ = τ } ; τ > 0. a normed linear space e is called uniformly smooth if lim τ→0 ρe(τ) τ = 0. https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 5 if there exist a constant c > 0 and a real number q > 1 such that ρe(τ) ≤ cτq, then e is said tobe q-uniformly smooth.a normed linear space e is said to be strictly convex if: ‖x‖ = ‖y‖ = 1, x 6= y ⇒ ∥∥∥x + y 2 ∥∥∥ < 1. the modulus of convexity of e is the function δe : (0, 2] → [0, 1] defined by: δe(�) := inf { 1 − 1 2 ‖x + y‖ : ‖x‖ = ‖y‖ = 1, ‖x −y‖≥ � } . e is uniformly convex if and only if δe(�) > 0 for every � ∈ (0, 2]. for p > 1, e is said to be p-uniformly convex if there exists a constant c > 0 such that δe(�) ≥ c�p for all � ∈ (0, 2]. observethat every p-uniformly convex space is uniformly convex.typical examples of such spaces are the lp, `p and wmp spaces for 1 < p < ∞ where, lp (or lp) or w m p is { 2 − uniformly smooth and p− uniformly convex if 2 ≤ p < ∞; 2 − uniformly convex and p− uniformly smooth if 1 < p < 2. remark 1. note also that duality mapping exists in each banach space.we recall from [11] someof the examples of this mapping in `p,lp,wm,p−spaces, 1 < p < ∞ • `p : jx = ‖x‖ 2−p `p y ∈ `q,x = (x1,x2, ...,xn, ...),y = (x1|x1|p−2,x2|x2|p−2, ...,xn|xn|p−2, ...) • lp : ju = ‖u‖ 2−p lp |u|p−2u ∈ lq • wm,p : ju = ‖u‖2−p wm,p ∑ |α≤m| (−1)|α|dα(|dαu|p−2dαu) ∈ w−m,p in lp,`p and wm,p spaces for 1 < p < ∞ are q−uniformly smooth real banach spaces with q, as q = min{2,p} and dq ≥ 1 (2.2) is given by dq = { 1+τq−1 (1+τ)q−1 , if 1 < p < 2; p− 1, if 2 ≤ p < ∞. (2.3)and τ(0, 1) as the unique solution of the equation (q − 2)tq−1 + (q − 1)tq−2 − 1 = 0 it is well known that • e is smooth if and only if j is single-valued. • if e is uniformly smooth then j is uniformly continuous on bounded subsets of e. • if e is reflexive and strictly convex dual then j−1 is single-valued, one-to-one, surjective,uniformly continuous on bounded subsets and it is the duality mapping from e∗ into e and j−1j = ie and jj−1 = ie. • j−1 is uniformly continuous if and only if it has a modulus of continuity. https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 6 lemma 2.1 (xu [32]). . let q > 1 be a real number and e be a banach space. then the following assertion are equivalent i): e is q−uniformly smooth ii): there exists a constant dn > 0, such that for all x,y ∈ e, then the following holds ‖x + q‖q ≤‖x‖q + q〈y,jq(x)〉 + dq‖y‖q. (2.4) 3. main result we now prove the following result theorem 3.1. let e be a 2 uniformly smooth and convex real banach space and let a mapping a : e → e∗ be lipschitz strongly monotone such that a−1(0) 6= ∅. for an arbitrary ({x1},{y1}) ∈ e, we define the sequences {xn} and {yn} by { yn = xn −θnj−1(axn), n ≥ 1 xn+1 = yn −λnj−1(ayn), n ≥ 1 (3.1) where λn and θn are positive real number and j is the duality mapping of e. letting (λn,θn) ∈ (0, 1) , then {xn} and {yn} converges strongly to ρ∗, a unique solution of the equation ax = 0. proof. letting ρ∗ = x∗ ∈ e be the unique solution of ax = 0. from inequality 2.4 in lemma 2.1with 3.1, knowingly that ‖j−1w‖ = ‖w‖ for all w ∈ e∗, then we have the following estimates: ‖xn+1 −ρ∗‖2 = ‖yn −ρ∗ −λnj−1(ayn)‖2 = ‖λnj−1(ayn)‖2 − 2〈yn −ρ∗,j(λnj−1(ayn))〉 + d2‖yn −ρ∗‖2 ≤ λ2n‖(ayn)‖ 2 − 2λn〈yn −ρ∗,ayn)〉 + d2‖yn −ρ∗‖2 ≤ λ2nl 2‖yn −ρ∗‖2 − 2λnk‖yn −ρ∗‖2 + d2‖yn −ρ∗‖2 = ( λ2nl 2 − 2kλn + d2 ) ‖yn −ρ∗‖2 (3.2) for the fact that 0 < (λ2nl2 − 2kλn + d2) < 1, we have the following ‖xn+1 −ρ∗‖2 ≤ δ(λ1)‖yn −ρ∗‖2 (3.3) where δ(λ1) = (λ2nl2 − 2kλn + d2).using 3.1 , lipschitz property of a, with the same computational we have the following: https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 7 ‖yn −ρ∗‖2 = ‖xn −ρ∗ −θnj−1(axn)‖2 = ‖θnj−1(axn)‖2 − 2〈xn −ρ∗,j(θnj−1(axn))〉 + d2‖xn −ρ∗‖2 ≤ θ2n‖(axn)‖ 2 − 2θn〈xn −ρ∗,axn)〉 + d2‖xn −ρ∗‖2 ≤ θ2nl 2‖xn −ρ∗‖2 − 2θnk‖xn −ρ∗‖2 + d2‖xn −ρ∗‖2 = ( θ2nl 2 − 2kθn + d2 ) ‖xn −ρ∗‖2 (3.4) again, with the fact that 0 < (θ2nl2 − 2kθn + d2) < 1, we have the following ‖yn −ρ∗‖2 ≤ δ(λ2)‖xn −ρ∗‖2 (3.5) where δ(λ2) = (θ2nl2 − 2kθn + d2)putting 3.5 in 3.3, we have the following ‖xn+1 −ρ∗‖2 ≤ δ(λ1)δ(λ2)‖xn −ρ∗‖2 (3.6) ‖xn+1 −ρ∗‖≤ √ δ(λ1)δ(λ2)‖xn −ρ∗‖ (3.7) ‖xn+1 −ρ∗‖≤ µ‖xn −ρ∗‖ where µ = √δ(λ1)δ(λ2).therefore the sequences {xn} and {yn} converges strongly to ρ∗. this complete the proof. � corollary 3.1. let e = lp, 2 ≤ p < ∞, and a : e → e∗ be a lipschitz strongly monotone mapping such that a−1(0) 6= ∅. for arbitrary (x1,y1) ∈ e, define the sequence {xn} and {yn} iteratively by{ yn = xn −θnj−1(axn), n ≥ 1 xn+1 = yn −λnj−1(ayn), n ≥ 1 (3.8) where λn and θn are positive real number and j is the duality mapping of e. letting (λn,θn) ∈ (0, 1), then xn and yn converges strongly to ρ∗, a unique solution of the equation ax = 0. proof. since e = lp spaces, 2 ≤ p < ∞, are 2−uniformly smooth and convex real banach spaces,then the proof follows from theorem 3.1. � 4. convergence in lp,`p or wm,p, 2 ≤ p < ∞ theorem 4.1. let e be a 2 uniformly smooth and convex real banach space either lp,`p or w m,p, 2 ≤ p < ∞ with it dual e∗. let a mapping a : e → e∗ be lipschitz and strongly monotone such that a−1(0) 6= ∅. for an arbitrary ({x1},{y1}) ∈ e, we define the sequences {xn} and {yn} by 3.1 converges strongly to ρ∗, a unique solution of the equation ax = 0. https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 8 proof. since lp,`p or wm,p, 2 ≤ p < ∞ are 2− uniformly smooth banach spaces, then with thesame computation in 3.1, the proof follows. � corollary 4.1. let e be a banach space either lp,`p or wm,p, 2 ≤ p < ∞ with it dual e∗. let a mapping a : e → e∗ be lipschitz and strongly monotone such that a−1(0) 6= ∅. for an arbitrary ({x1},{y1}) ∈ e, we define the sequences {xn} and {yn} by 3.1 converges strongly to ρ∗, a unique solution of the equation ax = 0. proof. since lp,`p or wm,p, 2 ≤ p < ∞ are 2− uniformly smooth banach spaces, then fromtheorem 4.1 with the same computation in 3.1, the proof follows. � 5. application to convex minimization problem now, we present a convex minimization problem for a convex function ∇ : e →r.the following results are well known. remark 2. let ∆ : e → r be a differentiable convex function and ρ∗ ∈ e, then the point ρ∗ is aminimizer of ∇ on e if and only if d∇(ρ∗) = 0. definition 5.1. a function ∇ : e →r is said to be strongly convex if there exists γ > 0 such thatthe following condition holds: ∇(βx + (1 −β)y) ≤ β∇x + (1 −β)∇y −γ‖x −y‖2 (5.1) for every x,y ∈ e with x 6= y and β ∈ (0, 1), lemma 5.2. let e be normed linear space and ∇ : e →r a convex differentiable function. suppose that ∇ is strongly convex. then the differential map d∇ : e → e∗ is strongly monotone, i.e., there exists k > 0 such that 〈d∇x −d∇y,x −y〉≥ k‖x −y‖2 ∀x,y ∈ e. (5.2) now we present the following result. theorem 5.3. let d∇ : e∗ → e be a l-lipschitz continuous and strongly monotone mapping such that d∇−1(0) 6= ∅. let e = lp,p ≥ 2 and ∇ : e → r be a differentiable, strongly convex real-valued function. for given x1,y1 ∈ e, define the sequence {xn} and {yn} as follows:{ yn = xn −θnd∇xn), n ≥ 1 xn+1 = yn −λnd∇yn), n ≥ 1 (5.3) where the sequences {λn} and {θn}, are in the interval [0, 1] . then ∇ has a unique minimizer ρ∗ ∈ e such that if ( λn,θn ) ∈ [0, 1], the sequence {xn} and {yn} converges strongly to ρ∗. https://doi.org/10.28924/ada/ma.3.18 eur. j. math. anal. 10.28924/ada/ma.3.18 9 proof. from remark 2 it follows that ∇ has a unique minimizer ρ∗ and is obtained by d∇(ρ∗) = 0.from lemma 5.2 and using the fact that the differential mapping d∇ : e → e∗ is lipschitz,considering the result of theorem 3.1, we can complete the proof. � 6. the proposed algorithm in lp(ω) now, from [14], the duality mapping j is known precisely in lp(ω) for 1 < p < ∞ by jv = ‖v‖2−p lp |v|p−2v,∀v ∈ lp(ω) and if lp(ω) is reflexive, smooth and strictly convex real banach space, for 1 < p < ∞, then theduality mapping j is surjective, one-to-one and its inverse j−1 is given by ju = ‖‖2−q l |u|q−2u,∀u ∈ lq(ω) with 1 p + 1 q = 1now from 3.1, we defined x1,y1 ∈ lq(ω){ yn = xn −θn‖axn‖ 2−q lq |axn| 2−q lq axn, n ≥ 1 xn+1 = yn −λn‖ayn‖ 2−q lq |ayn| 2−q lq ayn, n ≥ 1 (6.1) conclusion in this paper, we proposed and analyzed the strong convergence theorem of two step size of thenew krasnoselskii-type algorithm introduced by sene et al. [7] and prove a strong convergencetheorem to approximate the unique zero of a lipschitz strongly monotone mapping 2−uniformlysmooth and p−uniformly convex real banach space for p ≥ 2. this class of banach spaces containsall lp-spaces, 2 ≤ p < ∞ and sobolev space. then we apply our results to the convex minimizationproblem. we also complemented and generalized previous worked been done under this setting. references [1] ya. alber, i. ryazantseva, nonlinear ill posed problems of monotone type, springer, london, uk, 2006.[2] f.e. browder, nonlinear mappings of nonexpansive and accretive type in banach spaces, bull. amer. math. soc. 73(1967) 875-882. https://doi.org/10.1090/s0002-9904-1967-11823-8.[3] c.e. chidume, m.o. osilike, iterative solution of nonlinear integral equations of hammerstein-type, j. niger. math.soc. appl. anal. 11 (1992) 9-18.[4] c.e. chidume, iterative approximation of fixed points of lipschitzian strictly pseudocontractive mappings, proc. amer.math. soc. 99 (1987) 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spaces, num. funct. anal. optim.40 (2019) 1426-1447. https://doi.org/10.1080/01630563.2019.1606825.[37] h. zegeye, n. shahzad, an algorithm for a common minimum-norm zero of a finite family of monotone mappings inbanach spaces, j. ineq. appl. 2013 (2013) 566. https://doi.org/10.1186/1029-242x-2013-566. https://doi.org/10.28924/ada/ma.3.18 https://doi.org/10.1016/0022-247x(84)90019-2 https://doi.org/10.1016/0362-546x(91)90200-k https://doi.org/10.1080/01630563.2019.1606825 https://doi.org/10.1080/01630563.2019.1606825 https://doi.org/10.1186/1029-242x-2013-566 1. introduction 2. preliminaries 3. main result 4. convergence in lp,p or wm,p, 2p < 5. application to convex minimization problem 6. the proposed algorithm in lp() conclusion references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 20doi: 10.28924/ada/ma.3.20 on degree-based topological indices of petersen subdivision graph mukhtar ahmad1, saddam hussain2, ulfat parveen3, iqra zahid3, muhammad sultan3,ather qayyum3,∗ 1department of mathematics, khawaja fareed university of engineering and information technology rahim yar khan, pakistan itxmemuktar@gmail.com 2department of statistics, university of mian wali, pakistan saddamhussain.stat885@gmail.com 3department of mathematics, institute of southern punjab multan, pakistan uulfat05@gmail.com, iqraimran57@gmail.com, sultan.sadeeq7866127@gmail.com, atherqayyum@isp.edu.pk ∗correspondence: atherqayyum@isp.edu.pk abstract. in this paper, we adequately describe the generalised petersen graph, expanding to thecategories of graphs. we created a petersen graph, which is cyclic and has vertices that are arrangedin the centre and nine gons plus one vertex, leading to the factorization of regular graphs. petersengraph is still shown in graph theory literature, nevertheless. 1. introduction named after julius petersen, a danish mathematician, the graph of petersen is(from 1839 to1910). petersen researched factorizations of normal factorizations during the 1890s. in 1891, asignificant paper of graphs was published which is commemorated in that volume. petersen provedthat any graph of 3-regular with at a i-factor includes much of the two bridges. tait had writtena few years ago that he had shown i-factorable for each 3-regular graph,but that this outcomeit was not valid without restriction. but tait’s comment in 1898 was interpreted by petersen toimply that each 3-regular bridge less graph is l-factorable. if this outcome were valid, then itwould have been stronger than theorem for petersen. the key characteristics of the petersengraph were examined in detail in 1985. the graph of petersen continously to express in the entiregraph-theory education. we update our previous analysis in the present article by denoting extrarecently findings concerning the petersen graph.julius petersen’s ’die theorie der regulken graphs’ is an exceptional paper that developed a new received: 15 apr 2023. key words and phrases. atom-bond connectivity index; reduced zagreb; randic indices; general connectivity index;petersen graph. 1 https://adac.ee https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 2 theory in graph theory, based on the exchange property of trees spanning and the cyclomaticnumber of trees spanningresently resently zaib hassan niazi et.al[15]. 1.1. the graph of petersen. every petersen graph is cyclic graph and the graph g′ in general formconsists v having set of vertex and e having set of edge, if the natural number, there exist n thegraph with vertices are v (g′)=4n, edges are e(g′)=6n, and the specific of this graph is thatabout degree of every each vertex is p(k,t) = [d(x1),d(x2)]= 3. then this graphic which is said tobe petersen graphic. then petersen graphic is denoted by p[v (g′),e(g′)] =(4n,6n) petersen mapexplored by 1985, updated by new analysis.sylvester’s association to graphs of invariants and covariants requires interpretation of principleof invariants in 1880s. 1.2. graphical idea of petersen graph. if the set of natural number is tn = {1,2,3, ...}, if thereexists n then graph with vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form ofpetersen expressed by p[v (g′),e(g′)] =(4n,6n). this graph having a specification, that degree ofevery each vertex is p(k,t)= [d(x1),d(x2)]= 3.now we write; tn =1,2,3, .... v (g′) = 4n → [1] e(g′) = 6n → [2] k = d(x1)=3 t = d(x2)=3 then p(k,t) = [d(x1),d(x2)]= 3 p[v (g′),e(g′)] =(4n,6n). next we discuss the topological indices zagreb indices of the group were recognized in the early 1980s and are now known as thefirst and second zagreb indices. they are important molecular descriptors and have been closelycorrelated with chemical properties.[degree based topological indices]the first zagreb index m1(g) is equal to the sum of the squares of the degrees of the vertices for https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 3 the (molecular) graph g[1]. it can also be considered as the sum over the edges of g, and m1(g)is defined as:[the first and second zagreb indices of some graph operations] m1(g,x)= ∑ [x1,x2∈e(g)] [d(x1)+d(x2)] (1) the second zagreb index m2(g) is equal to the sum of the products of the degrees of the adjacentvertices for the pair of vertices for the (molecular) graph g, and m2(g) is defined as:[the first andsecond zagreb indices of some graph operations] m2(g,x)= ∑ [x1,x2∈e(g)] [d(x1)d(x2)] (2) in 1972, the first zagreb index, a very old topological index, was launched and several variants ofthe zagreb index were subsequently proposed, e.g. shirdel et al. described a novel index in 2013under the title of ’hyper-zagreb index’ and then it was identified as[2]: [a note on hyper-zagrebindex of graph operations] hm1(g)= ∑ [x1,x2∈e(g)] [d(x1)+d(x2)] 2 (3) e. deutshi and s. klavzar,in 2015, defined a new polynomial, m-polynomial in the followingway, based on the degree of the vertex[3]:[computing hyper zagreb index and m-polynomials] m1(g,y,z)= ∑ [x1,x2∈e(g)] y [d(x1)]z[d(x2)] (4) in shuxian defined two polynomials related to the first zagreb index as in the form: m∗1(g,x)= ∑ [xi∈v (g)] [d(xi)][x (xi)] (5) m0(g,x)= ∑ [xi∈v (g)] (x)[d(xi)] (6) two zagreb type polynomials are defined as follow: ma,b(g,x)= ∑ [xi,xj∈e(g)] (x)[a{d(xi)}+b{d(xi)}] (7) m′a,b(g,x)= ∑ [xi,xj∈e(g)] (x)([a+{d(xi)}][b+{d(xi)}]) (8) todeshine et al. introduced two updated models of the zagreb index for moleculargraphs[4]:[multiplicative zagreb indices of trees] first multiplicative zagreb index formolecular graph g defined as follows: pm1(g)= ∏ [x1,x2∈e(g)] [d(x1)+d(x2)] (9) https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 4 second multiplicative zagreb index for molecular graph g defined as follows: pm2(g)= ∏ [x1,x2∈e(g)] [d(x1)×d(x2)] (10) first multiplicative zagreb polynomial for molecular graph g defined as follows: pm1(g,x)= ∏ [x1,x2∈e(g)] x[d(x1)+d(x2)] (11) second multiplicative zagreb polynomial for molecular graph g defined as follows: pm2(g,x)= ∏ [x1,x2∈e(g)] x[d(x1)d(x2)] (12) the first degree-based topological index was proposed by milan randic in 1975[5]:[degree-basedtopological indices] r1(α)(g)= ∑ [x1,x2∈e(g)] [d(x1)+d(x2)] α (13) atom-bond connectivity index (abc) is a topological index used in chemistry, environmental sci-ences and pharmacology[6]: [estrada, torres, rodriguez, and gutman, 1998b] abc(g)= ∑ [x1,x2∈e(g)] √ [d(x1)+d(x2)]−2 d(x1)×d(x2) (14) first, second and third reduced zagreb indices[7] are described as follow: mr1(g)= ∑ [x1,x2∈e(g)] |(d(x1)−1)+(d(x2)−1)| (15) mr2(g)= ∑ [x1,x2∈e(g)] [(d(x1)−1)(d(x2)−1)] (16) mr3(g)= ∑ [x1,x2∈e(g)] |(d(x1)−1)− (d(x2)−1)| (17) rr(g)= ∑ [x1,x2∈e(g)] √ d(x1)×d(x2) (18) the reduced reciprocal randic index is defined as[8]: rrr(g)= ∑ [x1,x2∈e(g)] √ [d(x1)−1]× [d(x2)−1] (19) recently in 2015 furtula and gutman [8] introduced another topological index known as forgottenindex or f − index. for more detail on the f − index, we refer to the articles [9].the forgottenindex of a graph g is defined as[10, 11, 12]. f(g)= ∑ [x1,x2∈e(g)] [(dx1) 2 +(dx2) 2] (20) https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 5 the forgotten polynomial of a graph g is defined as: f(g,x)= ∑ [x1,x2∈e(g)] (x)[(dx1) 2+(dx2) 2] (21) the symmetric division degree index of a connected graph g is defined as: sdd(g)= ∑ [x1,x2∈e(g)] mini(d(x1),d(x2)) max(d(x1),d(x2)) + maxi(d(x1),d(x2)) mini(d(x1),d(x2)) (22) there are two types of general connectivity index. the general randic index (or product-connectivityindex) was proposed by bolloba and erdos and is defined as follows: m1(g)= ∑ [x2∈v (g)] [dg(x2)] 2 (23) where α is a real number. if α =−1 2 , then it becomes the randic index and if α =1 then it becomesthe second zagreb index. zhou and trinajstic developed the general sum-connectivity index: [onthe general sum-connectivity index of trees] m1(g)= ∑ [x1,x2∈e(g)] [d(x1)+d(x2)] α (24) where α is a real number. if α = 1, then the general sum connectivity index becomes the firstzagreb index resently asghar et.al[14]. 2. main results in this section, we established some results on degree based topological indices of petersengraph. theorem 2.1 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, first zagrebpolynomials indices are, m1(g,x) =[|6n|](x)(6) proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now first zagreb polynomials indices are i.e. , ⇒ ga(r) = ∑y1,y2∈e(r) 2√dy1dy2dy1+dy2now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in first zagrebtopological index of the general form, https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 6 ⇒ m1(g,x) =[|e(g)|](x)[(3)+(3)] ⇒ m1(g,x) =(6n)(x)(6) ⇒ m1(g,x) = (6n)(x)6. m1(g,x) = (x)6× [general edges of petersen graph] theorem 2.2 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, second zagrebpolynomials indices are, m2(g,x) = (6n)(x)9 proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now second zagreb polynomials indices are i.e. , m2(g,x) = ∑[x1,x2∈e(g)](x)[d(x1)×d(x2)] → [1] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in second zagrebtopological index of the general form, ⇒ m2(g,x) =∑x1,x2∈e(g)(x)[(3)(3)] ⇒ m2(g,x) =[|e(g)|](x)9 ⇒ m2(g,x) = (6n)(x)9. m2(g,x) = (x)9× [general edges of petersen graph] theorem 2.3 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, randic indicesare, r1(α)(g) = (6n)[6]α proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now randic indices are i.e. , r1(α)(g) = ∑[x1,x2∈e(g)][d(x1)+d(x2)]α now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in randic indices https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 7 topological index of the general form, r1(α)(g) = ∑[x1,x2∈e(g)][d(x1)+d(x2)]α in general form of topological index becomes; ⇒ r1(α)(g) = ∑[x1,x2∈e(g)][d(x1)+d(x2)]αnow putting values in above equation, ⇒ r1(α)(g) =[|e(g)|][(3)+(3)]α ⇒ r1(α)(g) =[|e(g)|][6]α ⇒ r1(α)(g) =[|6n|][6]α ⇒ r1(α)(g) = (6n)[6]α. r1(α)(g) = [6α]×[ the general edges of petersen graph] theorem 2.4 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, reducedreciprocal randic are, rrr(g) = 12n. proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now reduced reciprocal randic are i.e. , rr(g) = ∑[x1,x2∈e(g)] √d(x1)×d(x2) now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersengraph about every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in reducedreciprocal randic topological index of the general form, ⇒ rrr(g) = ∑[x1,x2∈e(g)] √[d(x1)−1]× [d(x2)−1]now puttings the values then; ⇒ rrr(g) = ∑[x1,x2∈e(g)] √(3−1)× (3−1) ⇒ rrr(g) = [|e(g)|]√(4) ⇒ rrr(g) = [|6n|]√(4) ⇒ rrr(g) = (6n)√(4) ⇒ rrr(g) = 6n(2) ⇒ rrr(g) = 12n. https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 8 theorem 2.5 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, hyper zagrebindex are, hm1(g) = 216n proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now hyper zagreb index are i.e. , hm1(g)= ∑[x1,x2∈e(g)][d(x1)+d(x2)]2 → [1] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in hyper zagrebindex topological index of the general form, ⇒ hm1(g) =[|e(g)|][(3+3)]2 ⇒ hm1(g) =[|6n|](6)2 ⇒ hm1(g) = (6n)(6)2 ⇒ hm1(g) = (6n)(36) ⇒ hm1(g) = 216n. hm1(g)= thirty six times to general edges of petersen graph. theorem 2.6 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, two polynomialrelated to the first zagreb index are, m∗1(g,x) =(12n)x4n m0(g,x) =4nx3 proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersengraph tn = {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. thedegree of each vertex in p(k,t) is 3 and now two polynomial related to the first zagreb index are i.e. , m∗1(g,x) =∑[xi∈v (g)][d(xi)][x[xi]] m0(g,x) =∑[xi∈v (g)](x)[d(xi)] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in two polynomialrelated to the first zagreb index topological index of the general form, https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 9 ⇒ m∗1(g,x) =∑[xi∈v (g)](3)[x[xi]] ⇒ m∗1(g,x) =∑[xi∈v (g)](3)x[4n] ⇒ m∗1(g,x) =[|v (g)|](3)x4n ⇒ m∗1(g,x) =4n(3)x4n m∗1(g,x) =(12n)x4n. m∗1(g,x) =[3x4n]× [general vertices of petersen graph]now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in two polynomialrelated to the first zagreb index topological index of the general form, ⇒ m0(g,x) =∑[xi∈v (g)](x)3 ⇒ m0(g,x) =[|v (g)|](x)3 m0(g,x) =4nx3. m0(g,x) =[x3]× [general vertices of petersen graph] theorem 2.7 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, zagreb typepolynomials are, ma,b(g,x) =6nx[3(a+b)] m′a,b(g,x) = (6n)(x)[(a+3)(b+3)] proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now zagreb type polynomials are i.e. , ma,b(g,x) = ∑[xi,xj∈e(g)](x)[a{d(xi)}+b{d(xi)}] m′a,b(g,x) = ∑[xi,xj∈e(g)](x)([a+{d(xi)}][b+{d(xi)}])now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in zagreb typepolynomials topological index of the general form, ⇒ ma,b(g,x) =∑[xi,xj∈e(g)](x)[a(3)+b(3)] ⇒ ma,b(g,x) =∑[xi,yj∈e(g)](x)[3(a+b)] ⇒ ma,b(g,x) =[|e(g)|](x)[3(a+b)] ma,b(g,x) =6nx[3(a+b)].now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in zagreb typepolynomials topological index of the general form, https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 10 ⇒ m′a,b(g,x) =∑[xi,xj∈e(g)](x)([(a+(3))][(b+(3))]) ⇒ m′a,b(g,x) =[|e(g)|](x)[(a+(3))(b+(3))] ⇒ m′a,b(g,x) =[|6n|](x)[(a+3)(b+3)] m′a,b(g,x) =(6n)(x)[(a+3)(b+3)]. m′a,b(g,x) =[(x)[(a+3)(b+3)]]× [general edges of petersen graph] theorem 2.8 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, atomic-bond-connectivity (abc) index are, abc(g) = (4n). proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now atomic-bond-connectivity (abc) index are i.e. , abc(g) = ∑[x1,x2∈e(g)] √[d(x1)+d(x2)]−2d(x1)×d(x2) → [1] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersengraph about every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values inatomic-bond-connectivity (abc) index topological index of the general form, ⇒ abc(g) = ∑[x1,x2∈e(g)] √[d(x1)+d(x2)]−2d(x1)×d(x2)putting values in above equation; ⇒ abc(g) =[|e(g)|]√[(3)+(3)−2] (3)×(3) ⇒ abc(g) =[|6n|]√ (4) (3)2 ⇒ abc(g) =(6n)√4 (3) ⇒ abc(g) = (4n). abc(g) = general vertices of petersen graph theorem 2.9 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, geometricarithmetic(ga) index are, ga(g) =6n proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now geometric arithmetic(ga) index are i.e. , ga(g) = ∑[x1,x2∈e(g)] 2√d(x1)×d(x2)d(x1)+d(x2) https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 11 now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in geometricarithmetic(ga) index topological index of the general form, ⇒ ga(g) =[|e(g)|]2√(3)×(3) (3)+(3) ⇒ ga(g) =[|6n|]2√(3)2 (6) ⇒ ga(g) =(6n)2√(3)2 6 ⇒ ga(g) =(6n)2(3) (6) ⇒ ga(g) =6n. ga(r)= general edges of petersen graph. theorem 2.10 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, first multiplezagreb index are, pm1(g) = (6)6n proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now first multiple zagreb index are i.e. , pm1(g) = ∏[x1,x2∈e(g)][d(x1)+d(x2)]now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in first multiplezagreb index topological index of the general form, ⇒ pm1(g) = ∏[x1,x2∈e(g)][(3+3)] ⇒ pm1(g) = (6)[|e(g)|] ⇒ pm1(g) = (6)[|6n|] ⇒ pm1(g) = (6)6n. pm1(g) = general edges of petersen graph to the power of six. theorem 2.11 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, secondmultiple zagreb index are, pm2(g) = (9)[6n] proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now second multiple zagreb index are i.e. , pm2(g) = ∏[x1,x2∈e(g)][d(x1)×d(x2)] https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 12 now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in second multiplezagreb index topological index of the general form, ⇒ pm2(g) = ∏[x1,x2∈e(r)][(3)× (3)] ⇒ pm2(g) = (9)[|e(r)|] ⇒ pm2(g) = (9)[6n]. pm2(g) = (9)6n pm2(g) = general edges of petersen graph to the power of nine. theorem 2.12 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, forgottenpolynomial are, f(r) = (6n)x18 proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now forgotten polynomial are i.e. , f(g,x) =∑[x1,x2∈e(g)](x)[(dx1)2+(dx2)2]now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in forgottenpolynomial topological index of the general form, ⇒ f(g,x) =∑[x1,x2∈e(g)](x)[(3)2+(3)2] ⇒ f(g,x) =[|e(g)|](x)[9+9] ⇒ f(g) =[6n](x)18 ⇒ f(r) =(6n)x18. f(r) =[x18]× [general edges of petersen graph] theorem 2.13 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, symmetricdivision deg. index are, sdd(g) = 12n proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now symmetric division deg. index are i.e. , sdd(g) = ∑[x1,x2∈e(g)][d(x1)2+d(x2)2d(x1)d(x2) ]now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersen graphabout every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in symmetric https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 13 division deg. index topological index of the general form, ⇒ sdd(g) = ∑[x1,x2∈e(g)] mini(3,3)max(3,3) + maxi(3,3)mini(3,3) ⇒ sdd(g) =[|e(g)|][3 3 + 3 3 ] ⇒ sdd(g) =[|e(g)|][(3)+(3) 3 ] ⇒ sdd(g) =[|6n|][(6) 3 ] ⇒ sdd(g) =(6n)[2] ⇒ sdd(g) =12n. sdd(g) = two times of general edges of petersen graph. theorem 2.14 let p(k,t) be petersen subdivision graph. then, for tn = {1,2,3, ...}, generalconnectivity index are, sdd(g) = 12n proof: the petersen graph tn = {1,2,3, ...} appears in figure(graph). the petersen graph tn= {1,2,3, ...} contains v (g′) = 4n no of vertices and e(g′) = 6n no of edges. the degree ofeach vertex in p(k,t) is 3 and now general connectivity index are i.e. , m1(g) = ∑[x2∈v (g)][dg(x2)]2 m2(g) = ∑[x1,x2∈e(g)][(dg(x1))× (dg(x2))] → [1] now we suppose vertices are v (g) = 4n, edges are e(g) = 6n and degree of petersengraph about every each vertices is p(k,t) =[d(x1),d(x2)] = 3. now putting the values in generalconnectivity index topological index of the general form, ⇒ m1(g) = ∑[x2∈v (g)] [(3)2] ⇒ m1(g) = [|v (g)|](3)2 ⇒ m1(g) = (4n)(9) ⇒ m1(g) = 36n. m1(g) = nine times to vertices of petersen graph. in general form of real number index is, in equation [1] becomes; ⇒ m2(g) = ∑[x1,x2∈e(g)][dg(x1)×dg(x2)]now putting values in above equation. ⇒ m2(g) = ∑[x1,x2∈e(g)](3)(3) ⇒ m2(g) = [|e(g)|](3)(3) ⇒ m2(g) = (6n)(9) https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 14 ⇒ m2(g) = 54n. m2(g) = nine times to edges of petersen graph. 3. numerical examples every petersen graph is cyclic graph and the graph g′ in general form consists v having set ofvertex and e having set of edge, if the natural number, there exist n the graph with vertices are v (g′)=4n, edges are e(g′)=6n, and the specific of this graph is that about degree of everyeach vertex is p(k,t) = [d(x1),d(x2)] = 3. then this graphic which is said to be petersen graphic.then petersen graphic is denoted by p[v (g′),e(g′)] =(4n,6n)the core features of petersen map explored by length in 1985. however, the petersen line contin-uously arise in literature of the theoretical graphing. by this article, we update previous analysisto introduce additionally fresh findings on the petersen mapping.sylvester’s association to graphs of invariants and covariants requires interpretation of principleof invariants in 1880s. example 3.1. if there exists n is positive natural number then tn = {1,2,3, ...}, so graphwith vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form of petersen expressed by p[v (g′),e(g′)] =(4n,6n). this graph having a specification, that degree of every each vertex is p(k,t)= [d(x1),d(x2)]= 3.now we write; tn =1,2,3, .... v (g′)=4n .........(1) e(g′)=6n .........(2) k = d(x1)=3 t = d(x2)=3put n =3 in equations (1) and (2) and these equations become; t3 =3 v (g′)=12 e(g′)=18 p(k,t) = [d(x1),d(x2)]= 3 p[v (g′),e(g′)] =(12,18)now figure is; example 3.2. if there exists n is positive natural number then tn = {1,2,3, ...}, so graphwith vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form of petersen expressed by p[v (g′),e(g′)] =(4n,6n). this graph having a specification, that degree of every each vertex is p(k,t)= [d(x1),d(x2)]= 3. https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 15 figure 1. petersen graph now we write; tn =1,2,3, .... v (g′)=4n .........(1) e(g′)=6n .........(2) k = d(x1)=3 t = d(x2)=3put n =4 in equations (1) and (2) and these equations become; t4 =4 v (g′)=16 e(g′)=24 p(k,t) = [d(x1),d(x2)]= 3 p[v (g′),e(g′)] =(16,24)now figure is; figure 2. petersen graph example 3.3. if there exists n is positive natural number then tn = {1,2,3, ...}, so graphwith vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form of petersen expressed by p[v (g′),e(g′)] =(4n,6n). this graph having a specification, that degree of every each vertex is p(k,t)= [d(x1),d(x2)]= 3.now we write; https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 16 tn =1,2,3, .... v (g′)=4n .........(1) e(g′)=6n .........(2) k = d(x1)=3 t = d(x2)=3put n =5 in equations (1) and (2) and these equations become; t5 =5 v (g′)=20 e(g′)=30 p(k,t) = [d(x1),d(x2)]= 3 p[v (g′),e(g′)] =(20,30)now figure is; figure 3. petersen graph example 3.4. if there exists n is positive natural number then tn = {1,2,3, ...}, so graphwith vertices are v (g′) = 4n, and edges e(g′) = 6n, in general form of petersen expressed by p[v (g′),e(g′)] =(4n,6n). this graph having a specification, that degree of every each vertex is p(k,t)= [d(x1),d(x2)]= 3.now we write; tn =1,2,3, .... v (g′)=4n .........(1) e(g′)=6n .........(2) k = d(x1)=3 t = d(x2)=3put n =10 in equations (1) and (2) and these equations become; t10 =10 v (g′)=40 https://doi.org/10.28924/ada/ma.3.20 eur. j. math. anal. 10.28924/ada/ma.3.20 17 e(g′)=60 p(k,t) = [d(x1),d(x2)]= 3 p[v (g′),e(g′)] =(40,60)now figures are; figure 4. petersen graph 4. conclusion and future studies frequently, graph theory is refuted using the petersen graph. in this paper, the general petersengraph was constructed, and the exact expressions of the first and second zagreb indices, the forgottentopological index, the hyper zagreb index, the reduced second zagreb index and the petersen graphin terms of cyclic graph were then examined. the future work will concentrate on topologicalindeces, then generalised petersen via graph operations. references [1] m.h. khalifeh, h. yousefi-azari, a.r. ashrafi, the first and second zagreb indices of some graph operations, discr.appl. math. 157 (2009) 804-811. https://doi.org/10.1016/j.dam.2008.06.015.[2] v. anandkumar, r.r. iyer, on the hyper-zagreb index of some operations on graphs, int. j. pure appl. math. 112(2017) 213-220. https://doi.org/10.12732/ijpam.v112i2.2.[3] s.m. sankarraman, a computational approach on acetaminophen drug using degree-based topological indices andm-polynomials, biointerface res. appl. chem. 12 (2021) 7249-7266. https://doi.org/10.33263/briac126. 72497266.[4] i. gutman, multiplicative zagreb indices of trees, bull. soc. math. banja luka. 18 (2011) 17-23.[5] e. estrada, l. torres, l. rodriguez, i. gutman, an atom-bond connectivity index: modelling the enthalpy of formationof alkanes, indian j. chem. 37 (1998) 849-855.[6] x. ren, x. hu, b. zhao, proving a conjecture concerning trees with maximal reduced reciprocal randic index, matchcommun. math. comput. chem. 76 (2016) 171-184.[7] a.r. bindusree, n. cangul i., v. lokesha, s. cevik a., zagreb polynomials of three graph operators, filomat. 30(2016) 1979-1986. https://doi.org/10.2298/fil1607979b. https://doi.org/10.28924/ada/ma.3.20 https://doi.org/10.1016/j.dam.2008.06.015 https://doi.org/10.12732/ijpam.v112i2.2 https://doi.org/10.33263/briac126.72497266 https://doi.org/10.33263/briac126.72497266 https://doi.org/10.2298/fil1607979b eur. j. math. anal. 10.28924/ada/ma.3.20 18 [8] b. furtula, i. gutman, a forgotten topological index, j. math. chem. 53 (2015) 1184-1190. https://doi.org/10. 1007/s10910-015-0480-z.[9] b. bollabas, p. erd, graphs of extremal weights, ars comb. 50 (1998) 225-233.[10] a.r. ashrafi, m. mirzargar, pi, szeged and edge szeged of an infinite family of nanostardendrimers, indian j. chem.47 (2008) 1656-1660.[11] z. chen, m. dehmer, f. emmert-streib, y. shi, entropy boundsfor dendrimers, appl. math. comput. 242 (2014)462-472.[12] m.v. diudea, a.e. vizitiu, m. mirzagar, a.r. ashrafi, sadhana polynomial in nano-dendrimers, carpathian j. math.26 (2010) 59-66.[13] b. zhou, n. trinajstic, on general sum-connectivity index, j. math. chem. 47 (2010) 210-218.[14] a. asghar, a. qayyum, n. muhammad, different types of topological structures by graphs, eur. j. math. anal. 3(2022) 3. https://doi.org/10.28924/ada/ma.3.3.[15] z.h. niazi, m.a.t. bhatti, m. aslam, y. qayyum, m. ibrahim, a. qayyum, d-lucky labelling of some special graphs,amer. j. math. anal. 10 (2022) 3-11. https://doi.org/10.12691/ajma-10-1-2. https://doi.org/10.28924/ada/ma.3.20 https://doi.org/10.1007/s10910-015-0480-z https://doi.org/10.1007/s10910-015-0480-z https://doi.org/10.28924/ada/ma.3.3 https://doi.org/10.12691/ajma-10-1-2 1. introduction 1.1. the graph of petersen 1.2. graphical idea of petersen graph 2. main results 3. numerical examples 4. conclusion and future studies references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 14doi: 10.28924/ada/ma.3.14 on the kolmogorov distance for the least squares estimator in the fractional ornstein-uhlenbeck process jaya p. n. bishwal department of mathematics and statistics, university of north carolina at charlotte,376 fretwell bldg, 9201 university city blvd. charlotte, nc 28223, usacorrespondence: j.bishwal@uncc.edu abstract. the paper shows that the distribution of the normalized least squares estimator of the driftparameter in the fractional ornstein-uhlenbeck process observed over [0,t] converges to the standardnormal distribution with an uniform optimal error bound of the order o(t−1/2) for 0.5 ≤ h ≤ 0.63and of the order o(t4h−3) for 0.63 < h < 0.75 where h is the hurst exponent of the fractionalbrownian motion driving the ornstein-uhlenbeck process. for the normalized quasi-least squaresestimator, the error bound is of the order o(t−1/4) for 0.5 ≤ h ≤ 0.69 and of the order o(t4h−3)for 0.69<h < 0.75. 1. introductionthe fractional ornstein-uhlenbeck process, is an extension of ornstein-uhlenbeck process withfractional brownian motion (fbm) driving term. in finance it is known as fractional vasicek model,and is being extensively used these days as one-factor short-term interest rate model which takesinto account the long memory effect of the interest rate. the model parameter is usually unknownand must be estimated from data.parameter estimation in stochastic differential equations is studied in bishwal [1]. for thestandard ornstein-uhlenbeck process, sufficiency and rao-blackwellization was studied in bish-wal [4] where also a time transformation to reduce the general problem to a fixed time case andthe asymptotics were studied in large parameter case. for the fractional ornstein-uhlenbeck pro-cess, berry-esseen inequalities of minimum contrast estimators based on continuous and discreteobservations was studied in bishwal [2]. hu et al. [11] studied parameter estimation for the frac-tional ornstein-uhlenbeck process of general hurst parameter. bishwal [5] studied berry-esseeninequalities for the fractional black-karasinski model of term structure of interest rates. usingfractional levy process as the driving term which include jumps, maximum quasi-likelihood estima-tion in fractional levy stochastic volatility model was studied in bishwal [3]. parameter estimationin partially observed stochastic differential system was studied in bishwal [6]. received: 12 nov 2022. key words and phrases. itô stochastic differential equation; fractional brownian motion; fractional ornstein-uhlenbeck process; long-memory; least squares estimator; quasi-least squares estimator; rate of weak convergence;kolmogorov distance; wiener chaos; fourier method; analytic continuation.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 2 let (ω,f,{ft}t≥0,p ) be a stochastic basis on which is defined the ornstein-uhlenbeck process xt satisfying the itô stochastic differential equation dxt = θxtdt + dw h t , t ≥ 0, x0 = 0 (1.1) where {wht } is a fractional brownian motion with h > 1/2 with the filtration {ft}t≥0 and θ < 0is the unknown parameter to be estimated on the basis of continuous observation of the process {xt} on the time interval [0,t ].recall that a fractional brownian motion (fbm) has the covariance c̃h(s,t) = 1 2 [ s2h + t2h −|s − t|2h ] , s,t > 0. (1.2) for h > 0.5 the process has long range dependence or long memory and the process is self-similar.for h 6= 0.5, the process is neither a markov process nor a semimartingale. for h = 0.5, theprocess reduces to standard brownian motion.note that the solution of the equation (1.1) is given by xt = ∫ t 0 eθ(t−s)dwhs . (1.3) let the realization {xt, 0 ≤ t ≤ t} be denoted by xt0 . let ptθ be the measure generatedon the space (ct ,bt ) of continuous functions on [0,t ] with the associated borel σ-algebra btgenerated under the supremum norm by the process xt0 and pt0 be the standard wiener measure.applying girsanov type formula for fbm, when θ is the true value of the parameter, ptθ is absolutelycontinuous with respect to pt0 and the radon-nikodym derivative (likelihood) of ptθ with respectto pt0 based on xt0 is given by lt (θ) := dptθ dpt0 (xt0 ) = exp { θ ∫ t 0 qtdzt − θ2 2 ∫ t 0 q2tdvt } . (1.4) consider the score function, the derivative of the log-likelihood function, which is given by yt (θ) := ∫ t 0 qtdzt −θ ∫ t 0 q2tdvt. (1.5) a solution of yt (θ) = 0 provides the maximum likelihood estimate (mle) θt := ∫t 0 qtdzt∫t 0 q2tdvt. (1.6) kleptsyna and le breton [13] showed that θt is strongly consistent. using the fourier method,bishwal [2] proved a berry-esseen type theorem for the estimator θt which gives the rate of weakconvergence in asymptotic normality.using the fractional itô formula, the score function yt (θ) can be written as yt (θ) = 1 2 [ λh (2 − 2h) zt ∫ t 0 t2h−1dzt −t ] −θ ∫ t 0 q2tdvt. (1.7) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 3 consider the contrast function kt (θ) := − thγ(h) 2 −θ ∫ t 0 q2tdvt (1.8) and the minimum contrast estimate (mce) θ̄t := −thγ(h) 2 ∫t 0 q2tdvt . (1.9) the least squares estimator (lse) of θ minimizes∫ t 0 |ẋt −θxt|2dt (1.10) and is given by θ̂t := ∫t 0 xtdxt∫t 0 x2t dt. = θ− ∫t 0 xtdw h t∫t 0 x2t dt. (1.11) based on ergodicity, quasi least squares estimate (qlse) θ̃t := ( −thγ(2h)∫t 0 x2t dt ) 1 2h (1.12) the lse and the qlse are strongly consistent and asymptotically norma as t →∞ √ t (θ̂t −θ) →d n(0,θσ2h), √ t (θ̃t −θ) →d n(0, θσ2h 4h2 ) (1.13) where σ2h := (4h − 1) ( 1 + γ(3 − 4h)γ(4h − 1) γ(2 − 2h)γ(2h) ) . (1.14) observe that h = 1/2, σ2h = 2. in this case the lse and the mle are identical. since θ̃t is aconsistent estimator of θ, we can derive the self normalized limit distributions immediately: ( t σ2 h θ̃t )1/2(θ̂t −θ) →d n(0, 1), 2h( t σ2 h θ̃t )1/2(θ̃t −θ) →d n(0, 1). (1.15) define mt := ∫ t 0 xt dw h t and it := ∫ t 0 x2t dt, nt := θ 2hit −thγ(2h). (1.16) vh,θ := θ −2hhγ(2h). (1.17) observe that ( t −σ2 h θ )1/2 (θ̂t −θ) = ( −σ2hθ t )1/2 mt( σ2 h θ t ) it (1.18) applying taylor’s formula to the function x− 12h at the point vh,θ, we have( it t )− 1 2h = v − 1 2h h,θ − 1 2h v −1+2h 2h h,θ ( it t −vh,θ ) + 1 + 2h 8h2 $ −1+4h 2h t ( it t −vh,θ )2 (1.19) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 4 where $t is a random point between vh,θ and itt . further θ̃t −θ = − θ1+2h 2h2γ(2h) ( it t −vh,θ ) + (1 + 2h)(hγ(2h)) 1 2h 8h2 $ −1+4h 2h t ( it t −vh,θ )2 . (1.20) thus 2h ( t −σ2 h θ )1/2 (θ̃t −θ) = ( −σ2hθ 4th2 )1/2 nt( σ2 h θ 4th2 ) it . (1.21) we study the large deviations, moderate deviations and berry-esseen bounds of the lse andthe qlse in this paper. we will use the following optimal fourth moment theorem from nourdinand peccati [14] in the sequel. see also douissi et al. [15]. theorem 1.1 (skewness kurtosis inequality) let (xn)n≥1 be a sequence of random variables in fixed wiener chaos of order q ≥ 2 such that v ar(xn) = 1. assume xn converges to normal distribution which is equivalent to limn e(xn)4 = 3, which is also known as the fourth moment theorem. then we have the following optimal rate for dtv (xn,n) known as the optimal fourth moment theorem: there exist two constants c,c > 0 depending only on the sequence (xn)n≥1 but not on n, such that c max{e(x4n ) − 3, |e(x 3 n )|}≤ dtv (xn,n) ≤ c max{e(x 4 n ) − 3, |e(x 3 n )|}. (1.22) let φ(·) denote the standard normal distribution function. throughout the paper, c denotes ageneric constant (which does not depend on t and x). we have not tried to estimate the constantin the bound on normal approximation.hu et al. [11] obtained limiting normal distribution of the lse and the qlse for the memoryrange 1 2 < h ≤ 3 4 with the rate √t for 1 2 < h < 3 4 and √t (log t )−1/2 for the case h = 3 4 , andlimiting rosenblatt distribution for the memory range 3 4 < h < 1.we only consider the memory range 1 2 < h < 3 4 . jiang et al. [12] used self-normalization alongwith the splitting method for the lse and the qlse in fractional ornstein-uhlenbeck process andobtained the rate t−1/2 log t for the range 1 2 ≤ h ≤ 5 8 for the lse and t−1/4 log t for the range 1 2 ≤ h ≤ 11 16 for the qlse. they obtained the rate t4h−3 for the range 5 8 < h < 3 4 for the lseand the same rate t4h−3 for the range 11 16 < h < 3 4 for the qlse.in this paper we improve the first rate to t−1/2 for the mle for the range 1 2 ≤ h ≤ 5 8 and t−1/4 for the range 1 2 ≤ h ≤ 11 16 for the qlse using the squeezing method as in chapter 1 inbishwal [1]. the main contribution of the paper is thus improvement in the rate my removing the log t term. note the critical points: 1 2 = 0.50, 5 8 = 0.63, 2 3 = 0.67, 11 16 = 0.69, 3 4 = 0.75. also 0.63 + 0.06 = 0.69, 0.69 + 0.06 = 0.75. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 5 remark on the critical point 5 8 : for the discrete observations case, es-sebaiy and viens [7]pointed out that if 0 < h < 5 8 , then the fourth moment is of the order n−1 and if 5 8 < h < 3 4 ,then the fourth moment is of the order n2(4h−3) where n is the number of observations. theberry-esseen rate for θ̂ is shown to be of the order n−1/4 for 0 < h < 5 8 and of the order n−(4h−3)/2 if 5 8 < h < 3 4 . for h = 3 4 , the rate is (log n)−1/4. the proofs also need large deviation results for the stochastic integral and the energy integral.these integrals can be represented by multiple wiener integrals. then their expectations andvariances as well as the fourth moment of their malliavin derivatives can be estimated.first we calculate bounds on the moments. let ϕt (s,t) := e −θ|t−s|, ψt (s,t) := e −2θt+θ(s+t), gt (s,t) := e −θ(t−s)i[0,t](s), (1.23) vh,θ := θ −2hhγ(2h), ch,θ := θ 1−4h(4h − 1)h2 ( γ2(2h) + γ(2h)γ(3 − 4h)γ(4h − 1) γ(2 − 2h) ) . (1.24) observe that xt = i1(gt (·,t)), (1.25) mt = ∫ t 0 xtdw h t = ∫ t 0 ∫ t 0 eθ(t−s)dwhs dw h t = 1 2 eθ|t−s|dwhs dw h t = 1 2 i2(ϕt ), (1.26) it = ∫ t 0 x2t dt = 1 2θ i2(ϕt ) + 1 2θ i2(ψt ) + ∫ t 0 ‖gt (·,t)‖2hdt (1.27)where i1 and i2 are first and second wiener chaos respectively. furthermore,∫ t 0 ‖gt (·,t)‖2hdt = vh,θt + o(t ). (1.28) for 1 2 < h < 3 4 , e(xtxs) ≤ c|t − s|2h−2, (1.29) ‖ϕt‖2h = 2t (ch,t + (o(1)), ‖ψt‖ 2 h = o(1). (1.30)for h = 1 2 , by the isometry of the itô integral, we obtain ‖ϕt‖2h = 2 ∫ t 0 ∫ t 0 e2θ(t−s)dtds = t θ + e2θt − 1 2θ2 = 2t (ch,t + (o(1)). (1.31) ‖ψt‖2h = e −4θt ∫ t 0 ∫ t 0 e−2θ(t+s)dtds = (e−2θt − 1)2 4θ2 = o(1). (1.32) for 1 2 < h < 3 4 ,, using lemma 5.3 in hu and nualart [10], we have ‖ψt‖2h ≤ γ2(2h) (2h − 1)2 θ−4h. (1.33) let υt = t for h = 12 and υt = t8h−4 for 12 < h < 34. we obtain the variances bounds on the malliavin derivative of mt and it . e(‖dmt‖2h −e‖dmt‖ 2 h) ≤ cυt , (1.34) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 6 e(‖dit‖2h −e‖dit‖ 2 h) ≤ cυt (1.35)where d is the malliavin derivative operator.we have the bound on the fourth moment e(‖di2(ϕt )‖2h −e‖di2(ϕt )‖ 2 h) 2 ≤ cυt . (1.36) for 1 2 < h < 3 4 ,, we have the bound on the fourth moment e(‖di2(ϕt )‖2h −e‖di2(ϕt )‖ 2 h) 2 ≤ ct8h−4. (1.38) we have the bound on the fourth moment e(‖di2(ψt )‖2h −e‖di2(ψt )‖ 2 h) 2 ≤ c. (1.39) dsi2(ψt ) = −2e−2θt+θs ∫ t 0 eθtdwht . (1.40) e‖dsi2(ψt )‖4h = 16e −8θt (∫ t 0 eθtdwht )4(∫ t 0 e2θtdt )2 = 48e−8θt (∫ t 0 e2θtdt )4 . (1.41) e‖dsi2(ψt )‖2h = 4e −4θt (∫ t 0 e2θtdt )2 . (1.42) therefore e(‖di2(ψt )‖2h −e‖di2(ψt )‖ 2 h) 2 = e‖di2(ψt )|4h − (e‖di2(ψt )‖ 2 h) 2 = 2(1 −e−2θt )4 θ4 . (1.43)similarly for the case 1 2 < h < 3 4 , it can be shown that e(‖di2(ψt )‖2h −e‖di2(ψt )‖ 2 h) 2 ≤ 32γ4(2h) (2h − 1)4 θ−8h. (1.44) first we have the berry-esseen bounds for the stochastic integral and adjusted energy integral. byusing the optimal fourth moment theorem (skewness-kurtosis inequality) from stein-malliavintheory, we have:for 1 2 ≤ h ≤ 5/8, we have sup x∈r ∣∣∣∣∣∣p  ( c−1 h,θ t )1/2 mt ≤ x − φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1 h,θ t )1/2 mt‖2h −e‖d ( c−1 h,θ t )1/2 mt‖2h )2 1/2 ≤ ct−1/2. (1.45) for 5 8 < h < 3 4 , sup x∈r ∣∣∣∣∣∣p  ( c−1 h,θ t )1/2 mt ≤ x − φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1 h,θ t )1/2 mt‖2h −e‖d ( c−1 h,θ t )1/2 mt‖2h )2 1/2 ≤ c t4h−3. (1.46) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 7 for 1 2 ≤ h ≤ 5/8, we have sup x∈r ∣∣∣∣∣∣p  ( c−1 h,θ t )1/2( θ̃tit − t −σ2 h ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1 h,θ t )1/2( θ̃tit − t−σ2 h ) ‖2h −e‖d ( c−1 h,θ t )1/2( θ̃tit − t−σ2 h ) ‖2h )2 1/2 ≤ ct−1/2. (1.47)for 5 8 < h < 3 4 , sup x∈r ∣∣∣∣∣∣p  ( c−1 h,θ t )1/2( θ̃tit − t −σ2 h ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1 h,θ t )1/2( θ̃tit − t−σ2 h ) ‖2h −e‖d ( c−1 h,θ t )1/2( θ̃tit − t−σ2 h ) ‖2h )2 1/2 ≤ c t4h−3. (1.48)for 1 2 ≤ h ≤ 5 8 , we have for |x| ≤ 2(log t )1/2, sup y∈r ∣∣∣∣∣∣p  ( −σ2hθ̃t t )1/2 mt − (( −σ2hθ̃t t ) it − 1 ) x ≤ y − φ(y) ∣∣∣∣∣∣ ≤ ct−1/2. (1.49) for 5 8 < h < 3 4 , we have for |x| ≤ 2(log t )1/2, sup y∈r ∣∣∣∣∣∣p  ( −σ2hθ̃t t )1/2 mt − (( −σ2hθ̃t t ) it − 1 ) x ≤ y − φ(y) ∣∣∣∣∣∣ ≤ ct4h−3. (1.50) for 1 2 ≤ h ≤ 11 16 , we have sup x∈r ∣∣∣∣∣∣p  ( c−1 h,θ t )1/2 nt ≤ x − φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1 h,θ t )1/2 nt‖2h −e‖d ( c−1 h,θ t )1/2 nt‖2h )2 1/2 ≤ ct−1/2. (1.51) for 11 16 < h < 3 4 , sup x∈r ∣∣∣∣∣∣p  ( c−1 h,θ t )1/2 nt ≤ x − φ(x) ∣∣∣∣∣∣ ≤ c e ( ‖d ( c−1 h,θ t )1/2 nt‖2h −e‖d ( c−1 h,θ t )1/2 nt‖2h )2 1/2 ≤ c t4h−3. (1.52) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 8 for 1 2 ≤ h ≤ 11 16 , we have for |x| ≤ 2(log t )1/2, sup y∈r ∣∣∣∣∣∣p 2h ( −σ2hθ̃t t )1/2 nt − (( −σ2hθ̃t t ) it − 1 ) x ≤ y − φ(y) ∣∣∣∣∣∣ ≤ ct−1/4. (1.53) for 11 16 < h < 3 4 , we have for |x| ≤ 2(log t )1/2, sup y∈r ∣∣∣∣∣∣p 2h ( −σ2hθ̃t t )1/2 nt − (( −σ2hθ̃t t ) it − 1 ) x ≤ y − φ(y) ∣∣∣∣∣∣ ≤ ct4h−3. (1.54) 2. main results we need the next two lemmas from jiang et al. [12] on large deviations to obtain bounds onthe tail probabilities of the estimators. the first lemma is on large deviations for stochastic integral. lemma 2.1 for every δ > 0, p {∣∣∣∣mtt ∣∣∣∣ ≥ δ} ≤ c exp ( − t1/2δ 4c 1/2 h,θ ) . remark for the case h = 0.5, there is a long history of work: for every δ > 0, p {∣∣∣∣mtt ∣∣∣∣ ≥ δ} ≤ c0 exp (−c1tδ2) . see gao and jiang [9].for any 0 ≤ α ≤ θ2/4, there exist constants c3 and c4 such that e(eαit ) ≤ c3ec4αt . (2.1) see gao and jiang [9]. by chebyshev inequality, we have p (|xt −e(xt )| ≥ δ) ≤ 2 exp(−θδ2). (2.2) the second lemma is on large deviations in the ergodic theorem. lemma 2.2 for every δ > 0, p {∣∣∣∣itt −vh,θ ∣∣∣∣ ≥ δ} ≤ c exp ( − t1/2δ 4c 1/2 h,θ ) . observe that by (1.11) θ̂t = θ− mt it . https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 9 using the elementary inequality p (| ξ η | ≥ u) ≤ p (|ξ| ≥ uv) + p (η − 2v| ≥ v), (2.3) we have p (|θ̂t −θ| ≥ δ) ≤ p (|it −vh,θt | ≥ 12vh,θt ) + p (|θ̂t −θ| ≥ δ, |it −vh,θt | < 1 2 vh,θt ) ≤ p (|it −vh,θt | ≥ 12vh,θt ) + p (|mt | ≥ 1 2 vh,θtδ). (2.4) combining lemma 2.1 and lemma 2.2, we obtain lemma 2.3 for every δ > 0 and large t > 0, we have a) p (|θ̂t −θ| ≥ δ) ≤ c0 exp(−c1t1/2δ) b) p (|θ̃t −θ| ≥ δ) ≤ c0 exp(−c1t1/2δ1/2). to obtain the rate of normal approximation for the lse and the qlse, we need the followingtail probability estimate of the estimators. lemma 2.4 (a) p  ( t −σ2 h θ̃t )1/2 |θ̂t −θ| ≥ 2(log t )1/2  ≤ ct−1/2. (b) p 2h ( t −σ2 h θ̃t )1/2 |θ̃t −θ| ≥ 2(log t )1/2  ≤ ct−1/4. proof : observe that p  ( t −σ2 h θ̃t )1/2 |θ̂t −θ| ≥ 2(log t )1/2  = p  ∣∣∣∣∣∣∣∣∣ ( −σ2hθ̃t t )1/2 mt ( −σ2 h θ̃t t )it ∣∣∣∣∣∣∣∣∣ ≥ 2(log t )1/2  ≤ p  ∣∣∣∣∣∣ ( −σ2hθ̃t t )1/2 mt ∣∣∣∣∣∣ ≥ (log t )1/2  + p {∣∣∣∣∣−σ2hθ̃tt it ∣∣∣∣∣ ≤ 12 } ≤ ∣∣∣∣∣∣p  ( −σ2hθ̃t t )1/2 |mt | ≥ (log t )1/2 − 2φ(−(log t )1/2) ∣∣∣∣∣∣ +2φ(−(log t )1/2) + p {∣∣∣∣∣σ2hθ̃tt it − 1 ∣∣∣∣∣ ≥ 12 } https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 10 ≤ sup x∈r ∣∣∣∣∣∣p  ( −σ2hθ̃t t )1/2 |mt | ≥ x − 2φ(−x) ∣∣∣∣∣∣ +2φ(−(log t )1/2) + p {∣∣∣∣∣ ( −σ2hθ̃t t ) it − 1 ∣∣∣∣∣ ≥ 12 } ≤ ct−1/2 + c(t log t )−1/2 + c exp ( − t1/2 8c 1/2 h,θ ) ≤ ct−1/2.the bounds for the first and the third terms come from lemma 2.2 and lemma 2.1 respectively andthat for the middle term comes from feller [8] (p. 166). proof of (b) is similar. now we are ready to obtain the uniform rate of normal approximation of the distribution of thelse and the qlse.recall that σ2h := (4h − 1) ( 1 + γ(3 − 4h)γ(4h − 1) γ(2 − 2h)γ(2h) ) . (2.5) theorem 2.5a) if 1 2 ≤ h ≤ 5 8 sup x∈r ∣∣∣∣∣∣p  ( t −σ2 h θ̃t )1/2 (θ̂t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ ct−1/2. b) if 5 8 < h < 3 4 sup x∈r ∣∣∣∣∣∣p  ( t −σ2 h θ̃t )1/2 (θ̂t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ ct4h−3. c) if 1 2 ≤ h ≤ 11 16 sup x∈r ∣∣∣∣∣∣p 2h ( t −σ2 h θ̃t )1/2 (θ̃t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ ct−1/4. d) if 11 16 < h < 3 4 sup x∈r ∣∣∣∣∣∣p 2h ( t −σ2 h θ̃t )1/2 (θ̃t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ ct4h−3. proof : first we prove (a). we shall consider two possibilities (i) and (ii). (i) |x| > 2(log t )1/2. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 11 we shall give a proof for the case x > 2(log t )1/2. the proof for the case x < −2(log t )1/2 runssimilarly. note that∣∣∣∣∣∣p  ( t −σ2 h θ̃t )1/2 (θ̂t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ p  ( t −σ2 h θ̃t )1/2 (θ̂t −θ) ≥ x +φ(−x). (2.6) but from feller [8] (p. 166) we have φ(−x) ≤ φ(−2(log t )1/2) ≤ ct−1. (2.7) moreover, by lemma 2.4 (a), we have p  ( t −σ2 h θ̃t )1/2 (θ̂t −θ) ≥ 2(log t )1/2  ≤ ct−1/2. (2.8) hence ∣∣∣∣∣∣p  ( t −σ2 h θ̃t )1/2 (θ̂t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ ct−1/2. (2.9) (ii) |x| ≤ 2(log t )1/2. let at :=  ( t −σ2 h θ̃t )1/2 |θ̂t −θ| ≤ 2(log t )1/2  and bt := { it t > c0 } (2.10) where 0 < c0 < 1−σ2 h θ . by lemma 2.4, we have p (act ) ≤ ct −1/2. (2.11) by lemma 2.1, we have p (bct ) = p {( −σ2hθ̃t t ) it − 1 < σ2hθc0 − 1 } < p {∣∣∣∣∣ ( −σ2hθ̃t t ) it − 1 ∣∣∣∣∣ > 1 −σ2hθc0 } ≤ c exp ( − t1/2(1 −σ2hθc0) 4c 1/2 h,θ ) . (2.12) let b0 be some positive number. on the set at∩bt for all t > t0 with 4b0(log t0)1/2(σ2hθt )1/2 ≤ c0, we have ( t −σ2 h θ̃t )1/2 (θ̂t −θ) ≤ x ⇒ it + b0t (θ̂t −θ) < it + ( t −σ2 h θ̃t )1/2 σ2hb0θx ⇒ ( t −σ2 h θ̃t )1/2 (θ̂t −θ)[it + b0t (θt −θ)] < x[it + ( t −σ2 h θ̃t )1/2 σ2hb0θx] ⇒ (θ̂t −θ)it + b0t (θt −θ)2 < ( −σ2hθ̃t t )1/2 itx + σ 2 hb0θx 2 https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 12 ⇒ −mt + (θ̂t −θ)it + b0t (θ̂t −θ)2 < −mt + ( σ2hθ̃t t )1/2 itx + σ 2 hb0θx 2 ⇒ 0 < −mt + ( −σ2hθ̃t t )1/2 itx + σ 2 hb0θx 2 since it + b0t (θ̂t −θ) > tc0 + b0t (θ̂t −θ) > 2σ2hb0(log t ) 1/2 ( −σ2hθ̃t t )1/2 −σ2hb0(log t ) 1−h ( −σ2hθ̃t t )1/2 = σ2hb0(log t ) 1/2 ( −σ2hθ̃t t )1/2 > 0. on the other hand, on the set at ∩bt for all t > t0 with 4b0(log t0)1/2(−σ2hθ̃tt0 )1/2 ≤ c0, wehave ( t −σ2 h θ̃t )1/2 (θ̂t −θ) > x ⇒ it −b0t (θ̂t −θ) < it − ( t σ2 h θ̃t )1/2 2b0θx ⇒ ( t −σ2 h θ̃t )1/2 (θ̂t −θ)[it −b0t (θ̂t −θ)] > x[it − ( t −σ2 h θ̃t )1/2 σ2hb0θx] ⇒ (θ̂t −θ)it −b0t (θ̂t −θ)2 > ( t −σ2 h θ̃t )−1/2 itx −σ2hb0θx 2 ⇒ −mt + (θ̂t −θ)it −b0t (θ̂t −θ)2 > −mt + ( t −σ2 h θ̃t )−1/2 itx −σ2hb0θx 2 ⇒ 0 > −mt + ( −σ2hθ̃t t )1/2 itx −σ2hb0θx 2 since it −b0t (θ̂t −θ) > tc0 −b0t (θ̂t −θ) > 2σ2hb0(log t ) 1/2 ( −σ2hθ̃t t )1/2 −σ2hb0(log t ) 1/2 ( −σ2hθ̃t t )1/2 = σ2hb0(log t ) 1/2 ( −σ2hθ̃t t )1/2 > 0. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 13 hence 0 < −mt + ( t −σ2 h θ̃t )1/2 itx −σ2hb0θx 2 ⇒ ( t −σ2 h θ̃t )1/2 (θ̂t −θ) ≤ x. letting d± t,x := −mt + ( σ2hθ̃t t )1/2 itx ±σ2hb0θx 2 > 0  , we obtain d− t,x ∩at ∩bt ⊆ at ∩bt ∩  ( t −σ2 h θ̃t )1/2 (θ̂t −θ) ≤ x  ⊆ d+t,x ∩at ∩bt . (2.13) if it is shown that ∣∣p {d± t,x } − φ(x) ∣∣ ≤ ct−1/2 (2.14) for all t > t0 and |x| ≤ 2(log t )1/2, then the theorem would follow from (2.11) (2.14).we shall prove (2.4) for d+ t,x . the proof for d− t,x is analogous. observe that∣∣p {d+t,x}− φ(x)∣∣ = ∣∣∣∣∣∣p  ( −σ2hθ̃t t )1/2 mt − (( −σ2hθ̃t t ) it − 1 ) x < x + σ2h ( −σ2hθ̃t t )1/2 b0θx 2 − φ(x) ∣∣∣∣∣∣ ≤ sup y∈r ∣∣∣∣∣∣p  ( −σ2hθ̃t t )1/2 mt − (( −σ2hθ̃t t ) it − 1 ) x ≤ y − φ(y) ∣∣∣∣∣∣ + ∣∣∣∣∣∣φ x + (−σ2hθ̃t t )1/2 b0θx 2 − φ(x) ∣∣∣∣∣∣ =: ∆1 + ∆2. (2.15)(1.50) immediately yields ∆1 ≤ ct−1/2. (2.16) on the other hand, for all t > t0, ∆2 ≤ 2 ( −σ2hθ̃t t )1/2 b0θx 2(2π)−1/2 exp(−x2/2) where |x −x| ≤ 2 ( −σ2hθ̃t t )1/2 b0θx 2. since |x| ≤ 2(log t )1/2, it follows that |x̄| > |x|/2 for all t > t0 and consequently ∆2 ≤ 2 ( −σ2hθ̃t t )1/2 b0θx 2(2π)−1/2x2 exp(−x2/8) ≤ ct−1/2. (2.17) https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 14 from (2.15) (2.17), we obtain ∣∣p {d+ t,x } − φ(x) ∣∣ ≤ ct−1/2. this completes the proof of part (a) of the theorem. next we prove (c). again we shall consider two possibilities (i) and (ii). (i) |x| > 2(log t )1−/2. we shall give a proof for the case x > 2(log t )1/2. the proof for the case x < −2(log t )1/2runs similarly. note that∣∣∣∣∣∣p 2h ( t −σ2 h θ̃t )1/2 (θ̃t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ p 2h ( t −σ2 h θ̃t )1/2 (θ̃t −θ) ≥ x  + φ(−x). by (2.7) and lemma 2.4 (b), we have p 2h ( t σ2 h θ̃t )1/2 (θ̃t −θ) ≥ 2(log t )1/2  ≤ ct−1/4. hence ∣∣∣∣∣∣p 2h ( t −σ2 h θ̃t )1/2 (θ̃t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ ct−1/4. (ii) |x| ≤ 2(log t )1/2. let a1,t := 2h ( t −σ2 h θ̃t )1/2 |θ̃t −θ| ≤ 2(log t )1/2  and b1,t := { it t > c0 } where 0 < c0 < 1−σ2 h θ . by lemma 2.4, we have p (ac1,t ) ≤ ct −1/4. (2.18) by lemma 2.1, we have p (bc1,t ) = p {( −σ2hθ 4th2 ) it − 1 < σ2hθc0 − 1 } < p {∣∣∣∣(−σ2hθ4th2 ) it − 1 ∣∣∣∣ > 1 −σ2hθc0} ≤ ct−1. (2.19)let b0 be some positive number. on the set a1,t ∩ b1,t for all t > t0 with 4b0(log t0) 1/2 ( −σ2hθ 4t0h2 )1/2 ≤ c0, we have 2h ( t −σ2 h θ̃t )1/2 (θ̃t −θ) ≤ x ⇒ it + b0t (θ̃t −θ) < it + 2h ( t −σ2 h θ̃t )1/2 σ2hb0θx https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 15 ⇒ 2h ( t −σ2 h θ̃t )1/2 (θ̃t −θ)[it + b0t (θt −θ)] < x it + 2h ( t −σ2 h θ̃t )1/2 σ2hb0θx  ⇒ (θ̃t −θ)it + b0t (θt −θ)2 < ( −σ2hθ 4th2 )1/2 itx + σ 2 hb0θx 2 ⇒ −nt + (θ̃t −θ)it + b0t (θ̃t −θ)2 < −nt + ( −σ2hθ 4th2 )1/2 itx + σ 2 hb0θx 2 ⇒ 0 < −nt + ( −σ2hθ 4th2 )1/2 itx + σ 2 hb0θx 2 since it + b0t (θt −θ) > tc0 + b0t (θt −θ) > 4b0(log t ) 1/2 ( −σ2hθ 4th2 )1/2 −σ2hb0(log t ) 1−h ( −σ2hθ 4th2 )1/2 = σ2hb0(log t ) 1/2 ( −σ2hθ 4th2 )1/2 > 0. on the other hand, on the set a1,t ∩b1,t for all t > t0 with 4b0(log t0)1/2(−σ2hθ4t0h2)1/2 ≤ c0,we have 2h ( t σ2 h θ̃t )1/2 (θt −θ) > x ⇒ it −b0t (θ̃t −θ) < it − 2h ( t −σ2 h θ̃t )1/2 σ2hb0θx ⇒ 2h ( t −σ2 h θ̃t )1/2 (θ̃t −θ)[it −b0t (θt −θ)] > x it − 2h ( t −σ2 h θ̃t )1/2 2b0θx  ⇒ (θ̃t −θ)it −b0t (θ̃t −θ)2 > ( −σ2hθ 4th2 )1/2 itx −σ2hb0θx 2 ⇒ −nt + (θ̃t −θ)it −b0t (θt −θ)2 > −nt + ( −σ2hθ 4th2 )1/2 itx −σ2hb0θx 2 ⇒ 0 > −nt + ( −σ2hθ 4th2 )1/2 itx −σ2hb0θx 2 since it −b0t (θ̃t −θ) > tc0 −b0t (θ̃t −θ) > 2σ2hb0(log t ) 1/2 ( −σ2hθ 4th2 )1/2 −σ2hb0(log t ) 1/2 ( −σ2hθ 4th2 )1/2 = σ2hb0(log t ) 1/2 ( −σ2hθ 4th2 )1/2 > 0. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 16 hence 0 < −nt + ( −σ2hθ 4th2 )1/2 itx −σ2hb0θx 2 ⇒ 2h( t −σ2 h θ̃t )1/2(θt −θ) ≤ x. letting d±1,t,x := { −nt + ( −σ2hθ 4th2 )1/2 itx ±σ2hb0θx 2 > 0 } , we obtain d−1,t,x ∩a1,t ∩b1,t ⊆ a1,t ∩b1,t ∩ 2h ( t −σ2 h θ̃t )1/2 (θ̃t −θ) ≤ x  ⊆ d+1,t,x ∩a1,t ∩b1,t . (2.20)if it is shown that ∣∣p {d±1,t,x}− φ(x)∣∣ ≤ ct−1/4 (2.21)for all t > t0 and |x| ≤ 2(log t )1/2, then the theorem would follow from (2.18) (2.21).we shall prove (2.21) for d+ 1,t,x . the proof for d− 1,t,x is analogous.observe that∣∣∣p {d+1,t,x}− φ(x)∣∣∣ = ∣∣∣∣∣p {( −σ2hθ 4th2 )1/2 nt − (( −σ2hθ 4th2 ) it − 1 ) x < x + 2 ( −σ2hθ 4th2 )1/2 b0θx 2 } − φ(x) ∣∣∣∣∣ ≤ sup y∈r ∣∣∣∣∣p {( −σ2hθ 4th2 )1/2 nt − (( −σ2hθ 4th2 ) it − 1 ) x ≤ y } − φ(y) ∣∣∣∣∣ + ∣∣∣∣∣φ ( x + ( −σ2hθ 4th2 )1/2 b0θx 2 ) − φ(x) ∣∣∣∣∣ =: ∆11 + ∆12. (2.22)(1.53) immediately yields ∆11 ≤ ct−1/4. (2.23)on the other hand, for all t > t0, ∆12 ≤ 2 ( −σ2hθ 4th2 )1/2 b0θx 2(2π)−1/2 exp(−x2/2) where |x −x| ≤ 2 ( −σ2hθ 4th2 )1/2 b0θx 2. since |x| ≤ 2(log t )1/2, it follows that |x̄| > |x|/2 for all t > t0 and consequently ∆12 ≤ 2 ( −σ2hθ 4th2 )1/2 b0θx 2(2π)−1/2x2 exp(−x2/8) ≤ ct−1/4. (2.24) from (2.12) (2.14), we obtain ∣∣p {d+1,t,x}− φ(x)∣∣ ≤ ct−1/4. https://doi.org/10.28924/ada/ma.3.14 eur. j. math. anal. 10.28924/ada/ma.3.14 17 this completes the proof of part (c) of the theorem. next we demonstrate the proof of (b) and (d).if 5 8 < h < 3 4 by following similar steps, one can show that sup x∈r ∣∣∣∣∣∣p  ( t σ2 h θ̃t )1/2 (θ̂t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ cθt4h−3. if 11 16 < h < 3 4 by following similar steps, one can show that sup x∈r ∣∣∣∣∣∣p 2h ( t σ2 h θ̃t )1/2 (θ̃t −θ) ≤ x − φ(x) ∣∣∣∣∣∣ ≤ cθt4h−3. this completes the proof of the theorem. concluding remark for the case 1 2 ≤ h ≤ 5 8 , our rate is o(t−1/2) is optimal. references [1] j.p.n. bishwal, parameter estimation in stochastic differential equations, springer-verlag, berlin, (2008).[2] j.p.n. bishwal, minimum contrast estimation in fractional ornstein-uhlenbeck process: continuous and discretesampling, fract. calc. appl. anal. 14 (2011) 375–410. https://doi.org/10.2478/s13540-011-0024-6.[3] j.p.n. bishwal, maximum quasi-likelihood estimation in fractional levy stochastic volatility model, j. math. finance.1 (2011) 58–62. https://doi.org/10.4236/jmf.2011.13008.[4] j.p.n. bishwal, sufficiency and rao-blackwellization of vasicek model, theory stoch. processes. 17 (2011) 12-15.[5] j.p.n. bishwal, berry–esseen inequalities for the fractional black–karasinski model of term structure of interestrates, monte carlo methods appl. 28 (2022) 111–124. https://doi.org/10.1515/mcma-2022-2111.[6] j.p.n. bishwal, parameter estimation in stochastic volatility models, springer nature, cham. 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[5] we will consider the transformations t� : x̄ i = ϕi (x i,uα,�), ūα = ψα(x i,uα,�), (2.1) in the euclidean space rn of x = x i independent variables and rm of u = uα dependent variables.the continuous parameter � ranges from a neighbourhood n ′ ⊂ n ⊂ r of � = 0 for ϕi and ψαdifferentiable and analytic in the parameter �. definition 2.1. let g be a set of transformations in (2.1) . then g is a local lie group if: (i). given t�1,t�2 ∈g, for �1,�2 ∈n ′ ⊂n , then t�1t�2 = t�3 ∈g, �3 = φ(�1,�2) ∈n (closure).(ii). there exists a unique t0 ∈g if and only if � = 0 such that t�t0 = t0t� = t�(identity).(iii). there exists a unique t�−1 ∈g for every transformation t� ∈g,where � ∈n ′ ⊂n and �−1 ∈n such that t�t�−1 = t�−1t� = t0 (inverse). remark 2.2. the condition (i ) is sufficient for associativity of g. prolongations. consider the system, ∆α ( x i,uα,u(1), . . . ,u(π) ) = ∆α = 0, (2.2) where uα are dependent variables with partial derivatives u(1) = {uαi }, u(2) = {uαij}, . . . ,u(π) = {u α i1...iπ }, of the first, second, . . . , up to the πth-orders. we shall denoteby di = ∂ ∂x i + uαi ∂ ∂uα + uαij ∂ ∂uα j + . . . , (2.3) the total differentiation operator with respect to the variables x i and δj i , the kronecker delta. then di (x j) = δ j i , ′, uαi = di (u α), uαij = dj(di (u α)), . . . , (2.4) where uαi defined in (2.4) are differential variables [6].(1) prolonged groups let g given by x̄ i = ϕi (x i,uα,�), ϕi ∣∣∣ �=0 = x i, ūα = ψα(x i,uα,�), ψα ∣∣∣ �=0 = uα, (2.5) where ∣∣∣ �=0 means evaluated on � = 0. definition 2.3. the construction of g in (2.5) is equivalent to the computation of infinitesimaltransformations x̄ i ≈ x i + ξi (x i,uα)�, ϕi ∣∣∣ �=0 = x i, ūα ≈ uα + ηα(x i,uα)�, ψα ∣∣∣ �=0 = uα, (2.6) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 3 obtained from (2.1) by a taylor series expansion of ϕi (x i,uα,�) and ψi (x i,uα,�) in � about � = 0 and keeping only the terms linear in �, where ξi (x i,uα) = ∂ϕi (x i,uα,�) ∂� ∣∣∣ �=0 , ηα(x i,uα) = ∂ψα(x i,uα,�) ∂� ∣∣∣ �=0 . (2.7) remark 2.4. by using the symbol of infinitesimal transformations, x, (2.6) becomes x̄ i ≈ (1 + x)x i, ūα ≈ (1 + x)uα, (2.8) where x = ξi (x i,uα) ∂ ∂x i + ηα(x i,uα) ∂ ∂uα , (2.9) is the generator g in (2.5). remark 2.5. the change of variables formula di = di (ϕ j)d̄j, (2.10) is employed to construct transformed derivatives from (2.1). the d̄j is total differentiation x̄ i . as a result ūαi = d̄i (ū α), ūαij = d̄j(ū α i ) = d̄i (ū α j ). (2.11) if we apply the change of variable formula given in (2.10) on g given by (2.5), we get di (ψ α) = di (ϕ j), d̄j(ū α) = ūαj di (ϕ j). (2.12) if we expand (2.12), we obtain( ∂ϕj ∂x i + u β i ∂ϕj ∂uβ ) ū β j = ∂ψα ∂x i + u β i ∂ψα ∂uβ . (2.13) the ūαi can be written as functions of x i,uα,u(1), meaning that, ūαi = φ α(x i,uα,u(1),�), φ α ∣∣∣ �=0 = uαi . (2.14) definition 2.6. the transformations in (2.5) and (2.14) give the first prolongation group g[1]. definition 2.7. infinitesimal transformation of the first derivatives is ūαi ≈ u α i + ζ α i �, where ζαi = ζαi (x i,uα,u(1),�). (2.15) remark 2.8. in terms of infinitesimal transformations, g[1] is given by (2.6) and (2.15). (2) prolonged generators https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 4 definition 2.9. by the relation (2.12) on g[1] from 2.6, we obtain [7] di (x j + ξj�)(uαj + ζ α j �) = di (u α + ηα�), which gives (2.16) uαi + ζ α j � + u α j �diξ j = uαi + diη α�, (2.17) and thus ζαi =di (η α) −uαj di (ξ j), (2.18) is the first prolongation formula. remark 2.10. analogously, one constructs higher order prolongations [7], ζαij = dj(ζ α i ) −u α iκdj(ξ κ), . . . , ζαi1,...,iκ = diκ(ζ α i1,...,iκ−1 ) −uαi1,i2,...,iκ−1jdiκ(ξ j). (2.19) remark 2.11. the prolonged generators of the prolongations g[1], . . . ,g[κ] of the group gare x[1] = x + ζαi ∂ ∂uα i , . . . ,x[κ] = x[κ−1] + ζαi1,...,iκ ∂ ∂ζα i1,...,iκ , κ ≥ 1, (2.20) for the group generator x in (2.9). group invariants. definition 2.12. a function γ(x i,uα) is said to be an invariant of g of in (2.1) if γ(x̄ i, ūα) = γ(x i,uα). (2.21) theorem 2.13. a function γ(x i,uα) is an invariant of the group g given by (2.1) if and only if it solves the following first-order linear pde: [8] xγ = ξi (x i,uα) ∂γ ∂x i + ηα(x i,uα) ∂γ ∂uα = 0. (2.22) from theorem (2.13), we have the following result. theorem 2.14. the lie group g in (2.1) [9] has precisely n−1 functionally independent invariants and one can take as the basic invariants, the left-hand sides of the first integrals ψ1(x i,uα) = c1, . . . ,ψn−1(x i,uα) = cn−1, (2.23) of the characteristic equations for (2.22): dx i ξi (x i,uα) = duα ηα(x i,uα) . (2.24) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 5 symmetry groups. definition 2.15. we define the vector field x (2.9) as a lie point symmetry of (2.2) if the determiningequations x[π]∆α ∣∣∣ ∆α=0 = 0, α = 1, . . . ,m, π ≥ 1, (2.25) are satisfied for the π-th prolongation of x, namely x[π]. definition 2.16. the lie group g is a symmetry group of (2.2) if (2.2) is form-invariant, that is ∆α ( x̄ i, ūα, ū(1), . . . , ū(π) ) = 0. (2.26) theorem 2.17. the lie group g (2.1) can be constructed from the infinitesimal transformations in (2.5) by integrating the lie equations dx̄ i d� = ξi (x̄ i, ūα), x̄ i ∣∣∣ �=0 = x i, dūα d� = ηα(x̄ i, ūα), ūα ∣∣∣ �=0 = uα. (2.27) lie algebras. definition 2.18. a vector space vr of operators [8] x (2.9) is a lie algebra if for any xi,xj ∈vr, [xi,xj] = xixj −xjxi, (2.28) is in vr for all i, j = 1, . . . , r . remark 2.19. the commutator is bilinear, skew symmetric and admits to the jacobi identity [5]. theorem 2.20. the set of solutions of (2.25) forms a lie algebra [10]. exact solutions. the methods of (g’/g)-expansion method [7], extended jacobi elliptic functionexpansion [9] and kudryashov [11] are usually applied after symmetry reductions. conservation laws. [11] fundamental operators. definition 2.21. the euler-lagrange operator δ δuα is δ δuα = ∂ ∂uα + ∑ κ≥1 (−1)κdi1, . . . ,diκ ∂ ∂uα i1i2...iκ , (2.29) and the liebäcklund operator in abbreviated form [11] is x = ξi ∂ ∂x i + ηα ∂ ∂uα + . . . . (2.30) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 6 remark 2.22. the liebäcklund operator (2.30) in its prolonged form is x = ξi ∂ ∂x i + ηα ∂ ∂uα + ∑ κ≥1 ζi1...iκ ∂ ∂uα i1i2...iκ , (2.31) for ζαi = di (w α) + ξjuαij , . . . ,ζ α i1...iκ = di1...iκ(w α) + ξjuαji1...iκ, j = 1, . . . ,n. (2.32)and the lie characteristic function wα = ηα −ξjuαj . (2.33) remark 2.23. the characteristic form of liebäcklund operator (2.31) is x = ξidi + w α ∂ ∂uα + di1...iκ(w α) ∂ ∂uα i1i2...iκ . (2.34) the method of multipliers. definition 2.24. a function λα (x i,uα,u(1), . . .) = λα, is a multiplier of (2.2) if [7] λα∆α = dit i, (2.35) where dit i is a divergence expression. definition 2.25. to find the multipliers λα, one solves the determining equations (2.36) [10], δ δuα (λα∆α) = 0. (2.36) ibragimov’s conservation theorem . the technique [5] enables one to construct conserved vectorsassociated with each lie point symmetry of (2.2). definition 2.26. the adjoint equations of (2.2) are ∆∗α ( x i,uα,vα, . . . ,u(π),v(π) ) ≡ δ δuα (vβ∆β) = 0, (2.37) for a new dependent variable vα. definition 2.27. the formal lagrangian l of (2.2) and its adjoint equations (2.37) is [8] l = vα∆α(x i,uα,u(1), . . . ,u(π)). (2.38) theorem 2.28. every infinitesimal symmetry xof (2.2) leads to conservation laws [6] dit i ∣∣∣ ∆α=0 = 0, (2.39) where the conserved vector t i = ξil + wα [ ∂l ∂uα i −dj ( ∂l ∂uα ij ) + djdk ( ∂l ∂uα ijk ) − . . . ] + dj(w α) [ ∂l ∂uα ij −dk ( ∂l ∂uα ijk ) + . . . ] + djdk(w α) [ ∂l ∂uα ijk − . . . ] . (2.40) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 7 3. main results 3.1. lie point symmetries of equal width equation(1.1). we start first by computing lie pointsymmetries of the equal width equation (1.1), which admits the one-parameter lie group of trans-formations with infinitesimal generator x = τ(t,x,u) ∂ ∂t + ξ(t,x,u) ∂ ∂x + η(t,x,u) ∂ ∂u (3.1) if and only if x[3]∆ ∣∣∣∣ ∆=0 = 0. (3.2) where x[3] = x + ζ1 ∂ ∂ut + ζ2 ∂ ∂ux + ζ122 ∂ ∂utxx , (3.3) is the third prolongation of the lie point symmetry x as defined in (2.20) and ζ1 = dt(η) −utdt(τ) −uxdt(ξ), (3.4) ζ12 = dx (ζ1) −uttdx (τ) −utxdx (ξ), (3.5) ζ2 = dx (η) −utdx (τ) −uxdx (ξ), (3.6) ζ122 = dx (ζ12) −uttxdx (τ) −utxxdx (ξ), (3.7) as defined in (2.19), and dt = ∂ ∂t + ut ∂ ∂u + utx ∂ ∂ux + utt ∂ ∂ut + · · · , (3.8) dx = ∂ ∂x + ux ∂ ∂u + uxx ∂ ∂ux + utx ∂ ∂ut + . . . . (3.9) applying the definitions of dt and dx given in (3.8) and (3.9), we obtain the expanded form of the ζs as ζ1 = ηt + ut(ηu −τt) + ux (−ξt) + utux (−ξu) + u2t (−τu), ζ12 = ηtx + ux (ηtu −ξtx ) + utx (ηu −τt −ξx ) + ut(ηxu −τtx ) + utux (ηuu −ξxu −τtu) + ututx (−2τu) + u2t (−τxu) + u 2 t ux (−τuu) + uxx (−ξt) + u 2 x (−ξtu) + uxutx (−2ξu) + utu 2 x (−ξuu) + utuxx (−ξu) + utt(−τx ) + uxutt(−τu) ζ2 = ηx + ux (ηu −ξx ) + ut(−τx ) + utux (−τu) + u2x (−ξu), https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 8 ζ122 = ηtxx + ux (2ηtxu −ξtxx ) + uxx (ηtu − 2ξtx ) + u2x (ηtuu − 2ξtxu) + utxx (ηu −τt − 2ξx ) + utux (2ηxuu −ξxxu − 2τtxu) + uxutx (2ηuu − 4ξxu −τtu) + utx (2ηxu − 2τtx −ξxx ) + ut(ηxxu −τtxx ) + utuxx (ηuu − 2ξxu −τtu) + utu2x (ηuuu − 2ξxuu −τtuu) + u 2 tx (−2τu) ututxx (−2τu) + ututx (−4τxu) + u2t (−τxxu) + uxu 2 t (−2τxuu) + utuxutx (−4τuu), + u2xu 2 t (−τuuu) + uxxx (−ξt) + u 2 t uxx (−τuu) + uxuxx (−4ξtu) + u 3 x (−ξtuu) + uxxutx (−3ξu) uxutxx (−2ξu) + u2xutx (−3ξuu) + uxutuxx (−3ξuu) + utu 3 x (−ξuuu) + utuxxx (−ξu) + uttx (−2τx ) + utt(−τxx ) + uxutt(−2τxu) + u2xutt(−τuu) + uxxutt(−τu) + uxuttx (−2τu −ξu) (3.10) now from equation (3.2), we have ζ1 + αηux + αζ2u + βζ122 ∣∣ utxx =− ut β −α β uux = 0, (3.11) if we substitute for ζ1, ζ2 and ζ122 in the determining equation (3.11), we obtain the following; ηt + ut(ηu −τt) + ux (−ξt) + utux (−ξu) + u2t (−τu) + αηux + αu{ηx + ux (ηu −ξx ) + ut(−τx ) + utux (−τu) + u2x (−ξu)} + β { ηtxx + ux (2ηtxu −ξtxx ) + uxx (ηtu − 2ξtx ) + u2x (ηtuu − 2ξtxu) + utxx (ηu −τt − 2ξx ) + utux (2ηxuu −ξxxu − 2τtxu) + uxutx (2ηuu − 4ξxu −τtu) + utx (2ηxu − 2τtx −ξxx ) + ut(ηxxu −τtxx ) + utuxx (ηuu − 2ξxu −τtu) + utu2x (ηuuu − 2ξxuu −τtuu) + u 2 tx (−2τu) ututxx (−2τu) + ututx (−4τxu) + u2t (−τxxu) + uxu 2 t (−2τxuu) + utuxutx (−4τuu), + u2xu 2 t (−τuuu) + uxxx (−ξt) + u 2 t uxx (−τuu) + uxuxx (−4ξtu) + u 3 x (−ξtuu) + uxxutx (−3ξu) uxutxx (−2ξu) + u2xutx (−3ξuu) + uxutuxx (−3ξuu) + utu 3 x (−ξuuu) + utuxxx (−ξu) + uttx (−2τx ) + utt(−τxx ) + uxutt(−2τxu) + u2xutt(−τuu) + uxxutt(−τu) + uxuttx (−2τu −ξu) } ∣∣∣∣∣ u txx=−ut β −α β uux = 0 (3.12) now replacing utxx by −utβ − αβuux in equation (3.12), we have https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 9 ηt + ut(ηu −τt) + ux (−ξt) + utux (−ξu) + u2t (−τu) + αηux + αu{ηx + ux (ηu −ξx ) + ut(−τx ) + utux (−τu) + u2x (−ξu)} + β { ηtxx + ux (2ηtxu −ξtxx ) + uxx (ηtu − 2ξtx ) + u2x (ηtuu − 2ξtxu)+[ − ut β − α β uux ] (ηu −τt − 2ξx ) + utux (2ηxuu −ξxxu − 2τtxu) + uxutx (2ηuu − 4ξxu −τtu) + utx (2ηxu − 2τtx −ξxx ) + ut(ηxxu −τtxx ) + utuxx (ηuu − 2ξxu −τtu) + utu2x (ηuuu − 2ξxuu −τtuu) + u 2 tx (−2τu) + ut [ − ut β − α β uux ] (−2τu) + ututx (−4τxu) + u2t (−τxxu) + uxu 2 t (−2τxuu) + utuxutx (−4τuu), + u2xu 2 t (−τuuu) + uxxx (−ξt) + u 2 t uxx (−τuu) + uxuxx (−4ξtu) + u 3 x (−ξtuu) + uxxutx (−3ξu) ux [ − ut β − α β uux ] (−2ξu) + u2xutx (−3ξuu) + uxutuxx (−3ξuu) + utu 3 x (−ξuuu) + utuxxx (−ξu) + uttx (−2τx ) + utt(−τxx ) + uxutt(−2τxu) + u2xutt(−τuu) + uxxutt(−τu) + uxuttx (−2τu −ξu) } = 0 (3.13) which can be written as ηt + αuηx + βηtxx + ut(βηxxu −βτtxx + 2ξx −αuτx ) + ux ( 2βηtxu −βξtxx −ξt + αuξx + αuτt + αη ) + utux (2ξu + 2βηxuu −βξxxu − 2βτtxu + αuτu) + u2t (τu −βτxxu)+ u2x (2αuξu + βηtuu − 2βξtxu) + β { uxx (ηtu − 2ξtx ) + uxutx (2ηuu − 4ξxu −τtu) + utx (2ηxu − 2τtx −ξxx ) + utuxx (ηuu − 2ξxu −τtu) + utu2x (ηuuu − 2ξxuu −τtuu) + u 2 tx (−2τu) + ututx (−4τxu) + uxu2t (−2τxuu) + utuxutx (−4τuu), + u2xu 2 t (−τuuu) + uxxx (−ξt) + u 2 t uxx (−τuu) + uxuxx (−3ξtu) + u 3 x (−ξtuu) + uxxutx (−3ξu) + u2xutx (−3ξuu) + uxutuxx (−3ξuu) + utu 3 x (−ξuuu) + utuxxx (−ξu) + uttx (−2τx ) + utt(−τxx ) + uxutt(−2τxu) + u2xutt(−τuu) + uxxutt(−τu) + uxuttx (−2τu) } = 0 (3.14) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 10 since the functions τ,ξ and η depend only on t,x and u and are independent of the derivativesof u, we can then split the above equation on the derivatives of u and obtain τx = τu = ξu = ξt = ξx = ηuu = ηtu =0, (3.15) η + uτt =0, (3.16) ηt + αuηx + βηtxx =0 (3.17) from equation (3.15), we find that τ =τ(t), (3.18) ξ =c1, (3.19) η =a(x)u + b(t,x). (3.20) now substituting η into equation (3.17) yields bt(t,x) + αu [ a(x)xu + bx (t,x) ] + βbtxx (t,x) = 0. (3.21) separation of (3.21) on powers of u gives the following equations u2 :a(x)x = 0, (3.22) u :bx (t,x) = 0, (3.23) u0 :bt(t,x) + βbtxx (t,x) = 0. (3.24) integration of equations (3.22) and (3.23) with respect to x gives that a(x) = c2 (3.25) b(t,x) = b(t). (3.26) now use equation (3.26) in equation (3.24) to obtain btxx (t,x) = 0 and as a result bt(t,x) = 0. (3.27) integrating equation (3.27) with respect to t gives b(t,x) = c3. (3.28) if we substitute η = c2u + c3 into equation (3.16), we have c2u + c3 + τtu = 0. (3.29) from equation (3.29), if we obtain τ(t) = −c2t −c3 t u + c4. (3.30) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 11 and finally; τ = −c2t −c3 t u + c4, (3.31) ξ =c1 (3.32) η =c2u + c3. (3.33) we have obtained a four-dimensional lie algebra of symmetries spanned by x1 = ∂ ∂x , (3.34) x2 =u ∂ ∂u − t ∂ ∂t , (3.35) x3 = ∂ ∂u − t u ∂ ∂t , (3.36) x4 = ∂ ∂t . (3.37) 3.2. commutator table for symmetries. we evaluate the commutation relations for the symmetrygenerators. by definition of lie bracket [9], for example, we have that [x1,x4] = x1x4 −x4x1 = ( ∂ ∂x ∂ ∂t ) − ( ∂ ∂t ∂ ∂x ) = 0. (3.38) remark 3.1. the remaining commutation relations are obtained analogously. we present all commutation relations in table (1) below. [xi,xj] x1 x2 x3 x4 x1 0 0 0 0 x2 0 0 -x3 x4 x3 0 −x3 0 1ux4 x4 0 -x4 1ux4 0 table 1. a commutator table for lie algebra of equal width equation. 3.3. group transformations. the corresponding one-parameter group of transformations can bedetermined by solving the lie equations [6]. let t�i be the group of transformations for each xi, i = 1, 2, 3, 4. we display how to obtain t�i from xi by finding one-parameter group for theinfinitesimal generator x1, namely, x1 = ∂ ∂x . (3.39) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 12 in particular, we have the lie equations dt̄ d� =0, t̄ ∣∣∣ �=0 = t, dx̄ d� =1, x̄ ∣∣∣ �=0 = x, dū d� =0, ū ∣∣∣ �=0 = u. (3.40) solving the system (3.40) one obtains, t̄ = t, x̄ = x + �, ū = u, (3.41) and hence the one-parameter group t�4 corresponding to the operator x1 is t�1 : (t̄, x̄, ū) = (t,x + �1,u). (3.42) all the five one-parameter groups are presented below : t�1 : (t̄, x̄, ū) = (t,x + �1,u) t�2 : (t̄, x̄, ū) = (te −�2,x,ue�2 ) t�3 : (t̄, x̄, ū) = (te −�3 u ,x,u + �3) t�4 : (t̄, x̄, ū) = (t + �4,x,u). (3.43) 3.4. symmetry transformations. we now show how the symmetries we have obtained can be usedto transform special exact solutions of the equal width equation into new solutions. the lie groupanalysis vouches for fundamental ways of e constructing exact solutions of pdes, that is, grouptransformations of known solutions and construction of group-invariant solutions. we will illustratethese methods with examples. if ū = g(t̄, x̄) is a solution of equation (1.1) φ(t,x,u,�) = g(f1(t,x,u,�), f2(t,x,u,�)), (3.44) is also a solution. the one parameter groups dictate to the following generated solutions: t�1 : u =g(t,x + �1) t�2 : u =g(te −�2,x)e−�2, t�3 : u =g(te −�3 u ,x) − �3, t�4 : u =g(t + �4,x). (3.45) 3.5. construction of group-invariant solutions. now we compute the group invariant solutions ofburger’s equation. (i) x1 = ∂∂xthe associated lagrangian equations https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 13 dt 0 = dx 1 = du 0 , (3.46) yield two invariants, j1 = t and j2 = u. thus using j2 = φ(j1), we have u(t,x) = φ(t). (3.47) the derivatives are given by : ut =φ ′(t), ux =0, utxx =0. if we substitute these derivatives into equation (1.1) , we obtain the first order ordinarydifferential equation φ′(t) = 0, whose space invariant solution is φ(t) = c1, (3.48) and the group-invariant solution associated to the x1 is u(t,x) = c1. (ii) x2 = u ∂∂u − t ∂∂t the lagrangian equations associated to this symmetry are dt −t = dx 0 = du u . (3.49) this gives the constants j1 = x and j2 = tu , giving the solution u = f (x) t . (3.50) we obtain the derivatives as follows: ut = − f (x) t2 , (3.51) ux = f ′(x) t (3.52) utxx = − f ′′(x) t2 (3.53) if we substitute the above derivatives in equation (1.1), we obtain the second order ordinarydifferential equation f (x) −αf (x)f ′(x) + βf ′′(x) = 0. (3.54) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 14 hence the group invariant solution to equation to (1.1) will be given by u(t,x) = f (x) t , (3.55) where f satisfies equation (3.54).(iii) x3 = ∂∂u − tu ∂∂tthe lagrangian system associated with the operator x3 is dt − t u = dx 0 = du 1 , (3.56) whose invariants are j1 = x and j2 = tu. so, u = g(x)t is the group-invariant solution.(iv) x4 = ∂∂tcharacteristic equations associated to the operator x4 are dt 1 = dx 0 = du 0 , (3.57) yieldsj1 = x and j2 = u. as a result, the group-invariant solution of (1.1) for this case is j2 = φ(j1), for some φ an arbitrary function. that is, u(t,x) = φ(x). (3.58) the derivatives of given function are ut = 0, (3.59) ux = φ ′(x), (3.60) utxx = 0. (3.61) substitution of the value of φ(x) into equation (1.1) yields a first order nonlinear ordinarydifferential equation φ(x)φ′(x) = 0. (3.62) from equation (3.62), either φ(x) = 0 or φ′(x) = 0. the case φ(x) = 0 =⇒ φ′(x) = 0,and the equation is satisfied. the case φ(x) 6= 0 implies that φ′(x) = 0 and by integration, φ(x) = c1, hence the group invariant solution is given by u(t,x) = c2. (3.63) 3.6. soliton. we obtain a traveling wave solution of the equal width equation(1.1) by consideringa linear combination of the symmetries x1 and x4, namely, [7] x = cx1 + x4 = c ∂ ∂x + ∂ ∂t , for some constant c. (3.64) the characteristic equations are dt 1 = dx c = du 0 (3.65) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 15 we get two invariants, j1 = x −ct and j2 = u. so the group-invariant solution is u(t,x) = q(x −ct), (3.66) for some arbitrary function ϕ and c the velocity of the wave.substitution of u into (1.1) yields a second order ordinary differential equation cq′ −αqq′ + βcq′′′ = 0, (3.67) which can be integrated with respect to q to give cq−α q2 2 + βcq′ = 0, (3.68) where we have used 0 as a constant of integration. equation (3.68) can be rearranged and variablesseparated to have dξ 2βc = dq αq2 − 2cq , ξ = x −ct. (3.69) the right hand side can be resolved into partial fractions to obtain ξ 2βc = 1 2c ∫ [ α αq− 2c − 1 q ] dq = 1 2c ln ∣∣∣∣∣αq− 2cq ∣∣∣∣∣ + ln |c3|, (3.70) where c3 is a constant of integration. after rewriting, we have q(x −ct) = 2cc3 αc3 −e x−ct β . (3.71) finally, the soliton solutions are given by u(t,x) = 2cc3 αc3 −e x−ct β . (3.72) 4. conservation laws of equation (1.1) we will employ multipliers in the construction of conservation laws. 4.1. the multipliers. we make use of the euler-lagrange operator defined as defined in [6] to lookfor a zeroth order multiplier λ = λ(t,x,u). the resulting determining equation for computing λ is δ δu [λ{ut + αuux + βutxx}] = 0. (4.1) where δ δu = ∂ ∂u −dt ∂ ∂ut −dx ∂ ∂ux −dtd2x ∂ ∂utxx + . . . (4.2) expansion of equation (4.1) yields λu(ut + αuux + βutxx ) + αux λ −dt(λ) −αdx (uλ) −βdtd2x (λ) = 0. (4.3) invoking the total derivatives https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 16 dt = ∂ ∂t + ut ∂ ∂u + utx ∂ ∂ux + utt ∂ ∂ut + · · · , (4.4) dx = ∂ ∂x + ux ∂ ∂u + uxx ∂ ∂ux + utx ∂ ∂ut + . . . . (4.5) on equation (4.3) produces λt + αuλx + βλtxx + 2β(λtxu)ux + β(λtu)uxx + β(λtuu)u 2 x + 2β(λxu)utx + 2β(λuu)uxutx + β(λxxu)ut + 2β(λxuu)uxux + β(λuu)utuxx + β(λuuu)utu 2 x = 0 (4.6) splitting equation (4.6) on derivatives of u produces an overdetermined system of four partialdifferentialequations, namely, λuu =0, (4.7) λxu =0, (4.8) λtu =0 (4.9) λt + αuλx + βλtxx =0 (4.10) by equation (4.7), we have λ = a(t,x)u + b(t,x), (4.11) which if used in equations (4.8-4.9), implies that λ = c1u + b(t,x). (4.12) if we substitute (4.12) into equation (4.10), we obtain bt(t,x) + αubx (t,x) + βbtxx (t,x) = 0. (4.13) separation of equation (4.13) into powers of u gives us u :bx (t,x) = 0, (4.14) u0 :bt(t,x) + βbtxx (t,x) = 0. (4.15) equation (4.14) insists that btxx (t,x) = 0 =⇒ bt(t,x) = 0 = bx (t,x), (4.16) and thus b(t,x) = c2. (4.17) as a result λ(t,x,u) = c1u + c2. (4.18) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 17 essentially, we extract the two multiplies λ1 =1 (4.19) λ2 =u. (4.20) remark 4.1. recall that a multiplier λ for equation(1.1) has the property that for the density tt = tt(t,x,u,ux ) and flux tx = tx (t,x,u,ux,utx ), λ (ut + αuux + βutxx ) = dtt t + dxt x. (4.21) we derive a conservation law corresponding to each of the multipliers. (i). conservation law for the multiplier λ1 = 1expansion of equation (4.21) gives 1{ut + αuux + βutxx} = ttt + utt t u + utxt t ux + txx + uxt x u + uxxt x ux + utxxt x utx . (4.22) splitting equation (4.22) on the third derivative of u yields utxx : t x utx = β, (4.23) rest : ut + αuux = ttt + utttu + utxttux + txx + uxtxu + uxxtxux. (4.24) the integration of equation (4.23) with respect to utx gives tx = βutx + a(t,x,u,ux ). (4.25) substituting the expression of tx from (4.25) into equation (4.22) we get {ut + αuux} =ttt + utt t u + utxt t ux + ax + uxau + uxxaux (4.26) which splits on second derivatives of u, to give uxx : aux = 0, (4.27) utx : t t ux = 0, (4.28) rest : {ut + αuux} = ttt + utttu + ax + uxau. (4.29) integrating equations (4.27) and (4.28) with respect to ux manifests that tt = tt(t,x,u) and a = a(t,x,u). using values of a and tt in equation (4.29), we have {ut + αuux} = ttt + utt t u + ax + uxau, (4.30) which separates on first derivatives to give us ut : t t u = 1, (4.31) ux : au = αu, (4.32) rest : ttt + ax = 0. (4.33) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 18 equations (4.31-4.32), can be integrated with respect u to obtain ttu = u + b(t,x), (4.34) a = α u2 2 + c(t,x), (4.35) if we use the obtained values in (4.33), we have bt(t,x) + cx (t,x) = 0. (4.36) since b(t,x) and c(t,x) contribute to the trivial part of the conservation law, we take b(t,x) = c(t,x) = 0 and obtain the conserved quantities tt =u, (4.37) tx =α u2 2 + βutx (4.38) from which the conservation law corresponding to the multiplier λ1 = 1 is given by dt(u) + dx ( α u2 2 + βutx ) = 0. (4.39) (ii). conservation law for the multiplier λ2 = u u{ut + αuux + βutxx} = ttt + utt t u + utxt t ux + txx + uxt x u + uxxt x ux + utxxt x utx . (4.40) splitting equation (4.40) on the third derivative of u yields utxx : t x utx = βu, (4.41) rest : ut + αuux = ttt + utttu + utxttux + txx + uxtxu + uxxtxux. (4.42) the integration of equation (4.41) with respect to utx gives tx = βuutx + a(t,x,u,ux ). (4.43) substituting the expression of tx from (4.43) into equation (4.40) we get u{ut + αuux} =ttt + utt t u + utxt t ux + ax + uxau + uxβutx + uxxaux. (4.44) which splits on second derivatives of u, to give uxx : aux = 0, (4.45) utx : t t ux = −βux, (4.46) rest : u{ut + αuux} = ttt + utttu + ax + uxau. (4.47) https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 19 integrating equations (4.45) and (4.46) with respect to ux manifests that tt = −βu 2 x 2 + b(t,x,u) and a = a(t,x,u). using values of a and tt in equation(4.47), we have u{ut + αuux} = ttt + utt t u + ax + uxau, (4.48) which separates on first derivatives to give us ut : b(t,x,u)u = u, (4.49) ux : au = αu 2, (4.50) rest : bt + ax = 0. (4.51) equations (4.49-4.50), can be integrated with respect u to obtain b = u2 2 + c(t,x), (4.52) a = α u3 3 + d(t,x), (4.53) if we use the obtained values in (4.51), we have ct(t,x) + dx (t,x) = 0. (4.54) since c(t,x) and d(t,x) contribute to the trivial part of the conservation law, we take c(t,x) = d(t,x) = 0 and obtain the conserved quantities tt = −β u2x 2 + u2 2 , (4.55) tx =βuutx + α u3 3 (4.56) from which the conservation law corresponding to the multiplier λ2 = u is given by dt ( −β u2x 2 + u2 2 ) + dx ( βuutx + α u3 3 ) = 0. (4.57) remark 4.2. it can be shown that the two sets of conserved quantities are conservation laws. giventhat λ1 = 1 , the verification reaffirms that the equal width equation is itself a conversation law. 5. conclusion in this manuscript, an infinite dimensional lie algebra of lie point symmetries has been appliedto study a third-order equal width equation. a commutator table has been constructed for theobtained lie algebra. we have also used symmetry reductions to compute exact group-invariantsolutions, including a soliton. conservation laws have also been derived for the model with the useof zeroth order multipliers. https://doi.org/10.28924/ada/ma.3.13 eur. j. math. anal. 10.28924/ada/ma.3.13 20 acknowledgement the author thanks referees and the editor for their careful reading and comments. author’s contribution the author wrote the article as a scholarly duty and passion to disseminate mathematical re-search and hereby declares that there is no conflict of interest. references [1] j.r. cannon, the one-dimensional heat equation, cambridge university press, cambridge, 1984.[2] p.j. morrison, j.d. meiss, j.r. cary, scattering of regularized-long-wave solitary waves, physica d: nonlinearphenomena. 11 (1984) 324–336. https://doi.org/10.1016/0167-2789(84)90014-9.[3] l.r.t. gardner, g.a. gardner, f.a. ayoub, n.k. amein, simulations of the ew undular bore, commun. numer. meth.engng. 13 (1997) 583–592. https://doi.org/10.1002/(sici)1099-0887(199707)13:7<583::aid-cnm90>3. 0.co;2-e.[4] l.r.t. gardner, g.a. gardner, solitary waves of the equal width wave equation, j. comput. phys. 101 (1992) 218–223. https://doi.org/10.1016/0021-9991(92)90054-3.[5] j.o. owino, group analysis on one-dimensional heat equation, int. j. adv. multidisc. res. stud. 2 (2022) 525-540.[6] j. owuor, exact symmetry reduction solutions of a nonlinear coupled system of korteweg-de vries equations, int.j. adv. multidisc. res. stud. 2 (2022) 76-87.[7] j. owuor owino, b. okelo, lie group analysis of a nonlinear coupled system of korteweg-de vries equations, eur.j. math. anal. 1 (2021) 133–150. https://doi.org/10.28924/ada/ma.1.133.[8] j.o. owino, a group approach to exact solutions and conservation laws of burger’s equation, int. j. math. comp.res. 10 (2022) 2894-2909. https://doi.org/10.47191/ijmcr/v10i9.03.[9] j. owuor, conserved quantities of a nonlinear coupled system of korteweg-de vries equations, int. j. math. comp.res. 10 (2022) 2673-2681. https://doi.org/10.47191/ijmcr/v10i5.02.[10] j. o. owino, an application of lie point symmetries in the study of potential burger’s equation, int. j. adv. multidisc.res. stud. 2 (2022) 191-207.[11] j. o. owino, group invariant solutions and conserved vectors for a special kdv type equation, int. j. adv. multidisc.res. stud. 2 (2022), 9-26. https://doi.org/10.28924/ada/ma.3.13 https://doi.org/10.1016/0167-2789(84)90014-9 https://doi.org/10.1002/(sici)1099-0887(199707)13:7<583::aid-cnm90>3.0.co;2-e https://doi.org/10.1002/(sici)1099-0887(199707)13:7<583::aid-cnm90>3.0.co;2-e https://doi.org/10.1016/0021-9991(92)90054-3 https://doi.org/10.28924/ada/ma.1.133 https://doi.org/10.47191/ijmcr/v10i9.03 https://doi.org/10.47191/ijmcr/v10i5.02 1. introduction 2. preliminaries local lie groups prolongations lie algebras conservation laws the method of multipliers ibragimov's conservation theorem 3. main results 3.1. lie point symmetries of equal width equation(1.1) 3.2. commutator table for symmetries 3.3. group transformations 3.4. symmetry transformations 3.5. construction of group-invariant solutions 3.6. soliton 4. conservation laws of equation (1.1) 4.1. the multipliers 5. conclusion acknowledgement author's contribution references ©2022 ada academica https://adac.eeeur. j. math. anal. 2 (2022) 1doi: 10.28924/ada/ma.2.1 convergence and stability of new approximation algorithms for certain contractive-type mappings imo kalu agwu∗, donatus ikechi igbokwe department of mathematics, micheal okpara university of agriculture, umudike, umuahia abia state, nigeria agwu.imoh@mouau.edu.ng, igbokwedi@yahoo.com ∗correspondence: agwu.imoh@mouau.edu.ng abstract. we present new fixed points algorithms called multistep h-iterative scheme and multistepsh-iterative scheme. under certain contractive-type condition, convergence and stability results wereestablished without any imposition of the ’sum conditions’, which to a large extent make some existingiterative schemes so far studied by other authors in this direction practically inefficient. our resultscomplement and improve some recent results in literature. 1. introduction there is an intimate connection existing between nonlinear problems and fixed point problemsof related contractive-type operators. as a result, researchers have focused more attention onfinding approximate fixed points of different contractive-type mappings in recent times; see, forexample, [5], [6], [7], [8], [9], [10], [11], [18], [25], [28], etc. and the reference contained in them. let xbe a normed linear space and γ : x −→ x a given of x. we represent the set of fixed points of γby f (γ) = {q ∈ x : q = γ(q)}.for the past forty years or so, some investigation of fixed points via iterative schemes have beena flourishing area of research for many mathematicians. mann [21], ishikawa [22] and noor [19]iterative schemes, with their modifications, have been studied by different authors and differentinteresting results were obtained. however, to meet up with the demand of the modern fixedpoint theory, researchers have continually renewed their efforts toward constructing more efficientiterative schemes. in this direction, following kirk’s introduction of his remarkable iterative schemein 1971, the results below have found thier place in the current literature.let x and γ be as earlier stated. received: 1 nov 2021. key words and phrases. strong convergence; multistep h-iterative scheme; multistep sh-iterative scheme; stability;contractive operator; fixed point; normed linear space. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 2 (a) for arbitrarily y0 ∈ x, let the sequence {yn}∞n=0 be defined iteratively as follows: yn+1 = ∑̀ j=0 αjγ jyn, ∑̀ j=0 αj = 1,n ≥ 0. (1.1) the iteration method defined by (1.1) is due to kirk [20]. (b) in [17], olatinwo presented the algorithms below:(i) for an arbitrary point y0 ∈ x and for αn,t ≥ 0,αn,0 6= 0,αn,t ∈ [0, 1] and ` as a fixedinteger, define the sequence {yn}∞n=0 by yn+1 = ∑̀ t=0 αn,tγ tyn, ∑̀ t=0 αn,t = 1,n ≥ 0 (1.2) (ii) for an arbitrary point y0 ∈ x and for ` ≥ m,αn,t βn,t ≥ 0,αn,0,βn,0 6= 0,αn,t,βn,t ∈ [0, 1] and `,m as fixed integers, define the sequence {yn}∞n=0 by yn+1 = αn,0yn + ∑̀ t=0 αn,tγ jzn, ∑̀ t=0 αn,t = 1; zn = m∑ t=0 βn,tγ tyn, ∑̀ t=0 βn,t = 1,n ≥ 0, (1.3) and called them kirk-mann and kirk-ishikawa algorithms, respectively. (c) chugh and kumar [25] presented the following iterative scheme: for an arbitrary point y0 ∈ x and for ` ≥ m ≥ p,αn,s,γn,r,βn,t ≥ 0,γn,0,αn,0,βn,0 6= 0,αn,s,γn,r,βn,t ∈ [0, 1] and `,m,p as fixed integers, define the sequence {yn}∞n=0 by yn+1 = γn,0yn + ∑̀ r=1 γn,rγ rzn, ∑̀ r=0 γn,r = 1; zn = αn,0yn + m∑ s=1 αn,sγ szn, m∑ s=0 αn,s = 1; zn = p∑ t=0 βn,tγ tyn, p∑ t=0 βn,t = 1,n ≥ 0, (1.4) (d) very recently, akewe, okeke and olayiwola [26] presented the following general iterativescheme in the sense of kirk [20]: (i) for an arbitrary point y0 ∈ x, for `1 ≥ `2 ≥ `3 ≥ ··· ≥ `u, for each i, αtn,s,γn,t ≥ 0,γn,0,αn,0, 6= 0, for each i, αin,s,γn,t ∈ [0, 1] and `1,`u as fixed integers for each u, https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 3 define the sequence {yn}∞n=0 by yn+1 = γn,0yn + `1∑ r=1 γn,rγ rz1n , `1∑ k=0 αn,r = 1; ztn = α t n,0yn + `t+1∑ s=1 αtn,sγ jzt+1n , `t+1∑ s=0 αtn,s = 1,t = 1, 2, · · · ,u − 2; zu−1n = `u∑ s=0 αu−1n,t γ syn, `u∑ s=0 αu−1n,t = 1,u ≥ 2,n ≥ 0, (1.5) (ii) for an arbitrary point y0 ∈ x, retaining the conditions in (i), define the sequence {yn}∞n=0 by yn+1 = γn,0z 1 n + `1∑ r=1 γn,kγ rz1n , `1∑ r=0 αn,r = 1; ztn = α t n,0z t+1 n + `t+1∑ s=1 αtn,sγ szt+1n , `t+1∑ s=0 αtn,s = 1,t = 1, 2, · · · ,u − 2; zu−1n = `u∑ s=0 αu−1n,t γ syn, `u∑ s=0 αu−1n,t = 1,u ≥ 2,n ≥ 0, (1.6) it is worthy to mention that in application, the stability of the iterative schemes studied aboveis quite invaluable. the first researcher to demonstrate this respecting the banach contractionconditions is ostrowski [13]. afterwards, several authors have developed this subject basicallybecause of its indispensable position in the current trend of computer programing. some recentworks in this direction could be seen in [1], [2], [3], [4], [12], [13], [14],[23], [24], [26] and the references therein. remark 1.1. the stability and the convergence results in the papers studied were made possible due to the sum conditions imposed on the control parameters; see, for example, [20], [17], [25], [26], etc and the references therein. but in application, especially for n large enough, the iterative schemes defined by (1.1), (1.2), (1.3), (1.4) (1.5) and (1.6) become practically inefficient due to the difficulties involved in generating a family of such control parameters, the windy process involved for each sum and the computational cost. base on the problems mentioned in remark 1.1, it becomes necessary to ask the followingquestions: question 1.1. is it possible to construct an alternative iterative scheme that would address the problems generated by the sum conditions imposed on the control parameters while maintaining, in particular, the results in [26], which in a larger sense contains the results of the other papers studied? https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 4 following the same argument as in [27] regarding the linear combination of the products ofcountably finite family of control parameters and the problems mentioned in remark 1.1, in thispaper, we provide an affirmative answer to question 1.1. 2. preliminary throughout the remaining sections, φ : r+ −→ r+,r+,n and h will denote monotone in-creasing subadditive function, the set of positive integers, the set of natural numbers and a realhilbert space, respectively. also, the following definition, lemmas and propositions will be neededestablish our results. definition 2.1. ( [13]) suppose y is a metric space and let γ : y −→ y be a self-map of y . let {xn}∞n=0 ⊆ y be a sequence generated by an iteration scheme xn+1 = g(γ,xn), (2.1) where x0 ∈ y is the initial approximation and g is some function. suppeose {xn}∞n=0 converges to a fixed point q of γ. let {tn}∞n=0 ⊆ y be an arbitrary sequence and set �n = d(tn,g(γ,tn)),n = 1, 2, · · · then, (2.1) is said to be γ-stable if and only if limn→∞�n = 0 implies limn→∞yn = q. note that in practice, the sequence {tn}∞n=0 could be obtained using the following approach: let x0 ∈ y . set xn+1 = g(γ,xn) and let t0 = x0. since, x1 = g(γ,x0) following the rounding in thefunction γ, the value t1 (which is estimated to be equal to x1) could be calculated to give t2, anapproximate value of g(γ,t1). the procedure is continued to yield the sequence {tn}∞n=0, which isapproximately tha same as the sequence {xn}∞n=0. lemma 2.1. (see, e.g., [26]) let {τn}∞n=0 ∈r + : τn → 0 as n →∞. for 0 ≤ δ < 1, let {wn}∞n=0 be a sequence of positive numbers satisfying wn+1 ≤ δwn +τn,n = 0, 1, 2, · · · then, wn → 0 as n →∞. lemma 2.2. (see, e.g., [17]) let (y, ‖ .‖) be a normed space, the self-map γ : y −→ y satisfies (1.13) and φ : r+ −→ r+ (retaining its usual meaning) be such that ψ(0) = 0,φ(mt) = mφ(t),m ≥ 0,t ∈r+. then, ∀i ∈n and ∀s,t ∈ y, we have ‖γjs − γjt‖≤ ρj‖s − t‖ + j∑ i=0 ( j i ) ρj−1φ(‖s − γs‖). (2.2) proposition 2.3. (see,e.g., [27]) let {αi}∞i=1 ⊆ n, where k ∈ [0,r +] is fixed and n ∈ n is any integer with k + 1 ≤ n. then, the following holds: αk + n∑ i=k+1 αi i−1∏ j=k (1 −αj) + n∏ j=k (1 −αj) = 1. (2.3) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 5 proposition 2.4. (see,e.g., [27]) let t,u,v ∈ h. let k ∈ [0,r+] be fixed and n ∈ n be such that k + 1 ≤ n. let {vi}n−1i=1 ⊆ h and {αi} n i=1 ⊆ [0, 1]. define y = αkt + n∑ i=k+1 αi i−1∏ j=k (1 −αj)vi−1 + n∏ j=k (1 −αj)v. then, ‖y −u‖2 = αk‖t −u‖2 + n∑ i=k+1 αi i−1∏ j=k (1 −αj)‖vi−1 −u‖2 + n∏ j=k (1 −αj)‖v −u‖2 −αk [ n∑ i=k+1 αi i−1∏ j=k (1 −αj)‖t −vi−1‖2 + i−1∏ j=k (1 −αj)‖t −v‖2 ] −(1 −αk) [ n∑ i=k+1 αi i−1∏ j=k (1 −αj)‖vi−1 − (αi+1 + wi+1)‖2 +αn i−1∏ j=k (1 −αj)‖v −vn−1‖2 ] , where wk = ∑n i=k+1αi ∏i−1 j=k(1 −αj)vi−1 + ∏i−1 j=k(1 −αj)v,k = 1, 2, · · · ,n and wn = (1 −cn)v . 3. main results i let h be a hilbert space and let γ : h −→ h be a self-map of x. for arbitrary x0 ∈ h definethe sequence {xn+1}∞n=0 iteratively, for s = 1, 2, · · · ,k − 2, as follows: xn+1 = δn,1xn + ∑`1 j=2δn,j ∏j−1 i=1(1 −δn,i)γ j−1y1n + ∏`1 i=1(1 −δn,i)γ `1y1n ; ysn = α s n,1xn + ∑`s+1 j=2 αsn,j ∏j−1 i=1(1 −α s n,i)γ j−1ys+1n + ∏`s+1 i=1 (1 −αsn,i)γ `1ys+1n ; yk−1n = ∑`k j=1 αk−1 n,j ∏j−1 i=1(1 −α k−1 n,i )γj−1xn + ∏`k i=1 (1 −αk−1 n,i )γ`kxn,k ≥ 2,n ≥ 1, (3.1) where `1 ≥ `2 ≥ `3 ≥ ··· ≥ `k , for each s, {{δn,i}∞n=0}`kj=1,{{αn,i}∞n=0}`kj=1 ∈ [0, 1] for each k and `1,`2, · · · ,`k are fixed integers (for each k). we shall call the iteration scheme defined by (3.1)the multistep ih-iteration scheme.again, for any x0 ∈ x, we shall call the sequence {xn}∞n=0 defined recursively, for s = 1, 2, · · · ,k − 2, by xn+1 = δn,1y 1 n + ∑`1 j=2δn,j ∏j−1 i=1(1 −δn,i)γ j−1y1n + ∏`1 i=1(1 −δn,i)γ `1y1n ; ysn = α s n,1y s+1 n + ∑`s+1 j=2 αsn,j ∏j−1 i=1(1 −α s n,i)γ j−1ys+1n + ∏`s+1 i=1 (1 −αsn,i)γ `1ys+1n ; yk−1n = ∑`k j=1 αk−1 n,j ∏j−1 i=1(1 −α k−1 n,i )γj−1xn + ∏`k i=1 (1 −αk−1 n,i )γ`kxn,k ≥ 2,n ≥ 1, (3.2) where `1 ≥ `2 ≥ `3 ≥ ··· ≥ `k , for each s, {{δn,i}∞n=0}`kj=1,{{αn,i}∞n=0}`kj=1 ∈ [0, 1] for each k and `1,`2, · · · ,`k are fixed integers (for each k), the multistep di-iteration scheme. https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 6 theorem 3.1. let h be a hilbert space, γ : h −→ h be a self-map of h satisfying the contractive condition ‖γjx − γjy‖≤ ρj‖x −y‖ + j∑ i=0 ( j i ) ρj−iφ(‖x − γx‖), (3.3) where x,y ∈ h, 0 ≤ ρj < 1, and let φ retain its usual meaning with φ(0) = 0 and φ(mt) = mφ(t),m ≥ 0,t ∈ r+. for arbitrary x0 ∈ h, let {ωn}∞n=0 be the multistep h-iteration scheme defined by (3.1). then, (i) γ defined by (3.3) has a fixed point q; (ii) the multistep ih-iteration scheme converges strongly to q ∈ γ. proof. firstly, we show that γ satisfying condition of (3.3) has a fixed point. assume there existstwo points q1,q2 ∈ f (γ) with 0 < ‖q1 −q2‖. then, we have 0 < ‖q1 −q2‖ = ‖γjq1 − γjq2‖ ≤ ρj‖q1 −q2‖ + j∑ i=0 ( j i ) ρj−iφ(‖q √ 1 − γq1‖) = ρj‖q1 −q2‖ + j∑ i=0 ( j i ) ρj−iφ(0) ⇒ (1 −ρj)ρj‖q1 −q2‖≤ 0. using the fact that ρj ∈ [[0, 1), we get 0 < 1 −ρj and ‖q1 −q2‖≤ 0.since the norm is a nonnegative function, we get ‖q1 −q2‖ = 0; q1 = q2 = q(say). therefore, γconverges uniquely to a point of f (γ).now, we show that the sequence defined by (3.1) converges strongly to q ∈ f (γ). using (3.3)and proposition 2.4 with xn+1 = y,u = q,xn = t, j = i,k = 1, γj−1y1n = vj−1 and γ`1y1n = v , wehave ‖xn+1 −q‖2 ≤ δn,1‖xn −q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖γj−1y1n − γ j−1q‖2 + `1∏ i=1 (1 −δn,i)‖γ`1y1n − γ `1q‖2 (3.4) but from (3.3), with y = y1n , we have ‖γj−1y1n − γ j−1q‖ ≤ ρj‖y1n −q‖ + j∑ i=0 ( j i ) ρj−1φ(‖q − γq‖) = ρj‖y1n −q‖ (3.5) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 7 proposition 2.3, (3.4) and (3.5) imply ‖xn+1 −q‖2 ≤ δn,1‖xn −q‖2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i)‖y1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖y1n −q‖ 2 = δn,1‖xn −q‖2 + ( 1 −δ1n,1 − `1∏ i=1 (1 −δn,i)(ρj)2 ) ‖y1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖y1n −q‖ 2 = δn,1‖xn −q‖2 + ( 1 −δ1n,1 ) ‖y1n −q‖ 2 (3.6) since `1,`k are fixed integers and αsn,i ∈ [0, 1] for each s, we have, using proposition 2.3, thefollowing estimates for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 : ‖y1n −q‖ 2 ≤ αn,1‖xn −q‖2 + `2∑ j=2 αn,j j−1∏ i=1 (1 −αn,i)‖γj−1y2n − γ j−1q‖2 + `2∏ i=1 (1 −αn,i)‖γ`2y2n − γ `2q‖2 ≤ α1n,1‖xn −q‖ 2 + `2∑ j=2 αn,j(ρ j)2 j−1∏ i=1 (1 −αn,i)‖y2n −q‖ 2 + `2∏ i=1 (1 −αn,i)(ρj)2‖y2n −q‖ 2 ≤ α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) [ α2n,1‖xn −q‖ 2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)‖y 3 n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖y3n −q‖ 2 ] + `2∏ i=1 (1 −α1n,i)(ρ j)2 [ α2n,1‖xn −q‖ 2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)‖y 3 n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖y3n −q‖ 2 ] https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 8 = α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ‖y3n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖y3n −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ‖y3n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ‖y3n −q‖ 2 ≤ α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) [α3n,1‖xn −q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖y4n −q‖ 2 ] + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖xn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖y4n −q‖ 2 ] +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )[ α3n,1‖xn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖y4n −q‖ 2 ] + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )[ α3n,1‖xn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖y4n −q‖ 2 ] https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 9 = α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖y4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖y4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖y4n −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖y4n −q‖2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 10 = α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖xn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2 ) ‖y4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2) ) ‖y4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2) ) ‖y4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2 ) ‖y4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 11 = α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖xn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  (1 −α3n,1)(ρj)2‖y4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2)‖y4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2)‖y4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2‖y4n −q‖ 2 ≤ α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖xn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 12 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1(1 −α3n,1)(ρj)2‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2)‖xn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2)‖xn −q‖2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2‖xn −q‖2 + · · · +  `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)3 j−1∏ i=1 (1 −α2n,i) ×···× `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i )  ×  `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) αsn,1‖xn −q‖2 + (ρj)2 ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ×(ρj)2 ( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) (ρj)2 ( `4∏ i=1 (1 −α3n,i)(ρ j)2 ) ×···× (ρj)2 `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2  ×(ρj)2 ( `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ‖xn −q‖2 < α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) α3n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i) ) ‖xn −q‖2 + ( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1‖xn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 13 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i)) )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + · · · + ( 1 −α1n,1 − `2∏ i=1 (1 −α2n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) × ( 1 −α3n,1 − `4∏ i=1 (1 −α3n,i) ) ×···× 1 −α`s−2n,1 − `s−1∏ i=1 (1 −α`s−2 n,i )  × ( 1 −α`s−1n,1 − `s∏ i=1 (1 −α`s−1 n,i ) ) αsn,1‖xn −q‖ 2 + ( `2∏ i=1 (1 −α1n,i) ) × ( `3∏ i=1 (1 −α2n,i) )( `4∏ i=1 (1 −α3n,i) ) ×···× `s−1∏ i=1 (1 −α`s−2 n,i )  × ( `s∏ i=1 (1 −α`s−1 n,i ) ) ‖xn −q‖2 (3.7) < α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) ) α3n,1 ( 1 −α2n,1 ) ‖xn −q‖2 +α3n,1 ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) ‖xn −q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + · · · + + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) αsn,1‖xn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 14 = α1n,1‖xn −q‖ 2 + α2n,1 ( 1 −α1n,1 − `2∏ i=1 (1 −α1n,i) ) ‖xn −q‖2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + ( (1 −α1n,1 − `2∏ i=1 (1 −α1n,i) ) α3n,1 ( 1 −α2n,1 ) ‖xn −q‖2 +α3n,1 ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) ‖xn −q‖2 + ( (1 −α1n,1 − `2∏ i=1 (1 −α1n,i) )( 1 −α2n,1 ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + · · · + + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) αsn,1‖xn −q‖ 2 < [α1n,1 + α 2 n,1 ( 1 −α1n,1 ) + (1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) + (1 −α1n,1) ( 1 −α2n,1 ) α3n,1(1 −α 3 n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) ]‖xn −q‖2 (3.8) (3.6) and (3.8) imply that ‖xn+1 −q‖2 ≤ {δn,1 + (1 −δn,1) [α1n,1 + α 2 n,1 ( 1 −α1n,1 ) + (1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) +(1 −α1n,1) ( 1 −α2n,1 ) (1 −α3n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) ]}‖xn −q‖2 (3.9) using lemma 2.3, we obtain (from (3.9)) that the sequence {xn}∞n=0 converges strongly to q ∈ f (γ);and this completes the proof. � theorem 3.2. let h be a hilbert space, γ : h −→ h be a self-map of h satisfying the contractive condition ‖γjx − γjy‖≤ ρj‖x −y‖ + j∑ i=0 ( j i ) ρj−1φ(‖x − γx‖), (3.10) where x,y ∈ h, 0 ≤ ρj < 1, and let φ retain its usual meaning with φ(0) = 0 and φ(mt) = mφ(t),m ≥ 0,t ∈ r+. for arbitrary x0 ∈ h, let {ωn}∞n=0 be the multistep di-iteration scheme defined by (3.2). then, (i) γ defined by (3.10) has a fixed point q; https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 15 (ii) the multistep sh-iteration scheme converges strongly tp q ∈ γ. proof. we first show that γ satisfying condition of (3.10) has a fixed point. assume there existstwo points q1,q2 ∈ f (γ) with 0 < ‖q1 −q2‖. then, we have 0 < ‖q1 −q2‖ = ‖γjq1 − γjq2‖ ≤ ρj‖q1 −q2‖ + j∑ i=0 ( j i ) ρj−1φ(‖q √ 1 − γq1‖) = ρj‖q1 −q2‖ + j∑ i=0 ( j i ) ρj−1φ(0) ⇒ (1 −ρj)ρj‖q1 −q2‖≤ 0. using the fact that ρj ∈ [[0, 1), we get 0 < 1 −ρj and ‖q1 −q2‖≤ 0.since the norm is a nonnegative function, we get ‖q1 −q2‖ = 0; q1 = q2 = q(say). therefore, γconverges uniquely to a point of f (γ).now, we show that the sequence defined by (3.1) converges strongly to q ∈ f (γ). using (3.2)and proposition 2.4 with xn+1 = y,u = q,y1n = t, j = i,k = 1, γj−1y1n = vj−1 and γ`1y1n = v , weget ‖xn+1 −q‖2 = δn,1‖y1n −q‖ 2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖γj−1y1n − γ j−1q‖2 + `1∏ i=1 (1 −δn,i)‖γ`1y1n − γ `1q‖2 (3.11) but from (3.10), with y = y1n , we have ‖γj−1y1n − γ j−1q‖ ≤ ρj‖y1n −q‖ + j∑ i=0 ( j i ) ρj−1φ(‖q − γq‖) = ρj‖y1n −q‖ (3.12) proposition 2.3, (3.11) and (3.12) imply ‖xn+1 −q‖2 ≤ δ1n,1‖y 1 n −q‖ 2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i)‖y1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖y1n −q‖ 2 = δ1n,1‖y 1 n −q‖ 2 + ( 1 −δ1n,1 − `1∏ i=1 (1 −δn,i)(ρj)2 ) ‖y1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖y1n −q‖ 2 = ‖y1n −q‖ 2 (3.13) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 16 since `1,`k are fixed integers and αsn,i ∈ [0, 1] for each s, we have (using proposition 2.3, (3.2)and (3.12)) the following estimates for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 : ‖y1n −q‖ 2 ≤ α1n,1‖y 2 n −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)‖γ j−1y2n − γ j−1q‖2 + `2∏ i=1 (1 −α1n,i)‖γ `2y2n − γ `2q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 ‖y2n −q‖2 ≤ α1n,1 + `2∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 [α2n,1‖y3n −q‖2 + `3∑ j=2 α2n,j j−1∏ i=1 (1 −α2n,i)‖γ j−1y3n − γ j−1q‖2 + `3∏ i=1 (1 −α2n,i)‖γ `3y3n − γ `3q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 [α2n,1‖y3n −q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i)‖y2n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖y3n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖y3n −q‖ 2 (3.14) ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖y 4 n −q‖ 2 + `4∑ j=2 α3n,j j−1∏ i=1 (1 −α3n,i)‖γ j−1y4n − γ j−1q‖2 + `4∏ i=1 (1 −α3n,i)‖γ `4y4n − γ `4q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 17 × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖y 4 n −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α3n,i)(ρ j)2‖y4n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) ‖y4n −q‖ 2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 )[ α4n,1‖y 5 n −q‖ 2 + `5∑ j=2 α4n,j j−1∏ i=1 (1 −α4n,i)‖γ j−1y5n − γ j−1q‖2 + `5∏ i=1 (1 −α4n,i)‖γ `5y5n − γ `5q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 )[ α4n,1‖y 5 n −q‖ 2 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i)‖y 5 n −q‖ 2 + `5∏ i=1 (1 −α4n,i)(ρ j)2‖y5n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 18 × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y5n −q‖ 2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖xn −q‖2 (3.15) (3.13) and (3.15) imply that ‖xn+1 −q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖xn −q‖2 (3.16) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 19 since ρj ∈ [0, 1], we obtain using proposition 2.3, for j = 1, 2, 3, · · · ,s − 1, that q ≤ p = 1, (3.17) where q = ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) and p = ( α3n,1 + `4∑ j=2 α3n,j j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i) ) × ( α4n,1 + `5∑ j=2 α4n,j j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i) ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i ) ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i ) ) applying (3.17) in (3.16), we obtain, using lemma 2.3 that the sequence {xn}∞n=0 defined by (3.2)converges strongly to the fixed point q in f (γ). thus, the proof is completed. � example 3.1. let the operator γ : [0, 1] −→ [0, 1] be defined as γz = z 3 ,∀z ∈ [0, 1]. clearly, γ is quasi-contractive satisfying (2.2) with a unique fixed point 0; see, for example, [26] for details. set α1n,1 = δ 1 n,1 = 1 √ n + 1 ,n = 1, 2, · · · ,n0, f or n0 ∈n; δn,i = 1 −δ1n,1, f or i = 1, 2, · · · ,`1 and αsn,i = 1 − 2α 1 n,1, f or i = 1, 2, · · · ,`s+1,s = 1, 2, · · · ,n0. it is not hard to see that all the conditions of theorem 3.1 and theorem 3.2 has been satisfied by example 3.1. https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 20 4. main results ii here, we consider stablity results for the multistep ih-iteration scheme and the multistep di-iteration scheme defined by (3.1) and (3.2) for operators satisfying (2.2), respectively. theorem 4.1. let h be a hilbert space, γ : h −→ h be a self-map of h satisfying the contractive condition ‖γjx − γjy‖≤ ρj‖x −y‖ + j∑ i=0 ( j i ) ρj−1φ(‖x − γx‖), (4.1) where x,y ∈ h, 0 ≤ ρj < 1, and let φ retains its usual meaning with φ(0) = 0 and φ(mt) = mφ(t),m ≥ 0,t ∈ r+. for arbitrary x0 ∈ h, let {xn}∞n=0 be the multistep di-iteration scheme defined by (3.2). assume f (γ) 6= ∅,q ∈ f (γ). then, the multisetp di-iterative scheme is γ-stable. proof. let {vn}∞n=0, be a real sequences in h. suppose {tn}∞n=0 ⊂ x is an arbitrary sequence, set �n = ‖tn+1 −δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n − `1∏ i=1 (1 −δn,i)γ`1v1n‖ 2 (4.2) where, for s = 1, 2, · · · ,k − 2, vsn = α s n,1v s+1 n + `s+1∑ j=2 αsn,j j−1∏ i=1 (1 −αsn,i)γ j−1vs+1n + `s+1∏ i=1 (1 −αsn,i)γ `1vs+1n (4.3) and, for k ≥ 2, vk−1n = `k∑ j=1 αk−1 n,j j−1∏ i=1 (1 −αk−1 n,i )γj−1tn + `k∏ i=1 (1 −αk−1 n,i )γ`ktn,n ≥ 1, (4.4) now, suppose �n → 0 as n →∞. then, we show that tn → q as n →∞ using contractive mappingdefined by (4.1).indeed, using proposition 2.4 with u = q,v1n = t, j = i,k = 1, γj−1v1n = vj−1 and γ`1v1n = v„we obtain ‖tn+1 −q‖2 = ‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q −[δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n − tn+1]‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 21 ≤ ‖− [tn+1 −δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n − `1∏ i=1 (1 −δn,i)γ`1v1n ]‖ 2 +‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q‖ 2 = ‖tn+1 −δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n − `1∏ i=1 (1 −δn,i)γ`1v1n‖ 2 +‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q‖ 2 = �n + ‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q‖ 2 ≤ �n + δn,1‖v1n,1 −q‖ 2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖γj−1v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)‖γ`1v1n −q‖ 2 ≤ �n + δn,1‖v1n,1 −q‖ 2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i)‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 ≤ �n + ( δn,1 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i) + `1∏ i=1 (1 −δn,i)(ρj)2 ) ×‖v1n −q‖ 2 (4.5) since `1,`k are fixed integers and αsn,i ∈ [0, 1] for each s, using (3.2) and (3.12), the estimationsbelow are obtained, for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 :, ‖v1n −q‖ 2 ≤ α1n,1‖v 2 n −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)‖γ j−1v2n − γ j−1q‖2 + `2∏ i=1 (1 −α1n,i)‖γ `2v2n − γ `2q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 ‖v2n −q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 22 ≤ α1n,1 + `2∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 [α2n,1‖v3n −q‖2 + `3∑ j=2 α2n,j j−1∏ i=1 (1 −α2n,i)‖γ j−1v3n − γ j−1q‖2 + `3∏ i=1 (1 −α2n,i)‖γ `3v3n − γ `3q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 [α2n,1‖v3n −q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i)‖v3n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖v3n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖v3n −q‖ 2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖v 4 n −q‖ 2 + `4∑ j=2 α3n,j j−1∏ i=1 (1 −α3n,i)‖γ j−1v4n − γ j−1q‖2 + `4∏ i=1 (1 −α3n,i)‖γ `4v4n − γ `4q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖v 4 n −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α3n,i)(ρ j)2‖v4n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) ‖v4n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 23 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 )[ α4n,1‖v 5 n −q‖ 2 + `5∑ j=2 α4n,j j−1∏ i=1 (1 −α4n,i)‖γ j−1v5n − γ j−1q‖2 + `5∏ i=1 (1 −α4n,i)‖γ `5v5n − γ `5q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 )[ α4n,1‖v 5 n −q‖ 2 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i)‖v 5 n −q‖ 2 + `5∏ i=1 (1 −α4n,i)(ρ j)2‖v5n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v5n −q‖ 2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 24 ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖tn −q‖2 (4.6) (4.5) and (4.6)imply that ‖tn+1 −q‖2 ≤ ( δn,1 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i) + `1∏ i=1 (1 −δn,i)(ρj)2 ) × α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖tn −q‖2 + �n (4.7) note that (4.7) is valid since γq = q and φ(0) = 0.now, since ρj ∈ [0, 1], we obtain using proposition 2.3, for j = 1, 2, 3, · · · ,s − 1, that τn < ηn = 1, (4.8) where τn = ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 25 × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) and ηn = ( α3n,1 + `4∑ j=2 α3n,j j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i) ) × ( α4n,1 + `5∑ j=2 α4n,j j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i) ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i ) ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i ) ) putting (4.8) in (4.7), we obtain, using lemma 2.3 that the sequence {tn}∞n=0 converges strongly tothe point q in f (γ).on the other hand, suppose tn → q as n → ∞. then, we show that � → 0 as n → ∞. indeed,from (3.5) with v1n = y1n , (4.2) and proposition 2.4 with u = q,v1n = t, j = i,k = 1, γj−1v1n = vj−1and γ`1v1n = v„ we have �n = ‖tn+1 −δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n − `1∏ i=1 (1 −δn,i)γ`1v1n‖ 2 = ‖tn+1 −q − δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q ‖2 ≤ ‖tn+1 −q‖2 + ‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q‖ 2 ≤ ‖tn+1 −q‖2 + δn,1‖v1n,1 −q‖ 2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖γj−1v1n − γ j−1q‖2 + `1∏ i=1 (1 −δn,i)‖γ`1v1n − γ `1q‖2 ≤ ‖tn+1 −q‖2 + δn,1‖v1n,1 −q‖ 2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 26 = ‖tn+1 −q‖2 + δn,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)(ρj)2 + `1∏ i=1 (1 −δn,i)(ρj)2  ×‖v1n −q‖ 2 (4.9) putting (4.6) into (4.9), and using (4.8), we get �n ≤ ‖tn+1 −q‖2 + δn,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)(ρj)2 + `1∏ i=1 (1 −δn,i)(ρj)2  × α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖tn −q‖2 ≤ ‖tn+1 −q‖2 + τn‖tn −q‖2 (4.10) thus, from our assumption, we obtain from (4.10) that �n → 0 as n → ∞. hence, the multistep di-iteration scheme (3.2) is γ-stable. thus, tje proof is completed. � theorem 4.2. let h be a hilbert space, γ : h −→ h be a self-map of h satisfying the contractive condition ‖γjx − γjy‖≤ ρj‖x −y‖ + j∑ i=0 ( j i ) ρj−1φ(‖x − γx‖), (4.11) where x,y ∈ h, 0 ≤ ρj < 1, and let φ retains its usual meaning with φ(0) = 0 and φ(mt) = mφ(t),m ≥ 0,t ∈ r+. for arbitrary x0 ∈ h, let {ωn}∞n=0 be the multistep ih-iteration scheme defined by (3.1). assume f (γ) 6= ∅,q ∈ f (γ). then, the multisetp ih-iteration scheme is γ-stable. https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 27 proof. let {tn}∞n=0 and {vn}∞n=0, for i = 1, 2, · · · ,s − 1, be two real sequences in h. set �n = ‖tn+1 −δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n − `1∏ i=1 (1 −δn,i)γ`1v1n‖ 2 (4.12) where, for s = 1, 2, · · · ,k − 2, vsn = α s n,1tn + `s+1∑ j=2 αsn,j j−1∏ i=1 (1 −αsn,i)γ j−1vs+1n + `s+1∏ i=1 (1 −αsn,i)γ `1vs+1n (4.13) and, for k ≥ 2, vk−1n = `k∑ j=1 αk−1 n,j j−1∏ i=1 (1 −αk−1 n,i )γj−1tn + `k∏ i=1 (1 −αk−1 n,i )γ`ktn,n ≥ 1, (4.14) now, suppose �n → 0 as n →∞. then, we show that tn → q as n →∞ using contractive mappingdefined by (4.1).indeed, using proposition 2.4 with u = q,tn = t, j = i,k = 1, γj−1v1n = vj−1 and γ`1v1n = v„we obtain ‖tn+1 −q‖2 = ‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q −[δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n − tn+1]‖ 2 ≤ ‖− [tn+1 −δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n − `1∏ i=1 (1 −δn,i)γ`1v1n ]‖ 2 +‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q‖ 2 = ‖tn+1 −δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n − `1∏ i=1 (1 −δn,i)γ`1v1n‖ 2 +‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q‖ 2 = �n + ‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q‖ 2 ≤ �n + δn,1‖tn −q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖γj−1v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)‖γ`1v1n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 28 ≤ �n + δn,1‖tn −q‖2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i)‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 = �n + δn,1‖tn −q‖2 + ( 1 −δn,1 − `1∏ i=1 (1 −δn,i) ) (ρj)2‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 < �n + δn,1‖tn −q‖2 + (1 −δn,1)‖v1n −q‖ 2 (4.15) since `1,`k are fixed integers and αsn,i ∈ [0, 1] for each s, the estimations below are obtained for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 : ‖v1n −q‖ 2 ≤ αn,1‖tn −q‖2 + `2∑ j=2 αn,j j−1∏ i=1 (1 −αn,i)‖γj−1v2n − γ j−1q‖2 + `2∏ i=1 (1 −αn,i)‖γ`2v2n − γ `2q‖2 ≤ α1n,1‖tn −q‖ 2 + `2∑ j=2 αn,j(ρ j)2 j−1∏ i=1 (1 −αn,i)‖v2n −q‖ 2 + `2∏ i=1 (1 −αn,i)(ρj)2‖v2n −q‖ 2 ≤ α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) [ α2n,1‖tn −q‖ 2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)‖v 3 n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖v3n −q‖ 2 ] + `2∏ i=1 (1 −α1n,i)(ρ j)2 [ α2n,1‖tn −q‖ 2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)‖v 3 n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖v3n −q‖ 2 ] = α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ‖v3n −q‖2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 29 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖v3n −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ‖v3n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ‖v3n −q‖ 2 ≤ α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) [α3n,1‖tn −q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖v4n −q‖ 2 ] + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖tn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖v4n −q‖ 2 ] +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )[ α3n,1‖tn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖v4n −q‖ 2 ] + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )[ α3n,1‖tn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖v4n −q‖ 2 ] = α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 30 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖v4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖v4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖v4n −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖v4n −q‖2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 = α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 31 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖tn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2 ) ‖v4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2) ) ‖v4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2) ) ‖v4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2 ) ‖v4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 32 = α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖tn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  (1 −α3n,1)(ρj)2‖v4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2)‖v4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2)‖v4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2‖v4n −q‖ 2 ≤ α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖tn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 33 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1(1 −α3n,1)(ρj)2‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2)‖tn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2)‖tn −q‖2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2‖tn −q‖2 + · · · +  `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)3 j−1∏ i=1 (1 −α2n,i) ×···× `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i )  ×  `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) αsn,1‖tn −q‖2 + (ρj)2 ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ×(ρj)2 ( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) (ρj)2 ( `4∏ i=1 (1 −α3n,i)(ρ j)2 ) ×···× (ρj)2 `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2  ×(ρj)2 ( `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ‖tn −q‖2 < α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i) ) ‖tn −q‖2 + ( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 34 + ( `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + ( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i)) )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + · · · + ( 1 −α1n,1 − `2∏ i=1 (1 −α2n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) × ( 1 −α3n,1 − `4∏ i=1 (1 −α3n,i) ) ×···× 1 −α`s−2n,1 − `s−1∏ i=1 (1 −α`s−2 n,i )  × ( 1 −α`s−1n,1 − `s∏ i=1 (1 −α`s−1 n,i ) ) αsn,1‖tn −q‖ 2 + ( `2∏ i=1 (1 −α1n,i) ) × ( `3∏ i=1 (1 −α2n,i) )( `4∏ i=1 (1 −α3n,i) ) ×···× `s−1∏ i=1 (1 −α`s−2 n,i )  × ( `s∏ i=1 (1 −α`s−1 n,i ) ) ‖tn −q‖2 (4.16) < α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) ) α3n,1 ( 1 −α2n,1 ) ‖tn −q‖2 +α3n,1 ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) ‖tn −q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 35 + ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + · · · + + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) αsn,1‖tn −q‖ 2 = α1n,1‖tn −q‖ 2 + α2n,1 ( 1 −α1n,1 − `2∏ i=1 (1 −α1n,i) ) ‖tn −q‖2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + ( (1 −α1n,1 − `2∏ i=1 (1 −α1n,i) ) α3n,1 ( 1 −α2n,1 ) ‖tn −q‖2 +α3n,1 ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) ‖tn −q‖2 + ( (1 −α1n,1 − `2∏ i=1 (1 −α1n,i) )( 1 −α2n,1 ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + · · · + + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) αsn,1‖tn −q‖ 2 < [ α1n,1 + α 2 n,1 ( 1 −α1n,1 ) + (1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) + (1 −α1n,1) ( 1 −α2n,1 ) α3n,1(1 −α 3 n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 )] ‖tn −q‖2 (4.17) (4.15) and (4.17) imply that ‖tn+1 −q‖2 ≤ {δn,1 + (1 −δn,1) [α1n,1 + α 2 n,1 ( 1 −α1n,1 ) + (1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) +(1 −α1n,1) ( 1 −α2n,1 ) (1 −α3n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) ]}‖tn −q‖2 (4.18) using lemma 2.3, we obtain (from (4.18)) that the sequence {xn}∞n=0 converges strongly to q ∈ f (γ).conversely, suppose tn → q as n → ∞. then, we show that � → 0 as n → ∞. indeed, from(3.5) with v1n = y1n , (4.12) and proposition 2.4 with u = q,tn = t, j = i,k = 1, γj−1v1n = vj−1 and https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 36 γ`1v1n = v„ we have �n = ‖tn+1 −δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n − `1∏ i=1 (1 −δn,i)γ`1v1n‖ 2 = ‖tn+1 −q − δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q ‖2 ≤ ‖tn+1 −q‖2 + ‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)γj−1v1n + `1∏ i=1 (1 −δn,i)γ`1v1n −q‖ 2 ≤ ‖tn+1 −q‖2 + δn,1‖tn −q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖γj−1v1n − γ j−1q‖2 + `1∏ i=1 (1 −δn,i)‖γ`1v1n − γ `1q‖2 ≤ ‖tn+1 −q‖2 + δn,1‖tn −q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 = ‖tn+1 −q‖2 + δn,1‖tn −q‖2 + ( 1 −δn,1 − `1∏ i=1 (1 −δn,i) ) (ρj)2‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 = ‖tn+1 −q‖2 + δn,1‖tn −q‖2 + (1 −δn,1)‖v1n −q‖ 2 (4.19) (4.17) and (4.19) imply �n ≤ ‖tn+1 −q‖2 + { δn,1 + (1 −δn,1) [ α1n,1 + α 2 n,1 ( 1 −α1n,1 ) +(1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) + (1 −α1n,1) ( 1 −α2n,1 ) α3n,1(1 −α 3 n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 )]} ‖tn −q‖2 (4.20) again, from our assumption, we obtain from (4.20) that �n → 0 as n → ∞. hence, the multistep ih-iteration scheme (3.1) is γ-stable, and this completes the proof. � remark 4.1. the following areas are still open: https://doi.org/10.28924/ada/ma.2.1 eur. j. math. anal. 10.28924/ada/ma.2.1 37 (i) to reconstruct, approximate the fixed points and the stability results of some existing iterative schemes in the current literature, other than the ones under study, for finite family of certain class of contractive-type map;(ii) to compare convergent rates of the iterative schemes defined by (3.1) and (3.2) with those of (1.5) and (1.6). competing interestthe authors declare that there is no conflict of interest. references [1] b. e. rhoade, fixed point theorems and stability results for fixed point iteration procedures, indian j. pure appl.math. 24(11) (1993) 691-03.[2] b. e. rhoade, fixed point theorems and stability results for fixed 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https://doi.org/10.1155/2020/3287968.[28] i. k. agwu, d. i. igbokwe, new iteration algorithm for equilibrium problems and fixed point problems of two finitefamilies of asymptotically demicontractive multivalued mappings, (in press). https://doi.org/10.28924/ada/ma.2.1 https://doi.org/10.2307/2032162 https://doi.org/10.2307/2032162 https://doi.org/10.1090/s0002-9939-1974-0336469-5 https://doi.org/10.1090/s0002-9939-1974-0336469-5 https://doi.org/10.1186/1687-1812-2014-45 https://doi.org/10.1186/1687-1812-2014-45 https://doi.org/10.1155/2020/3287968 1. introduction 2. preliminary 3. main results i 4. main results ii references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 7doi: 10.28924/ada/ma.3.7 a note on the stability of functional equations via a celebrated direct method dongwen zhang1, john michael rassias2, qi liu3, yongjin li4,∗ 1school of mathematics (zhuhai), sun yat-sen university, zhuhai 519082, p.r. china zhangdw25@mail2.sysu.edu.cn 2national and kapodistrian university of athens, department of mathematics and informatics, attikis 15342, greece jrassias@primedu.uoa.gr 3school of mathematics and physics, anqing normal university, anqing 246133, p.r. china liuq325@mail2.sysu.edu.cn 4department of mathematics, sun yat-sen university, guangzhou, 510275, p.r. china stslyj@mail.sysu.edu.cn ∗correspondence: stslyj@mail.sysu.edu.cn abstract. more than ten years after justyna sikorska [8] attempted to solve the heyers-ulam sta-bility of a single variable equation by using direct method. in this paper, we will improve the resultsof justyna sikorska by using a more efficient approach. relations between the generalized functionalequation, the dependence of their different parameters and several properties are also further ex-plored. to achieve the problem, we try to develop some new techniques to overcome the fundamentaldifficulties caused by the different properties of the function and the presence of several variables inthe equation. furthermore, we continue to construct and study a couple of functional equations bymaking a new direct method. 1. introduction the core idea of the hyers-ulam stability for functional equations has been dated back to awell-known problem concerning about group homomorphisms solved by s.m. ulam and d.h. hyers(see [1–3]). in the last decades, a great number of papers treating the stability problem aboutfunctional equations has already been achieved and a great deal of important problems about thisfield has been studied ( [4–7]). it follows that the most efficient methods have been stated in manypapers ( [10, 18–24, 27]) such as the direct approach, the shadowing approach, and invariant meanapproach and so on. in particular, the direct method is always the main studying tool on theinvestigation of functional equations of different types. received: 15 sep. 2022. key words and phrases. stability; several functional equations; approximations; odd function; even function.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 2 the stability problems for an appropriate simple variable functional equations have earlier beeninvestigated by direct method. the direct method is familiar with many readers to derive thesolutions of the equations. the author in [8] have made full use of quite a general way to solve thehyers-ulam stability problems on the functional equations under which many excellent outcomeshave been achieved without reduplicating the similar procedure in the whole process of computation.however, her results can only be used to derive the solutions of the equation where the mediatefunction is odd. this is exactly our contribution to the paper. in fact, a straightforward observationis that the inequality ‖f (x) −uf (e(x)) −vf (−e(x))‖6 δ(x) can be solved if the function h is even. next, the present studying approach calls us to investigatethe following functional inequality, by using a direct method, under which the result can not becovered by earlier works ‖f (x + y + z) + f (x) + f (y) + f (z) − f (x + y) − f (z + y) − f (x + z)‖ 6 k (‖x‖r + ‖y‖r + ‖z‖r ) . (1.1) in fact, the functional inequality (1.1) comes from some equivalent characterizations of hilbert spacein [15]. the investigator described several properties of an inner product space and applies theseresults to solve many interesting functional inequalities such as: zarantone’s inequality, hayashi’sinequality and so on. however, the more far reaching work can be done m. fréchet in [16] underwhich he ascertained that the corresponding equation is a necessary prerequisite condition whencomplex or real normed completed spaces become hilbert spaces. investigator in [17] studied thestability of fréchet functional equation from which a characterization of inner product spaces hadbeen achieved by using a stationary point theorem in banach spaces. compared with the beforestudying approaches, we further explored solutions of the equation (1.1) in this literature. of course,to the best of our knowledge, it has also already been solved by [8] under which a direct methodwas to derive solutions of the equation (1.1) and to look for some improvement approximations.however, in this literature we make a new direct method to achieve the solution of equation (1.1)must be close to the approximate solution, approximately satisfying the corresponding equation.besides this, we will consider that the functions on the functional equation of different typeshave been defined in a more general domain. for instance, the papers [11, 12] have defined anadditive ρ-functional inequalities in nonarchimedean normed spaces and banach spaces. however,this phenomenon can not attract enough attention to the study of functional equations in themore general and complex nonlinear structure of f-spaces (see the definition in [13, 14]). but,the nonlinear structure of space has always stood in a very important position of leadership infunctional analysis. based on the above analysis, it is of great significance that the functionalinequality is considered in β-homogeneous f-space. https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 3 in section 2, the counterpart of theorem 2.1 from [8] where the mediate function is odd willbe considered. in the subsequent part, a new direct method for solving (1.1) in f-space will bedescribed and some new extended results of theorem 2.1 from [8] will be presented. with it, twonew different applications of the results will be described in the final part. 2. a simple variable of abstract equation in theorem 2.1 from [8], sikorska solved the equation (2.1) where the mediate function e is oddand the related parameters u, v are restricted on the real field. for simplicity in notation, weprovide traditionally our first result with the studying mapping defined in banach space. by makinguse of small conjectures the more general form of the results will be provided in β-homogeneous f-space in section 3. therefore, our first result is simply considered in banach space. theorem 2.1 let (x, +) be a group, and (y,‖ ·‖) be a banach space, and assume the mapping f : x → y satisfying the inequality ‖f (x) −uf (e(x)) −vf (−e(x))‖6 δ(x), x ∈ x, (2.1) where u, v ∈ (−∞, +∞), and the mappings e : x → x, δ : x → [0,∞) satisfy that e is even ( i.e., e(−x) = e(x) for every x ∈ x). let the infinite progression ∑∞n=0 [|un|δ (en(x)) + |vn|δ (−en(x))]with u0 := 1, un := [ u(u + v)n−1 ] , v0 := 0, vn := [ v(u + v)n−1 ] , n ∈n(em states the m-th composition of function e ), be assumed convergence for every x ∈ x. thenthere has a unique even mapping g : x → y satisfying g(x) = ung(e n(x)) + vng(−en(x)), and ‖f (x) −g(x)‖6 ∞∑ i=0 [ |ui|δ ( ei (x) ) + |vi|δ ( −ei (x) )] , x ∈ x and n ∈n. (2.2) proof. we will prove that ‖f (x) −unf (en(x)) −vnf (−en(x))‖6 γn(x), x ∈ x, (2.3) where γn(x) := n−1∑ i=0 [ |ui|δ ( ei (x) ) + |vi|δ ( −ei (x) )] , x ∈ x,n ∈n. first of all, consider with every m,n ∈n and it is easy to observe that un+1 = uun + uvn, vn+1 = vvn + vun, and un+m = umun + vmun, vn+m = umvn + vmvn. (2.4) https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 4 from the definition of sequences (un) and (vn) we also have uvn = vun, vmun = umvn. first, (2.1) gives (2.3) with setting n = 1, and by mathematical induction, later we suppose that (2.3) establishes for some n ∈ n. we prove that in the case for n + 1 by virtue of (2.1) ‖f (x) −un+1f ( en+1(x) ) −vn+1f ( −en+1(x) ) ‖ 6‖f (x) −unf (en(x)) −vnf (−en(x))‖ + |un| ∥∥f (en(x)) −uf (en+1(x))−vf (−en+1(x))∥∥ + |vn| ∥∥f (−en(x)) −uf (en+1(x))−vf (−en+1(x))∥∥ 6 n−1∑ i=0 [ |ui|δ ( ei (x) ) + |vi|δ ( −ei (x) )] + |un|δ (en(x)) + |vn|δ (−en(x)) = n∑ i=0 [ |ui|δ ( ei (x) ) + |vi|δ ( −ei (x) )] . since the series ∑∞i=0 [|ui|δ(ei (x)) + |vi|δ(−ei (x))] is convergent for every x ∈ x, combinedwith (2.3) and by virtue of the completeness of y , the mapping can be well defined as in thefollowing: g(x) := lim n→∞ [unf (e n(x)) + vnf (−en(x))] , x ∈ x, (2.5) and we prove the following properties of the function g.an easy computation is to prove that ug(e(x)) + vg(−e(x)) = u lim n→∞ [ unf ( en+1(x) ) + vnf ( −en+1(x) )] + v lim n→∞ [ unf ( en+1(x) ) + vnf ( −en+1(x) )] = lim n→∞ [ (uun + vun) f ( en+1(x) ) + (uvn + vvn) f ( −en+1(x) )] = g(x). furthermore, we will prove the more general property of g g(x) = ung (e n(x)) + vng (−en(x)) , f or all x ∈ x and n ∈ n. (2.6) by induction, we assume that the equation is true for all natural number k with k ≤ n for some n ∈ n. let us calculate with k = n + 1 g(x) = ung (e n(x)) + vng (−en(x)) = un(ug ( en+1(x) ) + vg ( −en+1(x) ) ) + vn(ug ( en+1(x) ) + vg ( −en+1(x) ) ) = un+1g ( en+1(x) ) + vn+1g ( −en+1(x) ) ). in particular, we also have that the function g is even. an easy computation is to state that g(−x) = ung (en(x)) + vng (−en(x)) = g(x), f or every x ∈ x and n ∈n. https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 5 in order to achieve the uniqueness of g, suppose further that ḡ : x → y is the another mappingsuch that (2.2) and (2.6) hold. then ‖g(x) − ḡ(x)‖6 2 ∞∑ i=0 [ |ui|δ ( ei (x) ) + |vi|δ ( −ei (x) )] , x ∈ x. moreover, we have g(x) − ḡ(x) = un [g (en(x)) − ḡ (en(x))] + vn [g (−en(x)) − ḡ (−en(x))] , x ∈ x, and on account of (2.4) and (2.6) we can rewrite ‖g(x) − ḡ(x)‖6 |un + vn|‖g (en(x)) − ḡ (en(x))‖ 6 2|un + vn| ∞∑ i=0 [ |ui|δ ( ei+n(x) ) + |vi|δ ( −ei+n(x) )] = 2 ∞∑ i=0 [(|ui (un + vn)|) δ ( ei+n(x) ) + |vi (un + vn)|δ ( −ei+n(x) ) ] = 2 ∞∑ i=0 [ |ui+n|δ ( ei+n(x) ) + |vi+n|δ ( −ei+n(x) )] = 2 ∞∑ j=n [∣∣uj∣∣δ(ej(x)) + ∣∣vj∣∣δ(−ej(x))] for every x ∈ x and n ∈n, where it states that g = ḡ as n →∞. this proves the theorem. � the purpose of stating and proving this results is of particular interest and give out a solutionof a simple variable functional equation (2.1) at least. in section 3, we will extend the results oftheorem 2.1 form [8] to a more general setting. in particular, the related parameters u, v can beextended to complex numbers.according to the above analysis, we give out a corollary of theorem 2.1 (still quite general).first of all, we must state that the absolute of an element x ∈ x can be given out in the real fieldconsidering that the function h is even for the meaningful of the results, for example h(x) = a|x|.as a matter of fact, we can also present the absolute of x = (x1,x2, · · · ,xn) ∈ rn by |x| = (|x1|, |x2|, · · · , |xn|). thus, it worth stating the results. in particular, we can present the followingresults in the euclidean space if the more general setting can not be judged. corollary 2.1 assume that (x, +) is a real or complex normed linear space and set (y,‖ · ‖) isa banach space. suppose further that the mapping f : x → y fulfils the inequality∥∥∥∥f (x) − a + 12a2 f (a|x|) + a− 12a2 f (−a|x|) ∥∥∥∥ 6 δ(x), x ∈ x, (2.7) where a ∈ r with a > 1 and mappings e : x → x, δ : x → [0,∞) make that e is an even function ( i.e., e(−x) = e(x) for every x ∈ x). the infinite progression ∑∞i=0 1ai δ(ai|x|) is convergence for https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 6 every x ∈ x. then there has a unique mapping g : x → y fulfilling the following equations for all x ∈ x g(x) = a + 1 2a2 g(a|x|) − a− 1 2a2 g(−a|x|), and ‖f (x) −g(x)‖6 ∆(x) + λ(x), where ∆(x) := 1 2 ∑∞ i=0 1 ai [ δ ( ai|x| ) + δ ( −ai|x| )] , λ(x) := 1 2 ∑∞ i=0 1 a2i [ δ ( ai|x| ) −δ ( −ai|x| )] , x ∈ x. furthermore, g can be obtained in the following limiting equality g(x) := lim n→∞ ( an + 1 2a2n f (an|x|) − an − 1 2a2n f (−an|x|) ) , x ∈ x. proof. by using the results of theorem 2.1, u := 1+a 2a2 , v := 1−a 2a2 and together e(x) := a|x|, for all x ∈ x, a computation is to prove that un := 1 + a 2a2n , vn := 1 −a 2a2n , n ∈n. for the convergent series ∑∞i=0 1ai δ(ai|x|) with x ∈ x, therefore ∑∞i=0 1a2i δ(ai|x|) is convergence.applying theorem 2.1, there has a unique limiting function g : x → y fulfilling ‖f (x) −g(x)‖6 ∞∑ i=0 [∣∣∣∣1 + ai2a2i ∣∣∣∣δ(ai|x|) + ∣∣∣∣1 −ai2a2i ∣∣∣∣δ(−ai|x|)] = ∞∑ i=0 1 a2i [ δ ( ai|x| ) −δ ( −ai|x| )] + ∞∑ i=0 1 ai [ δ ( ai|x| ) + δ ( −ai|x| )] = λ(x) + ∆(x). function g has been dated back to derived from (2.5). we complete the proof. � remark 2.2 if u = 1, v = 0 and e(x) = |x|, the above results may be trivial and meaningless.in the above results, suppose that a ∈ (−∞,∞) which is not equal to −1, 0, 1. exchanging a with −a, this transformation may not be different from primary inequality (2.7). this is a basic factleaving to the reader to check it. assume that the convergent series ∑∞i=0 1|a|i δ(|a|i|x|) establishesfor every x ∈ x. in fact, the assertions with |a| exchanging for a has also been achieved by asimilar way. corollary 2.2 assume that (x, +) is a group and set (y,‖ · ‖) is a banach space. supposefurther that the mapping f : x → y fulfils the inequality∥∥∥∥f (x) − a + 12a2 f (a|x|) + a− 12a2 f (−a|x|) ∥∥∥∥ 6 δ, x ∈ x, https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 7 where a ∈ (−∞,∞) with |a| > 1 and δ > 0 is constant. then there is a unique limiting evenfunction g : x → y fulfilling ‖f (x) −g(x)‖6 |a|δ |a|− 1 . proof. since the function δ is a positive constant, thus ∆(x) = |a||a|−1δ and λ(x) = 0 for every x ∈ x. the mapping g has been stated in the following shape: g(x) := lim n→∞ ( |a|n + 1 2a2n f (|a|n|x|) − |a|n − 1 2a2n f (−|a|n|x|) ) , x ∈ x. this proves the proof. � remark 2.3 the above corollaries 2.1 and 2.2 will still establish in β-homogeneous f-spacewith a ∈ (−∞,∞) and |a| > 1. if we exchange a for 1 a in the equation f (x) − a + 1 2a2 f (a|x|) + a− 1 2a2 f (−a|x|) from (2.7), the second group of results will also be obtained with a is a positive constant stated inthe following results. corollary 2.3 assume that (x, +) is a group divisible by a with a ∈ (−∞,∞) and |a| > 1 andset (y,‖·‖) is a banach space. suppose further that the mapping f : x → y fulfils the inequality∥∥∥∥f (x) − a2 + a2 f ( 1 a |x| ) − a2 −a 2 f ( − 1 a |x| )∥∥∥∥ 6 δ(x), x ∈ x, with δ : x → [0,∞) is such that the convergent series ∑∞i=0a2iδ( 1ai |x|) holds for every x ∈ x.then there has a unique even limiting mapping g : x → y fulfilling for every x ∈ x, g(x) = a2 + a 2 g ( 1 a |x| ) + a2 −a 2 g ( − 1 a |x| ) , and ‖f (x) −g(x)‖6 ∆̃(x) + λ̃(x). furthermore, the mapping g can be stated in the following shape: g(x) := lim n→∞ [ a2n + an 2 f ( 1 an |x| ) + a2n −an 2 f ( − 1 an |x| )] , x ∈ x. proof. applying for theorem 2.1 for u := a2+a 2 , v := a2−a 2 and e(x) := 1 a |x|, for all x ∈ x, an easycomputation is to show that un := a2n + a2n−1 2 , vn := a2n −a2n−1 2 , n ∈n. according to the convergent series ∑∞i=0a2iδ( 1ai |x|) for all x ∈ x, hence the series∑∞ i=0a 2i−1δ ( 1 ai |x| ) is convergence, and there has a unique even limiting mapping g : x → y https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 8 fulfilling ‖f (x) −g(x)‖6 ∞∑ i=0 [∣∣∣∣a2i + ai2 ∣∣∣∣δ( 1ai |x| ) + ∣∣∣∣a2i −ai2 ∣∣∣∣δ(− 1ai |x| )] = ∞∑ i=0 a2i 2 [ δ ( 1 ai |x| ) + δ ( − 1 ai |x| )] + ∞∑ i=0 ai 2 [ δ ( 1 ai |x| ) −δ ( − 1 ai |x| )] = ∆̃(x) + λ̃(x). the definition of g is derived from (2.5). we complete the proof. � remark 2.4 corollary 2.3 can be used to investigate the function from which it could be splitinto even and odd parts. there is a good point of the approach achieved here where the functionssplit into two two parts of odd and even functions can give more concise approximations than thebefore approximations in theorem 2.1. the above results is the counterpart of the correspondingresults of sikorska’s paper. however, it is not copied word by word. it is the counterpart of evenfunction. 3. the stability of functional equations in f-space an f-space is called β-homogeneous if it satisfies ‖tx‖ = |t|β‖x‖ for every x ∈ x, t ∈ c. inthis section of the first two theorems, β1, β2 are to be 0 < β1 ≤ 1 and 0 < β2 ≤ 1. furthermore,we suppose x is β1-homogeneous f-space and y is β2-homogeneous f-space. before applyingtheorem 2.1 we would like to make an answer that all roads lead to rome. therefore anotherapproach to prove the following functional inequality has been stated in the following. in fact,there is also a similar solution about functional equation being stated in [8]. theorem 3.1 assume the mapping f : x → y fulfilling for some k ≥ 0 and r < β2 β1 ‖f (x + y + z) + f (x) + f (y) + f (z) − f (x + y) − f (z + y) − f (x + z)‖ 6 k (‖x‖r + ‖y‖r + ‖z‖r ) (3.1) for x,y,z ∈ x. then there has a unique limiting mapping ψ1 : x → y such that ‖f (x) −ψ1(x)‖6 (2 + 2rβ1 + 3 · 2β2)k (2β1r − 22β2)(2β1r − 2β2) ‖x‖r for x ∈ x. moreover, ψ1 satisfying the above inequality is also satisfying the following equation ψ1(x + y + z) + ψ1(x) + ψ1(z) + ψ1(y) = ψ1(x + y) + ψ1(z + y) + ψ1(x + z) (3.2) for all x,y,z ∈ x. proof. from (x,x,x) in place of (x,y,z) in (3.1) we have ‖f (3x) + 3f (x) − 3f (2x)‖6 3k (‖x‖r ) . https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 9 hence ‖2f (3x) + 6f (x) − 6f (2x)‖6 3 · 2β2k (‖x‖r ) . substitute (x,x, 2x) in place of (x,y,z) in (3.1), yielding that ‖f (4x) + 2f (x) − 2f (3x)‖6 (2 + 2rβ1)k (‖x‖r ) . and combining the above two inequalities, we get ‖f (4x) + 8f (x) − 6f (2x)‖6 (2 + 2rβ1 + 3 · 2β2)k (‖x‖r ) . (3.3) let us define g(x) = f (2x) − 4f (x) for all x ∈ x. hence ‖g(2x)/2 −g(x)‖6 (2 + 2rβ1 + 3 · 2β2)k (‖x‖r ) /2β2 (3.4) for all x ∈ x. therefore ‖g(2nx)/2n −g(2mx)/2m‖6 n−1∑ j=m (2 + 2rβ1 + 3 · 2β2)k 2jβ1r 2β22jβ2 (‖x‖r ) (3.5) for m, n ∈ n with n > m and all x ∈ x. since the sequence {g(2nx)/2n} is a cauchy sequencein y for all x ∈ x and y is complete, the mapping can be well defined as: φ(x) = lim n→∞ g(2nx)/2n for all x ∈ x. in particular, letting m = 0 and setting n →∞ in (3.4), we have ‖φ(x) −g(x)‖6 (2+2 rβ1+3·2β2)k 2β1r−2β2 (‖x‖ r ) . (3.6) now, we prove the mapping φ is additive and is unique. from (x,y,y + x) in (3.1) yields that ‖f (2x + 2y) + f (x) + f (y) − f (x + 2y) − f (2x + y)‖6 k (‖x‖r + ‖y‖r + ‖x + y‖r ) . from (x,x,y) in equation (3.1) yields that ‖f (2x + y) + 2f (x) + f (y) − f (2x) − 2f (x + y)‖6 k (2‖x‖r + ‖y‖r ) . from (x,y,y) in (3.1) we have ‖f (x + 2y) + f (x) + 2f (y) − f (2y) − 2f (x + y)‖6 k (‖x‖r + 2‖y‖r ) . https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 10 combining the above three inequalities, we have that for x,y,z ∈ x ‖φ(x + y) −φ(x) −φ(y)‖ = lim n→∞ 1 2β2n ‖f (2n+1x + 2n+1y) + 4f (2nx) + 4f (2ny) − f (2n+1x) − f (2n+1y) − 4f (2nx + 2ny)‖ 6 lim n→∞ 1 2β2n ∥∥f (2n+1x + 2n+1y) + f (2nx) + f (2ny) − f (2n+1x + 2ny) − f (2n+1y + 2nx)∥∥ + lim n→∞ 1 2β2n ∥∥f (2n+1x + 2ny) + 2f (2nx) + f (2ny) − f (2n+1x) − 2f (2nx + 2ny)∥∥ + lim n→∞ 1 2β2n ∥∥f (2nx + 2n+1y) + f (2nx) + 2f (2ny) − f (2n+1y) − 2f (2nx + 2ny)∥∥ 6 lim n→∞ 2β1rn 2β2n k (4‖x‖r + 4‖y‖r + ‖x + y‖r ) . so we have φ(x + y) = φ(x) + φ(y) for all x,y ∈ x.next, the uniqueness of the mapping φ will be proved. let u(x) be another additive mappingsuch that for some k2 ≥ 0 and r < β2β1 , ‖g(x) −u(x)‖6 k2‖x‖r2. hence ‖φ(x) −u(x)‖ =‖φ(nx) −u(nx)‖/nβ2 6‖φ(nx) −g(nx)‖/nβ2 + ‖g(nx) −u(nx)‖/nβ2 6 (2 + 2rβ1 + 3 · 2β2)k 2β1r − 2β2 ‖x‖rnrβ1−β2 + k2‖x‖r2nr2β1−β2 for all x ∈ x. therefore φ(x) = u(x) for all x ∈ x. by the condition r < β2 β1 . so there has a uniqueadditive limiting mapping φ fulfilling ‖(f (x) − 1 2 φ(x)) − (f (2x) − 1 2 φ(2x))/4‖≤ (2 + 2rβ1 + 3 · 2β2)k 2β1r − 2β2 ‖x‖r/22β2. hence ‖(f (x) − 1 2 φ(x)) − (f (2nx) − 1 2 φ(2nx))/4n‖≤ n−1∑ j=0 2β1rj 22β2j (2 + 2rβ1 + 3 · 2β2)k 22β2(2β1r − 2β2) ‖x‖r. then the mapping can be well defined as ψ(x) = lim n→∞ (f (2nx) − 1 2 φ(2nx))/4n for all x ∈ x, by the completeness of the space y . thus ‖ψ(x) − f (x) + φ(x)/2‖≤ (2 + 2rβ1 + 3 · 2β2)k (2β1r − 2β2)(2β1r − 22β2) ‖x‖r. let u(x) be another limiting mapping which has the same property to the function ψ(x) such that, ‖u(x) −φ(x)‖6 ‖u(x) − (f (2nx) − 1 2 φ(2nx))/4n‖ + ‖(f (2nx) − 1 2 φ(2nx))/4n −φ(x)‖ 6 2 ∞∑ j=n 2β1rj 22β2j (2 + 2rβ1 + 3 · 2β2)k 22β2(2β1r − 2β2) ‖x‖r https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 11 which shows that the approximation function φ(x) is unique. finally, it remains to prove that φ(x)satisfies (3.2) and we obtain 1 4n ‖f (2nx + 2ny + 2nz) + f (2nx) + f (2ny) + f (2nz)− f (2nx + 2ny) − f (2nz + 2ny) − f (2nx + 2nz)‖ 6 2 β1n 4n k (‖x‖r + ‖y‖r + ‖z‖r ) . (3.7) letting n →∞, and we get our assertion by using the additivity of φ(x). � in another direction, we will describe the similar stability results of the above theorem 3.1. theorem 3.2 let r > β2 β1 and assume that f : x → y is a mapping satisfying the equation (3.1).then there has a unique limiting mapping ψ1 : x → y satisfying ‖f (x) −ψ1(x)‖6 (2 + 2rβ1 + 3 · 2β2)k (2β1r − 22β2)(2β1r − 2β2) ‖x‖r for all x ∈ x. moreover, ψ1 solves also the following equation ψ1(x + y + z) + ψ1(x) + ψ1(z) + ψ1(y) = ψ1(x + y) + ψ1(z + y) + ψ1(x + z) (3.8) for all x,y,z ∈ x. proof. according to the equation (3.3), we obtain ‖g(x) − 2g( x 2 )‖6 (2 + 2rβ1 + 3 · 2β2)k (‖x‖r ) /2β1r. therefore ‖2ng( x 2n ) − 2mg( x 2m )‖6 n−1∑ j=m (2 + 2rβ1 + 3 · 2β2)k 2jβ2 2jβ1r 2β1r (‖x‖r ) for m,n ∈ n with n > m and x ∈ x. since the sequence {2ng( x 2n )} is a cauchy sequence in yfor all x ∈ x and y is complete, the mapping can be well defined as: φ(x) = lim n→∞ 2ng( x 2n ) for all x ∈ x. using a similar manner, we can complete the rest part. � if f (x) is odd, then (x,y,−x −y) in (3.2) can give a precise condition to ascertain the additiveproperty of the function f (x) (see [9]). obviously, the additive property is stronger than theproperty of the equation (3.2), but vice versa is not true. in contrast with the subadditive property,we can not get obvious strong or weak property temporarily. by using another approach to solvethe theorem 3.1, according to (3.6), we have ‖f (2nx)/22n − f (x) − n−1∑ j=0 φ(2jx)/22(j+1)‖≤ n−1∑ j=0 2β1rj 22β2j (2 + 2rβ1 + 3 · 2β2)k 22β2(2β1r − 2β2) ‖x‖r. then the mapping can be well defined as ψ(x) = lim n→∞ f (2nx)/22n https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 12 for all x ∈ x, by the completeness of the space y . thus ‖ψ(x) − f (x) −φ(x)/2‖≤ (2 + 2rβ1 + 3 · 2β2)k (2β1r − 2β2)(2β1r − 22β2) ‖x‖r. in a similar way, we can use two steps to prove that the mapping ψ(x) is unique. the first stepwe show that the mapping satisfies the property: ψ(kx) = k2ψ(x) for all k ∈ n, x ∈ x. we provethis by mathematical induction, for a fixed element x ∈ x. we will prove that the property is truefor k = 2. from (x,−x,x) in equation (3.1), we can get that ‖3f (x) + f (−x) − f (2x)‖6 3k (‖x‖r ) for all x ∈ x.thus ‖f (−x) − f (x) −φ(x)‖≤‖f (2x) − 4f (x) −φ(x)‖ + ‖3f (x) + f (−x) − f (2x)‖ 6 ( (2 + 2rβ1 + 3 · 2β2)k 2β1r − 2β2 + 3k) (‖x‖r ) for all x ∈ x. using the similar above argumentation together the above inequality and equation (3.1), yields ψ(−x) = ψ(x) + lim n→∞ φ(x) 2nand ψ(x + y + z) + ψ(x) + ψ(z) + ψ(y) = ψ(x + y) + ψ(z + y) + ψ(x + z) (3.9) for all x,y,z ∈ x. from (x,−x,x) in equation (3.7), we achieve ψ(2x) = 3ψ(x) + ψ(−x) = 4ψ(x). fixed x ∈ x, we prove this by induction. we have already proved that the property is true for n = 2. supposing that ψ(nx) = n2ψ(x) for all natural n ≤ 2k, with k ≥ 1, let us calculate ψ((2k + 1)x). from (kx,kx,x) in (3.7), we know ψ((2k + 1)x) = ψ(2kx) + 2ψ((k + 1)x) − 2ψ(kx) −ψ(x) = (4k2 + 2(k + 1)2 − 2k2 − 1)ψ(x) = (2k + 1)2ψ(x). now, we show the mapping ψ satisfies the property ψ(kx) = k2ψ(x) for all k ∈ n, x ∈ x. thesecond step, we claim that the mapping φ is unique. let u(x) be another limiting mapping suchthat for some k2 ≥ 0 and r < β2β1 , ‖u(x) − f (x) −φ(x)/2‖6 k2‖x‖r2 https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 13 which satisfies the property u(kx) = k2u(x) for all k ∈ n and x ∈ x. therefore ‖ψ(x) −u(x)‖ =‖ψ1(kx) −u(kx)‖/k2β2 6‖u(xk) − f (kx) −φ(kx)/2‖/k2β2 + |ψ(xk) − f (kx) −φ(kx)/2‖/k2β2 6 (2 + 2rβ1 + 3 · 2β2)k (2β1r − 22β2)(2β1r − 2β2) ‖x‖rkrβ1−2β2 + |k2‖x‖r2kr2β1−2β2. hence φ(x) = u(x) for all x ∈ x. this shows that ψ is unique. let ψ1(x) = ψ(x) −φ(x)/2. thiscompletes the uniqueness of ψ1(x). we have ‖ψ1(x) − f (x)‖6 (2 + 2rβ1 + 3 · 2β2)k (2β1r − 22β2)(2β1r − 2β2) ‖x‖r for all x ∈ x and also the equation (3.2) holds by using the additive property of φ and equation (3.7). we complete the proof. we may also assume that limn→∞ φ(x)2n = limn→∞ g(2nx)/2n2n = 0.otherwise, this limit may not be convergence to zero. conversely, we may add some similar smalladditional assumptions to guarantee the convergence in theorem 3.2. 4. the stability of functional equations in banach space in this section, we will prove the counterpart of the results of theorem 2.1 from [8] to moregeneral case. we generalize the results of sikorska in 2010. in particular, the related parameters u, v can be extended to complex numbers by using a more efficient approach. beyond that, westate that the first results in section 2 are presented and combined the first results in [8]. ourcontribution to the parameters u,v are complex numbers. the results is stated in this section inmore detail. theorem 4.1 suppose that (x, +) is a group, and (y,‖ · ‖) is a banach space, and let themapping f : x → y satisfy the inequality ‖f (x) −uf (e(x)) −vf (−e(x))‖6 δ(x), x ∈ x, where u,v ∈ c (c denotes the complex field.), and e : x → x, δ : x → [0,∞) are arbitrary givenfunctions.(1): if e is a even function ( i.e., e(−x) = e(x) for x ∈ x) and the convergent series ∞∑ n=0 [|un|δ (en(x)) + |vn|δ (−en(x))] with u0 := 1, un := [ u(u + v)n−1 ] , n ∈n, v0 := 0, vn := [ v(u + v)n−1 ] , n ∈n https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 14 (and where en states the n-th composition of the function e ), establishes for every x ∈ x. thenthere has a unique even limiting function g : x → y fulfilling g(x) = ung(e n(x)) + vng(−en(x)), x ∈ x and n ∈ n, (4.1) and ‖f (x) −g(x)‖6 ∞∑ i=0 [ |ui|δ ( ei (x) ) + |vi|δ ( −ei (x) )] , x ∈ x. (4.2) (2): if e is odd ( i·e., e(−x) = −e(x) for all x ∈ x) and the convergent series ∞∑ n=0 [|un|δ (en(x)) + |vn|δ (−en(x))] . with u0 := 1, un := 1 2 [(u + v)n + (u −v)n] , n ∈n, v0 := 0, vn := 1 2 [(u + v)n − (u −v)n] , n ∈nestablishes for all x ∈ x. then there has a unique limiting mapping g : x → y fulfilling (3.10)and (3.11). proof. we only need to prove the uniqueness of the approximation function. (1): let us supposethat g̃ : x → y is another approximating mapping. so let’s first prove the inequality together withthe equation (2.5) and g(−x) = g(x) ‖f (em(x)) −um(unf ( en+m(x) ) + vnf ( −em+n(x) ) ) −vm(unf ( en+m(x) ) + vnf ( −em+n(x) ) )‖ =‖f (em(x)) −un+mf ( en+m(x) ) −vn+mf ( −en+m(x) ) ‖ 6 n+m−1∑ j=m [∣∣uj∣∣δ(ej(x)) + ∣∣vj∣∣δ(−ej(x))] , and letting n →∞ we have for any m ∈n ‖f (em(x)) −umg (em(x)) −vmg (−em(x))‖6 ∞∑ j=m [∣∣uj∣∣δ(ej(x)) + ∣∣vj∣∣δ(−ej(x))] , and we can rewrite ‖g(x) − g̃(x)‖6‖f (km(x)) −umg (em(x)) −vmg (−em(x))‖ + ‖f (km(x)) −umg̃ (em(x)) −vmg̃ (em(x))‖ 62 ∞∑ j=m [∣∣uj∣∣δ(ej(x)) + ∣∣vj∣∣δ(−ej(x))] for any x ∈ x and m ∈n, which yields g = g̃ in x as m →∞.(2): combined with the results of theorem 2.1 from [8] where e is odd, we only need to prove theuniqueness of the approximation function. let us suppose that g̃ : x → y is another approximating https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 15 mapping. so let’s first prove the inequality together with the equation the results in theorem 2.1in [8] ‖f (em(x)) −um(unf ( en+m(x) ) + vnf ( −em+n(x) ) ) −vm(unf ( −en+m(x) ) + vnf ( em+n(x) ) )‖ =‖f (em(x)) −un+mf ( en+m(x) ) −vn+mf ( −en+m(x) ) ‖ 6 n+m−1∑ j=m [∣∣uj∣∣δ(ej(x)) + ∣∣vj∣∣δ(−ej(x))] , and letting n →∞ we have for any m ∈n ‖f (em(x)) −umg (em(x)) −vmg (−em(x))‖6 ∞∑ j=m [∣∣uj∣∣δ(ej(x)) + ∣∣vj∣∣δ(−ej(x))] , and we can rewrite ‖g(x) − g̃(x)‖6‖f (km(x)) −umg (em(x)) −vmg (−em(x))‖ + ‖f (km(x)) −umg̃ (em(x)) −vmg̃ (em(x))‖ 62 ∞∑ j=m [∣∣uj∣∣δ(ej(x)) + ∣∣vj∣∣δ(−ej(x))] for any x ∈ x and m ∈n, which yields g = g̃ in x as m →∞. this completes the proof. � for the euler-lagrange equation, we provide another method to solve it in contrast with [10]. theorem 4.2 suppose that (x, +) is a group, and (y,‖ · ‖) is a banach space and let themapping f : x → y satisfy the inequality for all x,y,z ∈ x and some ε > 0 ‖f (x + y + z) + f (x −y + z) + f (x + y −z) + f (x −y −z) − 4f (x) − 4f (y) − 4f (z)‖6 ε. (4.3) then there has a unique limiting function g : x → y such that g(x) = 2 9 g(3x) − 1 9 g(−3x), x ∈ x and ‖f (x) −g(x)‖6 3ε 8 x ∈ x. in particular, if x is abelian, then g is a solution of the equation in the following f (x + y + z) + f (x −y + z) + f (x + y −z) + f (x −y −z) = 4f (x) + 4f (y) + 4f (y), (4.4) for all x,y ∈ x. proof. from (x,x,−x) in (4.3), we obtain ‖6f (x) + 3f (−x) − f (3x)‖6 ε, x ∈ x. replacing x by −x in the above inequality we obtain ‖6f (−x) + 3f (x) − f (−3x)‖6 ε, x ∈ x. https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 16 consequently, combining the above two inequalities yield that ‖9f (x) + f (−3x) − 2f (3x)‖6 3ε, x ∈ x. by using the results second part of theorem 4.1, a computation is to prove that un := 3n + 1 2 · 9n , vn := 1 − 3n 2 · 9n , n ∈n. and the convergent series can be described as ∞∑ n=0 [|un|δ (en(x)) + |vn|δ (−en(x))] = 3ε 8 . and we show that if x is commutative, by using (x,y,z) = (3nx, 3ny, 3nz), then ‖un[f (3n(x + y + z)) + f (3n(x −y + z)) + f (3n(x + y −z)) + f (3n(x −y −z)) − 4f (3nx) − 4f (3ny) − 4f (3ny)] + vn[f (3n(x + y + z)) + f (3n(x −y + z)) + f (3n(x + y −z)) + f (3n(x −y −z)) − 4f (3nx) − 4f (3ny) − 4f (3ny)]‖ 6 ε 9nwhich we achieve our result (3.13) by letting n →∞. � theorem 4.3 suppose that x is a group, and (y,‖ · ‖) is a banach space and let the mapping f : x → y satisfy the inequality for all x,y ∈ x and some ε > 0 ‖f (x + y) + f (x −y) − 2f (x) − f (y) − f (−y)‖6 ε, x,y ∈ x. (4.5) then there has a unique limiting function g : x → y fulfilling g(x) = 3 8 g(2x) − 1 8 g(−2x), x ∈ x and ‖f (x) −g(x)‖6 2ε 3 x ∈ x. in particular, if x is commutative, then g also fulfils g(x + y) + g(x −y) = 2g(x) + g(y) + g(−y), x,y ∈ x. proof. substituting in the sequel (x,x) in (4.5), we obtain ‖f (2x) + f (0) − 3f (x) − f (−x)‖6 ε, x ∈ x. (4.6) replacing x by −x in (4.6) we have ‖f (−2x) + f (0) − 3f (−x) − f (x)‖6 ε, x ∈ x. (4.7) consequently, (4.6) and (4.7) yield that ‖8f (x) + f (−2x) − 3f (2x)‖6 4ε, x ∈ x. https://doi.org/10.28924/ada/ma.3.7 eur. j. math. anal. 10.28924/ada/ma.3.7 17 by using the results of theorem 3.3, a computation is to prove that un := 2n + 1 2 · 4n , vn := 1 − 2n 2 · 4n , n ∈n. and the convergent series ∞∑ n=0 [|un|δ (en(x)) + |vn|δ (−en(x))] = 2ε 3 . and we show that if x is commutative, by using (x,y) = (2nx, 2ny), then ‖un[f (2nx + 2ny) + f (2nx − 2ny) − 2f (2nx) − f (2ny) − f (−2ny)] + vn[f (2 nx + 2ny) + f (2nx − 2ny) − 2f (2nx) − f (2ny) − f (−2ny)]‖ 6 ε 4nwhich we achieve our result (∗) by letting n →∞. � if we can not set f (0) = 0, then the approximate constat is 5 6 ε. acknowledgments the authors express their gratitude to the anonymous reviewers and editor for their carefulreading the manuscript and for many valuable remarks and suggestions. conflict of interest the author(s) declare(s) that there is no conflict of interest regarding this manuscript. data availability 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functional equations with a new direct method, mathematics.10 (2022) 1188. https://doi.org/10.3390/math10071188. https://doi.org/10.28924/ada/ma.3.7 https://doi.org/10.4134/bkms.2005.42.1.057 https://doi.org/10.3390/math10071188 1. introduction 2. a simple variable of abstract equation 3. the stability of functional equations in f-space 4. the stability of functional equations in banach space acknowledgments conflict of interest data availability funding statement references ©2021 ada academica https://adac.eeeur. j. math. anal. 1 (2021) 45-67doi: 10.28924/ada/ma.1.45 convergence of a three-step iteration scheme to the common fixed points of mixed-type total asymtotically nonexpansive mappings in uniformly convex banach spaces imo kalu agwu1,∗, donatus ikechi igbokwe1, nathenial c. ukeje2 1department of mathematics, micheal okpara university of agriculture, umudike, umuahia abia state nigeria agwuimo@gmail.com, igbokwedi@yahoo.com 2department of mathematics and statistics, university of port-harcourt, port-harcourt rivers state nigeria ukejechukwuebuanathan@gmail.com ∗correspondence: agwuimo@gmail.com abstract. we propose a three-step iteration scheme of hybrid mixed-type for three total asymptoti-cally nonexpansive self mappings and three total asymptotically nonexpansive nonself mappings. inaddition, we establish some weak convergence theorems of the scheme to the common fixed point ofthe mappings in uniformly convex banach spaces. our results extend and generalize numerous resultscurrently in literature. 1. introduction let k be a nonempty subset of a real banach space e. let t : k −→ k be a nonlinear mapping,we denote the set of all fixed points of t by f (t ). the set of common fixed points of six mappings s1,s2,s3,t1,t2 and t3 will be denoted by f = ∩3i=1(f (ti ) ∩f (si )). definition 1.1. a mapping t : k −→ k is said to asymptotically nonexpansive [6] if there exists a sequence {kn} in [1,∞) with limn→∞kn = 1 such that ‖tn(x) −tn(y)‖≤ kn‖x −y‖,∀x,y ∈n (1.1). ln 1972, the class of asymptotically nonexpansive mapping was introduced by goebel and kirk [6].they proved that if k is a nonempty closed convex subset of a uniformly convex banach space and t is an asymptotically nonexpansive mapping of k, then t has a fixed point. received: 29 aug 2021. key words and phrases. asymtotically nonexpansive mapping; total asymptotically nonexpansive nonself mapping;hybrid mixed type iteration scheme; common fixed point; uniformly convex banach space; weak convergence.45 https://adac.ee https://doi.org/10.28924/ada/ma.1.45 eur. j. math. anal. 1 (2021) 46 definition 1.2. a mapping t is said to be total asymptotically nonexpansive [1] if ‖tn(x) −tn(y)‖≤‖x −y‖ + µnφ(‖x −y‖) + νn,∀x,y ∈ k,∀n ∈n, (1.2) where {µn}and {νn} are nonnegative real sequences such that µn → 0 and νn → 0 as n →∞ and φ is a strictly increasing continuous function φ : [0,∞) → [0,∞) with φ(0) = 0. from the above definitions, we see that the class of total asymptotically nonexpansive mappingsincludes the class of asymptotically nonexpansive mapping as a special case; see [4] for moredetails. each asymptotically nonexpansive mapping is total asymptotically nonexpansive mappingwith νn = 0,µn = kn − 1 f or all n ≥ 1,φ(t) = t,t ≥ 0. definition 1.3. a subset k of a banach space e is said to be a retract of e if there exists a continuous mapping p : e −→ k (called retraction) such that p (x) = x for all x ∈ k. if, in addition p is nonexpansive, then p is said to be nonexpansive retraction of e. if p : e −→ k is a retraction, then p2 = p. a retract of a hausdorff space must be a closed subset. every closed convex subset of a uniformly convex banach space is a retract. in 2012, yolacan and kiziltune [18] defined the following: definition 1.4. let k be a nonempty and closed convex subset of a banach space e. a nonself mapping t : k → e is said to be total asymptotically nonexpansive mapping if there exist sequences k(1)n and k (2) n in [0,∞) with k (1) n → 0 and k (2) n → 0 as n →∞ and a strictly increasing function φ : [0,∞) → [0,∞) with φ(0) = 0 such that ‖t (pt )n−1(x) −t (pt )n−1(y)‖≤‖x −y‖ + k1(n)φ(‖x −y‖) + k (2) n ,∀x,y ∈ k,n ∈n. (1.3) chidume et al. [3] studied the following iterative scheme in 2004: x1 = x ∈ k xn+1 = p (αnt (pt ) n−1xn + (1 −αn)xn),n ≥ 1, (1.4) where {αn} is a sequence in (0, 1), k is a nonempty closed convex subset of of a real uniformlyconvex banach space e, p is a nonexpansive retraction of e onto k, and proved some strong andweak convergence theorems for asymptotically nonexpansive nonself mappings in the intermediatesense in the framework of uniformly convex banach spaces.ln 2006, wang [17] generalised the iteration process (1.4) as follows: x1 = x ∈ k, xn+1 = p ((1 −αn)xn + αnt1(pt1)n−1yn), yn = p ((1 −βn)xn + βnt2(pt2)n−1xn),n ≥ 1, (1.5) eur. j. math. anal. 1 (2021) 47 where t1,t2 : k −→ e are two asymptotically nonexpansive nonself mappings, {αn} and {βn} arereal sequences in [0, 1), and proved some weak and strong convergence theorems for asymptoticallynonexpansive nonself mappings.ln 2012, guo et al [8] generalised the iteration process (1.5) as follows: x1 = x ∈ k, xn+1 = p ((1 −αn)sn1xn + αnt1(pt1) n−1yn), yn = p ((1 −βn)sn2xn + βnt2(pt2) n−1xn),n ≥ 1, (1.6) where t1,t2 : k −→ e are two asymptotically nonexpansive nonself mappings, s1,s2 : k −→ eare two asymptotically nonexpansive self mappings and {αn},{βn} are real sequences in [0, 1),and proved some strong and weak convergence theorems for mixed-type asymptotically nonexpan-sive mappings. hybrid mixed-type iteration schemelet e be a real uniformly convex banach space, k a nonempty closed convex subset of e and p : e −→ k a nonexpansive retraction of e onto k. let s1,s2,s3 : k −→ k be three totalasymptotically nonexpansive self mappings and t1,t2,t3 : k −→ e be three total asymptoti-cally nonexpansive nonself mappings. then, the hybrid iteration scheme for the above mentionedmappings is as follows: x1 = x ∈ k; xn+1 = p ((1 −αn)sn1xn + αnt1(pt1) n−1yn); yn = p ((1 −βn)sn2xn + βnt2(pt2) n−1zn); zn = p ((1 −γn)sn3xn + γnt3(pt3) n−1xn), (1.7) where {αn},{βn}, and {γn} are real sequences in [0, 1).the aim of this paper is to study this new hybrid mixed-type iteration scheme (1.7), prove demi-closedness principle for total asymptotically nonexpansive nonself map and establish some conver-gence theorems for mixed-type mappings in the setting of uniformly convex banach spaces. 2. preliminary for the sake of convenience, we restate the following concepts and results:let e be a banach space with its dimension greater than or equal to 2. the modulus of convexityof e is a function δe(ε) : (0, 2] −→ (0, 2] defined by δe(ε) = inf{1 −‖ 1 2 (x + y)‖ : ‖x‖ = 1,‖y‖ = 1,ε = ‖x −y‖}. eur. j. math. anal. 1 (2021) 48 a banach space e is uniformly convex if and if δe(ε) > 0, for all ε ∈ (0, 2].we recall the following: definition 2.1. (see [19]: let % = {x ∈ e : ‖x‖ = 1} and let e? be the dual of e. the space e has gateaux differentiable norm if limn→∞ ‖x+ty‖−‖x‖ t exists ∀x,y ∈ %. definition 2.2. (see [19]: the space e has frechet differentiable norm [15] if for each x ∈ %, the limit of the norm above exists and is attained uniformly for all y ∈ %, and in this case, it is also well known that 〈h,j(x)〉 + 1 2 ‖x‖2 ≤ 1 2 ‖x + h‖2 ≤〈h,j(x)〉 + 1 2 ‖x‖2 + b(‖x‖), (2.1) ∀x,y ∈ e, where j is the frechet derivative of the functional 1 2 ‖ · |2 at x ∈ e,〈·〉 is the pairing between e and e? and b is an increasing function defined on [0,∞) such that limt→∞ b(t) t = 0. definition 2.3. : the space e has opial condition [10] if for any sequence {xn} in e, xn converges to x weakly, then it follows that lim supn→∞‖xn − x‖ < lim supn→∞‖xn − y‖ for all y ∈ e with x 6= y . examples of banach spaces satisfying opial conditions are hilbert spaces and all spaces lp(1 < p < ∞). on the other hand, lp[0,π] with 1 < p 6= 2 fails to satify opial condition. definition 2.4. : a mapping t : k −→ k is said to be demiclosed at 0, if for any sequence {xn} in k, the condition that xn converges weakly to x ∈ k and txn converges strongly to 0 implies tx = 0. definition 2.5. : a banach space has the kadec-klec property [14] if for every sequence xn in e,xn → x weakly and ‖xn‖→‖x‖, then it follows that ‖xn −x‖→ 0. next, we state the following useful lemmas which will be needed in order to prove our mainresults. lemma 2.1. (see [16]): let {αn}∞n=1,{βn} ∞ n=1 and {γn} ∞ n=1 be sequences of nonnegative numbers satisfying the inequality: αn+1 ≤ (1 + βn)αn + γn,∀n ≥ 1. (2.2) if ∑∞ n=1βn < ∞ and ∑∞ n=1γn < ∞, then(1) limn→∞αn exists(2) ln particular, if {αn}∞n=1 has a subsequence which converges strongly to 0, then limn→∞αn = 0. eur. j. math. anal. 1 (2021) 49 lemma 2.2. (see [14]): let e be a uniformly convex banach space and 0 < p ≤ tn ≤ q < 1 for each n ≥ 1. suppose that {xn} and {yn} are sequences in e such that lim sup n→∞ ‖xn‖≤ r, lim sup n→∞ ‖yn‖≤ r and lim n→∞ ‖tnxn + (1 − tn)yn‖ = r, (2.3) hold for some r ≥ 0. then limn→∞‖xn −yn‖ = 0. lemma 2.3. (see [14]): let e be a real reflexive banach space such that its dual e? has the kadec-klec property. let {xn} be a bounded sequence in e and p,q ∈ ωω(xn) ( where ωω(xn) denotes the set of all weak subsequential limits of {xn}). suppose limn→∞‖txn + (1 − t)p −q‖ exists for all t ∈ [0, 1]. then, p = q. lemma 2.4. (see [14]): let k be a nonempty convex subset of a uniformly convex banach space e. then, there exists a strictly incraesing continous convex function φ : [0,∞) → [0,∞) with φ(0) = 0 such that for each lipshitizian mapping t : c −→ c with the lipschiz constant l, ‖ttx − (1 − t)ty −t (tx − (1 − t)y)‖≤ lφ−1(‖x −y‖− 1 l ‖tx −ty‖) (2.4) for all x,y ∈ k and for all t ∈ [0.1]. lemma 2.5. (see [2]) let e be a uniformly convex banach space, k a nonempty bounded close convex subset of e. then, there exists a strictly increasing continous convex function φ : [0,∞) −→ [0,∞) with φ(0) = 0 such that for any lipschitizian mapping t : k −→ e with lipschitz constant l ≥ 1 and elements {xn}nj=i in k and any nonnegative numbers {tj} n j=1 with ∑n j=1 tj = 1, the following inequality holds: ‖t ( n∑ j=1 tjxj) − n∑ j=1 tjtxj‖≤ lφ−1{max1≤j,k≤n(‖xj −xk‖−l−1‖txj −txk‖)} lemma 2.6. (see [21]) if the sequence {xn}∞n=1 converges weakly to x, then there exists a sequence of convex combination yj = ∑n(j) k=1 λ (j) k xk+j , λ (j) k ≥ 0 and ∑n(j) k=1 λ(j) = 1, such that ‖yj − x‖ → 0. as n →∞. 3. main results lemma 3.1. ( demiclosedness principle f or nonself total asymptotically nonexpansive maps ) let k be a nonempty closed convex and bounded subset of a uniformly convex banach space e and t : k −→ e be l-lipschitz continuous and total asymptotically nonexpansive mapping with the function φ : [0,∞) −→ [0,∞) (such that φ(0) = 0) and nonnegative sequences {k(1)n },{k (2) n } such that k(1)n ,k (2) n → 0 as n →∞. then, i −t is demiclosed at 0. proof. let {xn} converge weakly to ω ∈ k and {xn −txn} converge strongly to 0. we prove that (i −t )ω = 0. clearly, {xn} is bounded. so, there exists ρ > 0 such that {xn}⊂ c = k ∩bρ(0),where bρ(0) is a closed ball in e with centre 0 and radius ρ. thus, c is nonempty, closed , eur. j. math. anal. 1 (2021) 50 bounded and convex subset in k.claim: t (pt )n−1ω → ω as n → ∞. in fact, since {xn} converges weakly to ω, by lemma6(see [21]), we have for all n > 1, there exists a convex combination yn = m(n)∑ i=1 t (n) i xi+n, t (n) i ≥ 0 and m(n)∑ i=1 t (n) i = 1 such that ‖yn −ω‖→ 0 as n →∞. (3.1) also, since {xn−txn} converges to 0, then for any � > 0 and a positive integer m ≥ 1, there exists n1 = n(�) > 0 such that ‖(i −t )xn‖ < � 1 + m ,∀n ≥ n1. (3.2) hence, ∀n ≥ n1, using definition 1.4 and the fact that p is nonexpansive , we have the followingestimates:for arbitrary but fixed j ≥ 1, we have ‖xn −t (pt )(j−1)xn‖ ≤ ‖(i −t )xn‖ + ‖(t −t (pt ))xn‖ +‖(t (pt ) −t (pt )2)xn‖ +‖(t (pt )2 −t (pt )3)xn‖ + · · · + ‖(t (pt )j−2 −t (pt )j−1))xn‖ ≤ ‖(i −t )xn‖ + (‖(i −t )xn‖ + µ (1) n φ(‖(i −t )xn‖) +ξ (1) n ) + (‖(i −t )xn‖ + µ (2) n φ(‖(i −t )xn‖) + ξ (2) n ) +(‖(i −t )xn‖ + µ (3) n φ(‖(i −t )xn‖) + ξ (3) n ) + · · · + (‖(i −t )xn‖ + µ (j−1) n φ(‖(i −t )xn‖) + ξ (j−1) n ) = ‖(i −t )xn‖ + m−1∑ j=1 ‖(i −t )xn‖ + m−1∑ j=1 µ (j) n φ(‖(i −t )xn‖) + m−1∑ j=1 ξ (j) n ≤ m‖xn −txn‖ + mµnφ(‖(i −t )xn‖) + mξn, (3.3) where µn = max1≤j≤m−1{µ(j)n } and ξn = max1≤j≤m−1{ξ(j)n }.from (3.2) and (3.3), we get ‖xn −t (pt )j−1xn‖ < �. (3.4) now, since t : k −→ e is l-lipschitizian and total asymptotically nonexpansive , so is t : c −→ e. therefore, ∀j ≥ 1,t (pt )j−1 : c −→ e is lipschitizian mapping with the lipschitz constant µj ≥ 1. eur. j. math. anal. 1 (2021) 51 in addition, ‖t (pt )j−1yn −yn‖ = ‖t (pt )j−1yn − m(n)∑ i=1 t (n) i t (pt )j−1xi+n + m(n)∑ i=1 t (n) i t (pt )j−1xi+n − m(n)∑ i=1 t (n) i xi+n‖ ≤ ‖t (pt )j−1yn − m(n)∑ i=1 t (n) i t (pt )j−1xi+n‖ + m(n)∑ i=1 t (n) i ‖t (pt )j−1xi+n −xi+n‖. (3.5) using (3.4), we get m(n)∑ i=1 t (n) i ‖t (pt )j−1xi+n −xi+n‖ < �,∀n ≥n. (3.6) furthermore, by lemma 2.5, there exists a strictly increasing continous function φ : [0,∞) −→ [0,∞) with φ(0) = 0 such that for all n ≥n, we have ‖t (pt )j−1yn − m(n)∑ i=1 t (n) i t (pt ) j−1xi+n‖ = ‖t (pt )j−1( m(n)∑ i=1 t (n) i xi+n) − m(n)∑ i=1 t (n) i t (pt ) j−1xi+n‖ ≤ µjφ−1{max1≤j,k≤n(‖xi+n −xi+k‖ −µ−1j ‖t (pt ) j−1xi+n −t (pt )j−1xk+n‖)} = µjφ −1{max1≤j,k≤n(‖xi+n −t (pt )j−1xi+n +t (pt )j−1xi+n −t (pt )j−1xk+n +t (pt )j−1xk+n −xi+k‖ −µ−1j ‖t (pt ) j−1xi+n −t (pt )j−1xk+n‖)} ≤ µjφ−1{max1≤j,k≤n(‖xi+n −t (pt )j−1xi+n‖ +‖t (pt )j−1xi+n −t (pt )j−1xk+n‖ +‖t (pt )j−1xk+n −xi+k‖ −µ−1j ‖t (pt ) j−1xi+n −t (pt )j−1xk+n‖)} ≤ µjφ−1{max1≤j,k≤n(� + � + (1 −µ−1j ) ×‖t (pt )j−1xi+n −t (pt )j−1xk+n‖)} ≤ µjφ−1{max1≤j,k≤n(� + � + (1 −µ−1j )µj ×‖xi+n −xk+n‖} ≤ µjφ−1{max1≤j,k≤n(� + � + (1 −µ−1j )µj ×(‖xi+n‖ + ‖xk+n‖}. eur. j. math. anal. 1 (2021) 52 thus, ‖t (pt )j−1yn − m(n)∑ i=1 t (n) i t (pt )j−1xi+n‖≤ µjφ−1(� + � + 2r(1 −µ−1j )µj), (3.7) since xi+n and xk+n are both in c.also, (3.5), (3.6) and (3.7) imply that ‖t (pt )j−1yn −yn‖≤ µjφ−1(� + � + 2r(1 −µ−1j )µj). (3.8) taking lim supn→∞ on both sides of (3.8) and noting that � > 0 is arbitrary, we have that lim sup n→∞ ‖t (pt )j−1yn −yn‖≤ µjφ−1(2r(1 −µ−1j )µj). (3.9) on the other hand, for any j ≥ 1, it follows from (3.1) that ‖t (pt )j−1ω −ω‖ ≤ ‖t (pt )j−1ω −t (pt )j−1yn‖ + ‖t (pt )j−1yn −yn‖ + ‖yn −ω‖ ≤ µj‖yn −ω‖ + ‖t (pt )j−1yn −yn‖ + ‖yn −ω‖. (3.10) taking lim supn→∞ on both sides of the above inequality and using (3.1) and (3.9), we have ‖t (pt )j−1ω −ω‖≤ µjφ−1(2r(1 −µ−1j )µj). again, taking lim supj→∞ on both sides of the above inequality, we have lim sup j→∞ ‖t (pt )j−1ω −ω‖≤ φ−1(0) = 0, which implies that ‖t (pt )j−1ω−ω‖→ 0 as j →∞, and hence proving our claim. by continuityof tp, we have that lim j→∞ tp (t (pt )j−1ω) = tpω = tω = ω. this completes the proof. � lemma 3.2. let e be a uniformly convex banach space and k a nonempty closed convex subset of e. let s1,s2,s3 : k −→ k be three total asymptotically nonexpansive self mapping with sequences {k(1)n },{k (2) n },{k (3) n }∈ [1,∞), {w(1)n },{w(2)}n,{w (3) n } ∈ [1,∞) and t1,t2,t3 : k −→ e are three total asymptotically nonexpansive nonself mappings with sequences {µ(1)n },{µ (2) n },{µ (3) n }∈ [1,∞),{ν (1) n },{ν (2) n }, {ν(3)n } ∈ [1,∞). let {xn} be the sequence defined by (1.7), where {αn} and {βn} are real sequences ∈ [0, 1). suppose f = (f (ti ) ∩f (si )) 6= ∅. if the following conditions hold:i. ∑∞n=1k(1)n < ∞, ∑∞n=1k(2)n < ∞, ∑∞n=1k(3)n < ∞, ∑∞n=1µ(1)n < ∞, ∑∞n=1µ(2)n < ∞,∑∞ n=1µ (3) n < ∞, ∑∞ n=1ν (1) n < ∞, ∑∞ n=1ν (2) n < ∞, ∑∞ n=1ν (3) n < ∞,ii. there exists a constant m > 0 such thatψ(t) = φ(t) ≤ mt,t ≤ 0. then, limn∞‖xn −q‖ and limn∞d(xn −f )both exist for all q ∈ f . eur. j. math. anal. 1 (2021) 53 proof. set hn = max(k(1)n ,k(2)n ,k(3)n ,µ(1)n ,µ(2)n ,µ(3)n ),m = max(m1,m2,m3,m4,m5,m6) and θn =max(ν(1)n ,ν(2)n ,ν(3)n ,ω(1)n ,ω(2)n ,ω(3)n ). then, ∑∞n=1hn < ∞ and ∑∞n=1θn < ∞. for any q ∈ f , itfollows from (3.1) that ‖zn −q‖ = |p ((1 −βn)sn3xn + βnt3(pt3) n−1xn) −p (q)‖ ≤ ‖(1 −βn)sn3xn + βnt3(pt n−1 3 xn −q‖ = ‖(1 −βn)sn3xn + βnq −q −βnq + βnt3(pt3) n−1xn‖ = ‖(1 −βn)sn3xn − (1 −βn)q + βn(t3(pt3) n−1xn −q)‖ = ‖(1 −βn)(sn3xn −q) + βn(t3(pt3) n−1xn −q)‖ (3.11) ≤ (1 −βn)‖sn3xn −q‖ + βn‖t3(pt3) n−1xn −q‖ ≤ (1 −βn)[‖xn −q‖ + k (3) n ψ(‖xn −q‖) + ω (3) n ] + βn[‖xn −q‖ + µ (3) n φ(‖xn −q‖) +ν (3) n ] = (1 −βn)‖xn −q‖ + (1 −βn)hnψ(‖xn −q‖) + (1 −βn)θn + βn‖xn −q‖ +βnhnφ(‖xn −q‖) + βnθn ≤ (1 −βn)(1 + hnm5)‖xn −q‖ + βn(1 + hnm6)‖xn −q‖ + θn ≤ (1 −βn)(1 + hnm)‖xn −q‖ + βn(1 + hnm)‖xn −q‖ + θn ≤ (1 + hnm)‖xn −q‖ + θn. (3.12) also, form (1.7), we get ‖yn −q‖ = |p ((1 −βn)sn2xn + βnt2(pt2) n−1zn) −p (q)‖ ≤ ‖(1 −βn)sn2xn + βnt2(pt2) n−1xn −q‖ = ‖(1 −βn)sn2xn + βnq −q −βnq + βnt2(pt2) n−1zn‖ = ‖(1 −βn)sn2xn − (1 −βn)q + βn(t2(pt2) n−1zn −q)‖ (3.13) = ‖(1 −βn)(sn2xn −q) + βn(t2(pt2) n−1zn −q)‖ ≤ (1 −βn)‖sn2xn −q‖ + βn‖t2(pt2) n−1zn −q‖ ≤ (1 −βn)[‖xn −q‖ + k (2) n ψ(‖xn −q‖) + ω (2) n ] + βn[‖zn −q‖ + µ (2) n φ(‖xn −q‖) +ν (2) n ] = (1 −βn)‖xn −q‖ + (1 −βn)hnψ(‖xn −q‖) + (1 −βn)θn + βn‖xn −q‖ +βnhnφ(‖zn −q‖) + βnθn ≤ (1 −βn)(1 + hnm3)‖xn −q‖ + βn(1 + hnm4)‖zn −q‖ + θn ≤ (1 −βn)(1 + hnm)‖xn −q‖ + βn(1 + hnm)‖zn −q‖ + θn. (3.14) eur. j. math. anal. 1 (2021) 54 putting (3.12) into (3.14), we have ‖yn −q‖ ≤ (1 −βn)(1 + hnm)‖xn −q‖ + βn(1 + hnm)[(1 + hnm)‖xn −q‖ + θn] + θn = (1 + hnm)[(1 −βn)‖xn −q‖ + βn((1 + hnm)‖xn −q‖ + θn)] + θn = (1 + hnm)[(1 −βn + βn + βnhnm))‖xn −q‖ + θn)] + θn ≤ (1 + hnm)[1 + hnm))‖xn −q‖ + θn)] + θn = (1 + hnm) 2‖xn −q‖ + (2 + hnm)θn. (3.15) again, using (1.7), we have ‖xn+1 −q‖ = |p ((1 −αn)sn1xn + αnt1(pt1) n−1yn) −p (q)‖ ≤ ‖(1 −αn)sn1xn + αnt1(pt1) n−1yn −q‖ = ‖(1 −αn)sn1xn + αnq −q −αnq + αnt1(pt1) n−1yn‖ = ‖(1 −αn)sn1xn − (1 −αn)q + αn(t1(pt1) n−1yn −q)‖ = ‖(1 −αn)(sn1xn −q) + αn(t1(pt1) n−1yn −q)‖ (3.16) ≤ (1 −αn)‖sn1xn −q‖ + αn‖t1(pt1) n−1yn −q‖ ≤ (1 −αn)[‖xn −q‖ + k (1) n ψ(‖xn −q‖) + ω (1) n ] + αn[‖yn −q‖ +µ (1) n φ(‖yn −q‖) + ν (1) n ] ≤ (1 −αn)‖xn −q‖ + (1 −αn)hnψ(‖xn −q‖) + (1 −αn)θn + αn‖yn −q‖ +αnhnφ(‖yn −q‖) + αnθn ≤ (1 −αn)(1 + hnm1)‖xn −q‖ + αn(1 + hnm2)‖yn −q‖ + θn ≤ (1 −αn)(1 + hnm)‖xn −q‖ + αn(1 + hnm)‖yn −q‖ + θn. (3.17) putting (3.15) into (3.17), we obtain ‖xn+1 −q‖ ≤ (1 −αn)(1 + hnm)‖xn −q‖ + αn(1 + hnm)[(1 + hnm)2‖xn −q‖ +(2 + hnm)θn] + θn] = (1 + hnm)‖xn −q‖−αn(1 + hnm)‖xn −q‖ + αn(1 + hnm)3‖xn −q‖ +αn(1 + hnm)(2 + hnm)θn + θn ≤ [1 + (3 + 3hnm + h2nm 2)hnm]‖xn −q‖ + [1 + (1 + hnm)(2 + hnm]θn = (1 + δn)‖xn −q‖ + ρn. (3.18) where δn = 1 + (3 + 3hnm +h2nm2)hnm and ρn = [1 + (1 +hnm)(2 +hnm]θn. since ∑∞n=1δn < ∞and ∑∞n=1ρn < ∞, it follows from lemma 2.1 that limn→∞‖xn −q‖ exists. eur. j. math. anal. 1 (2021) 55 now taking the infimum over all q ∈ f in (3.18), we get d(xn+1,f ) ≤ (1 + δn)d(xn,f ) + ρn,∀n ∈n. (3.19) again, since ∑∞n=1δn < ∞ and ∑∞n=1ρn < ∞, it follows from lemma 2.1 and (3.19) that limn→∞d(xn,f ) exists. this completes the proof. � lemma 3.3. let e be a uniformly convex banach space and k a nonempty closed convex subset of e. let s1,s2,s3 : k −→ k be three total asymptotically nonexpansive self mapping with sequences {k(1)n },{k (2) n },{k (3) n }∈ [1,∞),{w (1) n },{w(2)}n,{w (3) n }∈ [1,∞) and t1,t2,t3 : k −→ e are three total asymptotically nonexpansive nonself mappings with sequences {µ(1)n },{µ (2) n },{µ (3) n } ∈ [1,∞),{ν (1) n },{ν (2) n },{ν (3) n } ∈ [1,∞). let {xn} be the sequence defined by (1.7), where {αn} and {βn} are real sequences ∈ [0, 1). suppose f = (f (ti )∩f (si )) 6= ∅. if the following conditions hold: i. ∑∞n=1k(1)n < ∞, ∑∞n=1k(2)n < ∞, ∑∞n=1k(3)n < ∞, ∑∞n=1µ(1)n < ∞, ∑∞n=1µ(2)n < ∞,∑∞ n=1µ (3) n < ∞, ∑∞ n=1ν (1) n < ∞, ∑∞ n=1ν (2) n < ∞, ∑∞ n=1ν (3) n < ∞,ii. ‖x−t1(pt1)n−1y‖≤‖sn1x−t1(pt1)n−1y‖,‖x−t2(pt2)n−1y‖≤‖sn2x−t2(pt2)n−1y‖, ‖x −t3(pt3)n−1y‖≤‖sn3x −t3(pt3) n−1y‖iii. there exists a constant m1,m2 > 0 such that ψ(t) ≤ m1t, φ(t) ≤ m2t, t ≥ 0. then, limn∞‖xn −sixn‖ = 0 and limn∞‖xn −tixn‖ = 0, for i = 1, 2, , 3. proof. set hn = max(k(1)n ,k(2)n ,k(3)n ,µ(1)n ,µ(2)n ,µ(3)n ),m = max(m1,m2,m3,m4,m5,m6) and θn =max(ν(1)n ,ν(2)n ,ν(3)n ,ω(1)n ,ω(2)n ,ω(3)n ). then, ∑∞n=1hn < ∞ and ∑∞n=1θn < ∞. for any given q ∈ f , limn∞‖xn −q‖ exists by lemma 3.2. now, assume that limn∞‖xn −q‖ = c. it follows from (3.15),(3.16) and the fact that ∑∞n=1hn < ∞ and ∑∞n=1θn < ∞ that lim‖(1 −αn)(sn1xn −q) + αnt1(pt1) n−1yn −q)‖ = c. (3.20) also, we have ‖sn1xn −q‖ ≤ ‖xn −q‖ + k (1) n ψ(‖xn −q‖) + ω (1) n ≤ ‖xn −q‖ + k (1) n m‖xn −q‖) + ω (1) n ≤ (1 + k(1)n m)‖xn −q‖ + ω (1) n ≤ (1 + hnm)‖xn −q‖ + θn ⇒ lim sup‖sn1xn −q‖ ≤ lim sup[(1 + hnm)‖xn −q‖ + θn] = c. (3.21) eur. j. math. anal. 1 (2021) 56 furthermore, ‖t1(pt1)yn −q‖ ≤ ‖yn −q‖ + µ (1) n φ(‖yn −q‖) + ν (1) n ≤ ‖yn −q‖ + µ (1) n m‖yn −q‖) + ν (1) n ≤ (1 + µ(1)n m)‖yn −q‖ + ν (1) n ≤ (1 + hnm)‖yn −q‖ + θn taking limsup on both sides of (3.15), we obtain lim sup‖yn−q‖≤ c and so lim sup‖t1(pt1)yn−q‖≤ lim sup[(1 + hnm)‖yn−q‖+ θn] ≤ c. thus, lim sup‖t1(pt1)yn −q‖≤ lim sup[(1 + hnm)‖yn −q‖ + θn] = c. (3.22) using lemma 2.2, we get lim n→∞ ‖sn1xn −t1(pt1) n−1yn‖ = 0. (3.23)by condition (ii), it follows that ‖xn −t1(pt1)n−1yn‖≤‖sn1xn −t1(pt1) n−1yn‖, and so from (3.23), we have lim n→∞ ‖xn −t1(pt1)n−1yn‖ = 0. (3.24)also, we have ‖sn2xn −q‖ ≤ ‖xn −q‖ + k (2) n ψ(‖xn −q‖) + ω (2) n ≤ ‖xn −q‖ + k (2) n m‖xn −q‖) + ω (2) n ≤ (1 + k(2)n m)‖xn −q‖ + ω (2) n ≤ (1 + hnm)‖xn −q‖ + θn ⇒ lim sup‖sn2xn −q‖ ≤ lim sup[(1 + hnm)‖xn −q‖ + θn] = c. (3.25)furthermore, ‖t2(pt2)zn −q‖ ≤ ‖zn −q‖ + µ(2)n φ(‖zn −q‖) + ν (2) n ≤ ‖zn −q‖ + µ(2)n m‖zn −q‖) + ν (2) n ≤ (1 + µ(2)n m)‖zn −q‖ + ν (2) n ≤ (1 + hnm)‖zn −q‖ + θn eur. j. math. anal. 1 (2021) 57 taking lim sup on both sides of (3.12), we obtain lim supn→∞‖zn −q‖≤ c and so lim sup‖t2(pt1)zn −q‖≤ lim sup[(1 + hnm)‖zn −q‖ + θn] ≤ c. (3.26) (3.13), (3.25), (3.26) and lemma 2.2 imply lim n→∞ ‖sn2xn −t2(pt2) n−1zn‖ = 0. (3.27) (3.27) and condition (ii) yields lim n→∞ ‖xn −t2(pt2)n−1zn‖ = 0. (3.28) from (3.11), using the same argument as was used in obtaining (3.27) above, we get lim n→∞ ‖sn3xn −t3(pt3) n−1xn‖ = 0. (3.29) now, we prove that lim n→∞ ‖xn −t1(pt1)n−1xn‖ = lim n→∞ ‖xn −t2(pt2)n−1xn‖ lim n→∞ ‖xn −t3(pt3)n−1xn‖ = 0. indeed, since ‖xn −t3(pt3)n−1xn‖ ≤ ‖sn3xn −t3(pt3)n−1xn‖, (by condition (ii)), it follows from(3.29) that lim n→∞ ‖xn −t3(pt3)n−1xn‖ = 0. (3.30) since, p (snxn) = snxn and p : e −→ k is a nonexpansive retraction of e onto k, we get ‖zn −sn3xn‖ = ‖p ((1 −γn)s n 3xn + γnt3(pt3) n−1xn) −sn3xn‖ ≤ ‖(1 −γn)sn3xn + γnt3(pt3) n−1xn −sn3xn‖ = ‖−γn(sn3xn −γnt3(pt3) n−1xn)‖ = γn‖(sn3xn −γnt3(pt3) n−1xn)‖, which by (3.29) gives lim n→∞ ‖zn −sn3xn‖ = 0. (3.31)observe that ‖zn −xn‖ = ‖zn −sn3xn + s n 3xn −t3(pt3) n−1xn + t3(pt3) n−1xn −xn‖ ≤ ‖zn −sn3xn‖ + ‖s n 3xn −t3(pt3) n−1xn‖ +‖t3(pt3)n−1xn −xn‖. (3.32) thus, it follows from (3.29), (3.30),(3.31) and (3.32) that lim n→∞ ‖zn −xn‖ = 0. (3.33) eur. j. math. anal. 1 (2021) 58 again, observe that ‖sn2xn −t2(pt2) n−1xn‖ ≤ ‖sn2xn −t2(pt2) n−1zn‖ + ‖t2(pt2)n−1zn −t2(pt2)n−1xn‖ ≤ ‖sn2xn −t2(pt2) n−1zn‖ + (‖zn −xn‖ + k (2) n φ(‖zn −xn‖) + ν (2) n ≤ ‖sn2xn −t2(pt2) n−1zn‖ + ‖zn −xn‖ + mhn(‖zn −xn‖) + θn = ‖sn2xn −t2(pt2) n−1zn‖ + (1 + mhn)‖zn −xn‖ + θn. (3.34) from (3.27),(3.33), (3.34) and the fact that ∑∞n=1θn < ∞, we get lim n→∞ ‖sn2xn −t2(pt2) n−1xn‖ = 0. (3.35) since ‖xn−t2(pt2)n−1xn‖≤‖sn2xn−t2(pt2)n−1xn‖ (by condition (ii), it follows from (3.35) that lim n→∞ ‖xn −t2(pt2)n−1xn‖ = 0. (3.36) also, since p (snxn) = snxn and p : e −→ k is a nonexpansive retraction of e onto k, we get ‖yn −sn2xn‖ = ‖p ((1 −βn)s n 2xn + βnt2(pt2) n−1zn) −sn2xn‖ ≤ ‖(1 −βn)sn2xn + βnt2(pt2) n−1zn −sn2xn‖ = ‖−βn(sn2xn −βnt2(pt2) n−1zn)‖ = βn‖(sn2xn −βnt2(pt2) n−1zn)‖, which by (3.27) gives lim n→∞ ‖yn −sn2xn‖ = 0. (3.37)moreover, since ‖yn −xn‖ = ‖yn −sn2xn + s n 2xn −t2(pt2) n−1zn + t2(pt2) n−1xn −zn‖ ≤ ‖yn −sn2xn‖ + ‖s n 2xn −t2(pt2) n−1zn‖ + ‖t2(pt2)n−1zn −xn‖, it follows from (3.27), (3.28) and (3.37) that lim n→∞ ‖yn −xn‖ = 0. (3.38) observe that ‖sn1xn −t1(pt1) n−1xn‖ ≤ ‖sn1xn −t1(pt1) n−1yn‖ + ‖t1(pt1)n−1yn −t1(pt1)n−1xn‖ ≤ ‖sn1xn −t1(pt1) n−1yn‖ + (‖yn −xn‖ + k (1) n ψ(‖yn −xn‖) + ν (1) n ≤ ‖sn2xn −t2(pt2) n−1zn‖ + ‖yn −xn‖ + mhn(‖zn −xn‖) + θn = ‖sn1xn −t1(pt1) n−1yn‖ + (1 + mhn)‖yn −xn‖ + θn. (3.39) from (3.23), (3.38), (3.39) and the fact that ∑∞n=1θn < ∞ lim n→∞ ‖sn1xn −t1(pt1) n−1xn‖ = 0. (3.40) eur. j. math. anal. 1 (2021) 59 now, since ‖xn−t1(pt1)n−1xn‖≤‖sn1xn−t1(pt1)n−1xn‖ (by condition (ii), it follows from (3.40)that lim n→∞ ‖xn −t1(pt1)n−1xn‖ = 0. (3.41) from ‖xn+1 −sn1xn‖ = ‖p [(1 −αn)s n 1xn + αnt1(pt1) n−1yn] −sn1xn‖ ≤ ‖(1 −αn)sn1xn + αnt1(pt1) n−1yn −sn1xn‖ = ‖−αn(sn1xn −t1(pt1) n−1yn])‖ = αn‖sn1xn −t1(pt1) n−1yn]‖ and (3.23), we obtain lim n→∞ ‖xn+1 −sn1xn‖ = 0. (3.42) from ‖xn+1 −t1(pt1)n−1yn‖≤‖xn+1 −sn1xn‖ + ‖s n 1xn −t1(pt1) n−1yn‖, (3.23) and (3.42), we get lim n→∞ ‖xn+1 −t1(pt1)n−1yn‖ = 0. (3.43) also, from (3.23), (3.24) and the inequality ‖sn1xn −xn‖≤‖s n 1xn −t1(pt1) n−1yn‖ + ‖t1(pt1)n−1yn −xn‖, we have lim n→∞ ‖sn1xn −xn‖ = 0. (3.44) again, from (3.41), (3.44) and the inequality ‖sn1xn −t2(pt2) n−1xn‖≤‖sn1xn −xn‖ + ‖xn −t2(pt2) n−1xn‖, we have lim n→∞ ‖sn1xn −t2(pt2) n−1xn‖ = 0. (3.45) eur. j. math. anal. 1 (2021) 60 since ‖xn+1 −t2(pt2)n−1yn‖ ≤ ‖xn+1 −sn1xn‖ + ‖s n 1xn −t2(pt2) n−1xn‖ +‖t2(pt2)n−1xn −t2(pt2)n−1yn‖ ≤ ‖xn+1 −sn1xn‖ + ‖s n 1xn −t2(pt2) n−1xn‖ + (‖xn −yn‖ +k (2) n φ(‖xn −yn‖) + ν (2) n ) ≤ ‖xn+1 −sn1xn‖ + ‖s n 1xn −t2(pt2) n−1xn‖ + ‖xn −yn‖ +mhn‖xn −yn‖) + θn = ‖xn+1 −sn1xn‖ + ‖s n 1xn −t2(pt2) n−1xn‖ +(1 + mhn)‖xn −yn‖) + θn, it follows from (3.38), (3.42), (3.45) and the fact that ∑∞n=1θn < ∞ that lim n→∞ ‖xn+1 −t2(pt2)n−1yn‖ = 0. (3.46) now, from (3.30), (3.41) and the inequality ‖sn1xn −t3(pt3) n−1xn‖≤‖sn1xn −xn‖ + ‖xn −t3(pt3) n−1xn‖, we obtain lim n→∞ ‖sn1xn −t3(pt3) n−1xn‖ = 0. (3.47) since ‖xn+1 −t3(pt3)n−1yn‖ ≤ ‖xn+1 −sn1xn‖ + ‖s n 1xn −t3(pt3) n−1xn‖ +‖t3(pt3)n−1xn −t3(pt3)n−1yn‖ ≤ ‖xn+1 −sn1xn‖ + ‖s n 1xn −t3(pt3) n−1xn‖ + (‖xn −yn‖ +k (3) n φ(‖xn −yn‖) + ν (3) n ) ≤ ‖xn+1 −sn1xn‖ + ‖s n 1xn −t3(pt3) n−1xn‖ + ‖xn −yn‖ +mhn‖xn −yn‖) + θn = ‖xn+1 −sn1xn‖ + ‖s n 1xn −t3(pt3) n−1xn‖ +(1 + mhn)‖xn −yn‖) + θn it follows from (3.38), (3.42), (3.47) and the fact that ∑∞n=1θn < ∞ that lim n→∞ ‖xn+1 −t3(pt3)n−1yn‖ = 0. (3.48) eur. j. math. anal. 1 (2021) 61 again, since (pti)(pti)n−2yn−1,xn ∈ k for i = 1, 2, 3 and t1,t2,t3 are three total asymptoti-cally nonexpansive nonself mappings, we have ‖ti (pti )n−1yn−1 −tixn‖ = ‖ti (pti )(pti )n−2yn−1 −ti (pxn)‖ ≤ ‖(pti )(pti )n−2yn−1 −p (xn)‖ +k (i) n φ(‖(pti )(pti )n−2yn−1 −p (xn)‖) + ν (i) n ≤ ‖(pti )(pti )n−2yn−1 −p (xn)‖ +mhn‖(pti)(pti)n−2yn−1 −p (xn)‖ + θn = (1 + mhn)‖(pti )(pti )n−2yn−1 −p (xn)‖ + θn = (1 + mhn)‖ti (pti )n−2yn−1 −xn‖ + θn. (3.49) for i = 1.2.3,, it follows from (3.43), (3.46) and (3.48) that lim n→∞ ‖ti(pti)n−1yn−1 −tixn‖ = 0. (3.50) observe that ‖xn+1 −yn‖≤‖xn+1 −t1(pt1)n−1yn‖ + ‖t1(pt1)n−1yn −xn‖ + ‖xn −yn‖, so that, by (3.24), (3.38) and (3.43), we get lim n→∞ ‖xn+1 −yn‖ = 0. (3.51) next,observe, for i = 1, 2, 3, that ‖xn −tixn‖ ≤ ‖xn −ti (pti )n−1xn‖ + ‖ti (pti )n−1xn −ti (pti )n−1yn−1‖ +‖ti (pti )n−1yn−1 −tixn‖ ≤ ‖xn −ti (pti )n−1xn‖ + [‖xn −yn−1‖ + k (i) n φ(‖xn −yn−1‖) +ν (i) n ] + ‖ti (pti )n−1yn−1 −tixn‖ ≤ ‖xn −ti (pti )n−1xn‖ + ‖xn −yn−1‖ + k (i) n m‖xn −yn−1‖ +ν (i) n + ‖ti (pt1)n−1yn−1 −tixn‖ = ‖xn −ti (pti )n−1xn‖ + (1 + k (i) n m)‖xn −yn−1‖ + ν (i) n ] +‖ti (pti )n−1yn−1 −tixn‖ ≤ ‖xn −ti (pti )n−1xn‖ + max[supn≥1(1 + k (i) n m)]‖xn −yn−1‖ +max[supn≥1]ν (i) n ] + ‖ti (pti )n−1yn−1 −tixn‖ thus, it follows from (3.30), (3.36), (3.41), (3.50) and (3.51) that limn→∞‖xn − tixn‖ = 0, for i = 1, , 2, 3. eur. j. math. anal. 1 (2021) 62 finally, we prove that limn→∞‖xn −sni xn‖ = 0, for i = 1, , 2, 3.infact, by condition (ii), we have for i = 1, 2, 3, that ‖xn −sni xn ≤‖xn −ti (pti ) n−1xn‖ + ‖sni xn −ti (pti ) n−1xn‖ thus, it follows from (3.29), (3.30), (3.36), (3.40), (3.41) and (3.45) that lim n→∞ ‖xn −sni xn‖ = 0, f or i = 1, 2, 3. (3.52) this completes the proof of lemma 3.3. � lemma 3.4. under the assumption of lemma 3.2, for all p1,p2 ∈ ∩3i1(f (si ) ∩ f (ti )), the limit limn→∞‖xn + (1 − t)p1−p2‖ exists for all t ∈ [0, 1], where {xn} is the sequence defined by (1.7). proof. by lemma 3.2, limn→∞‖xn − q‖ exists for all q ∈ f and therefor {xn} is bounded. let an(t) = ‖xn + (1 − t)p1 − p2‖ exists for all t ∈ [0, 1]. then, limn→∞a(0) = ‖p1 − p2‖ and limn→∞a(1) = ‖xn −p2‖ exist by lemma 3.2. it remains therefor to prove lemma 3.4 for t ∈ (0, 1).for all x ∈ k, we define the mapping  rn(x) = p [(1 −γn)sn3 + γt3(pt3) n−1xn]; wn(x) = p [(1 −βn)sn2 + βt2(pt2) n−1xn]; vn(x) = p [(1 −αn)sn1 + αt1(pt1) n−1xn],n ≥ 1. (3.53) then, it follows that xn+1 = vnxn,vnp = p,∀p ∈ f . now, from (3.12), (3.15) and (3.18) of lemma3.2, we see that  ‖rn(x) −rn(y)‖≤ (1 + hn)m‖x −y‖ + θn; ‖wn(x) −wn(y)‖≤ (1 + rn)m‖x −y‖ + δnθn; ‖vn(x) −vn(y)‖≤ (1 + en)m‖x −y‖ + θn = fn‖x −y‖ + gn, (3.54) where rn = 2hn+h2nm2,δn = 2+hnm,en = 3hnm+3h2nm2+h3nm3 and gn = (1+hnm)(2+hnm)θnwith ∑∞n=1en < ∞, ∑∞n=1gn < ∞ and fn = 1 + en. since ∑∞n=1en < ∞, it follows that fn → 1 as n →∞. set  sn,m = vn+m−1vn+m−2 · · ·vn, m ∈n; bn,m = ‖sn,m(txn + (1 − t)p1) −sn.m(txm + (1 − t)p2‖. (3.55) eur. j. math. anal. 1 (2021) 63 from (3.54) and (3.55), we have ‖sn,m(x) −sn,m(y)‖ = ‖vn+m−1vn+m−2 · · ·vn(x) −vn+m−1vn+m−2 · · ·vn(y)‖ ≤ fn+m−1‖vn+m−2vn+m−3 · · ·vn(x) −vn+m−2vn+m−3 · · ·vn(y)‖ +gn+m−1 ≤ (fn+m−1)(fn+m−2)‖vn+m−3vn+m−4 · · ·vn(x) −vn+m−3vn+m−4 · · ·vn(y)‖ + gn+m−1 + gn+m−2... ≤ ( n+m−1∏ i=n fi )‖x −y‖ + n+m−1∑ i=n gi = bn‖x −y‖ + n+m−1∑ i=n gi, (3.56) for all x,y ∈ k, where bn = ∏n+m−1i=n fi,sn,mxn = xn and sn,mp = p for all p ∈ f . thus, an+m(t) = ‖txn + (1 − t)p1 −p2‖ = ‖sn,m(txn + (1 − t)p1 −p2‖ ≤ bn,m + ‖sn,m(txn + (1 − t)p1 −p2‖. (3.57) by using theorem 2.3 in [5], we have bn,m ≤ ψ−1(‖(xn −u‖−‖xn+1 −sn,mu‖) = ψ−1(‖(xn −u‖−‖xn+1 −u + u −sn,mu‖) ≤ ψ−1(‖(xn −u‖− (‖xn+1 −u‖ + ‖sn,mu −u‖)), (3.58) so that the sequence {bn.m} converges uniformly to 0, i.e, bn,m → 0 as n →∞. since limn→bn = 1and limn→∞bn,m = 0, it follows from (3.57) that lim supn→∞an(t) ≤ lim infb→∞bn.m ≤ lim infn→∞an(t).this shows that limn→∞an(t) exists, i.e, limn→∞‖txn + (1 − t)p1 − p2‖ exists for all t ∈ [0, 1].this completes the proof lemma 3.4. � lemma 3.5. under the assumption of lemma 3.2, if e has frechet differentiable norm,then for all p1,jp2 ∈ f = ∩3i=1(f (ti ) ∩ f (si )), the limn→∞(〈xn,j(p1 − p2)〉 exists, where {xn} is the sequence defined by (1.7). if ωω(xn) denotes the set of all weak subsequential limits of {xn}, then 〈q1 −q2,j(p1 −p2〉 = 0 for all p1,p2 ∈ f and for all q1,q2 ∈ ωω(xn). proof. suppose that x = p1 −p2 with p1 6= p2 and h = t(xn −p1) in(2.1). then, we have t(〈xn,j(p1 −p2)〉 + 1 2 ‖p1 −p2‖2 ≤ 1 2 ‖txn + (1 − t)p1 −p2‖2 ≤ t(〈xn,j(p1 −p2)〉 + 1 2 ‖p1 −p2‖2 + b(t‖xn −p1‖) eur. j. math. anal. 1 (2021) 64 since supn≥1‖xn −p‖≤ q for some q > 0, we have t lim n→∞ sup(〈xn,j(p1 −p2)〉 + 1 2 ‖p1 −p2‖2 ≤ 1 2 lim n→∞ sup‖txn + (1 − t)p1 −p2‖2 ≤ t lim n→∞ inf (〈xn,j(p1 −p2)〉 + 1 2 ‖p1 −p2‖2 +b(tq) that is, t limn→∞ sup(〈xn,j(p1 − p2)〉 ≤ t lim infn→∞(〈xn,j(p1 − p2)〉 + b(tq). if t → 0, then limn→∞〈xn − p1,j(p1 − p2)〉 exists for all p1,p2 ∈ f and for all q2,q2 ∈ ωω(xn); in particular, (〈q1 −q2,j(p1 −p2)〉 = 0 for all q2,q2 ∈ ωω(xn). this completes the proof lemma 3.5. � theorem 3.6. under the assumption of lemma 3.2, if e has frechet differentiable norm, then the sequence {xn} defined by (1.7) converges weakly to a common fixed point in f = ∩3i=1f (ti )∩f (si ). proof. by lemma 3.5, (〈q1−q2,j(p1−p2)〉 = 0 for all q2,q2 ∈ ωω(xn). therefore, ‖q?−p ?‖2 = 〈q? −p?,j(q? −p?)〉 = 0. this implies that p? = q?. consequently, {xn} converges to a commonfixed point of f = ∩3i=1f (ti ) ∩f (si ). this completes the proof theorem 3.6. � theorem 3.7. under the assumption of lemma 3.2, if the dual space e? of e has the kadec klec (kk) property and the mappings i − si and i − ti for i = 1, 2, 3, where i denotes the identity mapping, are demiclosed at zero, then the sequence {xn} defined by (1.7) converges weakly to a common fixed point in f = ∩3i=1(f (ti ) ∩f (si )). proof. by lemma 3.2 {xn} is bounded and since e is reflexive, there exists a subsequence {xnk} of {xn} which converges weakly to some q? ∈ k. by lemma 3.3, we have limn→∞‖xnk −sixnk‖ = 0and limn→∞‖xnk −tixnk‖ = 0 for i = 1, 2, 3. since by hypothesis, the mappings i −si and i −tifor i = 1, 2, 3, where i denotes the identity mapping, are demiclosed at zero, siq? = q? and tiq ? = q? for i = 1, 2, 3.; which means q? ∈ f = ∩3i=1(f (ti ) ∩f (si )). now, we show that {xn}converges weakly to q?. suppose {xnj} is another subsequence of {xn} which converges weakly to p? ∈ k. by the same method as above, we have p? ∈ f and q? ∈ ωω(xn). by lemma 3.4, the limit limn→∞‖txn + (1 − t)q? −p?‖ exists for all t ∈ [0, 1] and so q? = p?. thus, the sequence {xn}converges weakly to q? ∈ f . this completes the proof. � theorem 3.8. under the assumption of lemma 3.2, if e satisfies opial’s condition and the mappings i−si and i−ti for i = 1, 2, 3, where i denotes the identity mapping, are demiclosed at zero, then the sequence {xn} defined by (1.7) converges weakly to a common fixed point in f = ∩3i=1(f (ti )∩ f (si )). proof. let q? ∈ f . from lemma 3.2, the squence {‖xn −p ?‖} is convergent and hence bounded.since, e is uniformly convex , every bounded subset of e is weakly compact. thus, the existsa subsequence {xnk} of {xn} which converges weakly to some q? ∈ k. by lemma 3.3, we have eur. j. math. anal. 1 (2021) 65 limn→∞‖xnk −sixnk‖ = 0 and limn→∞‖xnk −tixnk‖ = 0 for i = 1, 2, 3. since by hypothesis, themappings i −si and i −ti for i = 1, 2, 3, where i denotes the identity mapping, are demiclosedat zero, siq? = q? and tiq? = q? for i = 1, 2, 3.; which means q? ∈ f = ∩3i=1(f (ti ) ∩ f (si )).finally, we show that {xn} converges weakly to q?. suppose on the contrary that {xnj} is anothersubsequence of {xn} which converges weakly to p? ∈ k and q? 6= p? by lemma 3.2, limn→∞‖xn− q?‖ and limn→∞‖xn −p?‖ exist. by virtue of opial’s condition on e, we obtain lim n→∞ ‖xn −q?‖ = lim n→∞ ‖xnk −q ?‖ < lim n→∞ ‖xnk −p ?‖ = lim n→∞ ‖xn −p?‖ = lim n→∞ ‖xnj −p ?‖ < lim n→∞ ‖xnj −q ?‖ = lim n→∞ ‖xn −q?‖, (3.59) which is a contradiction, so q? = p? therefore, the sequence {xn} defined by (1.7) converges weaklyto q? ∈ f . this completes the proof. � corollary 3.9. let e be a uniformly convex banach space and k a nonempty closed convex subset of e. let s1,s2,s3 : k −→ k be three generalize asymptotically nonexpansive self mapping with sequences {k(1)n },{k (2) n },{k (3) n }∈ [1,∞), {w(1)n },{w(2)}n,{w (3) n } ∈ [1,∞) and t1,t2,t3 : k −→ e are three generalize asymptotically nonexpansive nonself mappings with sequences {µ(1)n },{µ (2) n },{µ (3) n }∈ [1,∞),{ν (1) n },{ν (2) n }, {ν(3)n }∈ [1,∞). let {xn} be the sequence defined by (1.7), where {αn} and {βn} are real sequences ∈ [0, 1).. suppose f = ∩3i=1(f (ti ) ∩f (si )) 6= 0. if the following conditions hold:i. ∑∞n=1k(1)n < ∞, ∑∞n=1k(2)n < ∞, ∑∞n=1k(3)n < ∞, ∑∞n=1µ(1)n < ∞, ∑∞n=1µ(2)n < ∞, ∑∞n=1µ(3)n < ∞, ∑∞ n=1ν (1) n < ∞, ∑∞ n=1ν (2) n < ∞, ∑∞ n=1ν (3) n < ∞,ii. there exists a constant m > 0 such that ψ(t) = φ(t) ≤ mt,t ≤ 0. then, limn∞‖xn −q‖ and limn∞d(xn −f ) both exist for all q ∈ f . corollary 3.10. let e be a uniformly convex banach space and k a nonempty closed convex subset of e. let s1,s2,s3 : k −→ k be three generalize asymptotically nonexpansive self mapping with sequences {k(1)n },{k (2) n },{k (3) n }∈ [1,∞), {w(1)n },{w(2)}n,{w (3) n } ∈ [1,∞) and t1,t2,t3 : k −→ e are three generalize asymptotically nonexpansive nonself mappings with sequences {µ(1)n },{µ (2) n },{µ (3) n }∈ [1,∞),{ν (1) n },{ν (2) n }, {ν(3)n } ∈ [1,∞). let {xn} be the sequence defined by (1.7), where {αn} and {βn} are real sequences ∈ [0, 1). suppose f = ∩3i=1(f (ti ) ∩f (si )) 6= 0. if the following conditions hold: eur. j. math. anal. 1 (2021) 66 i. ∑∞n=1k(1)n < ∞, ∑∞n=1k(2)n < ∞, ∑∞n=1k(3)n < ∞, ∑∞n=1µ(1)n < ∞, ∑∞n=1µ(2)n < ∞, ∑∞n=1µ(3)n < ∞, ∑∞ n=1ν (1) n < ∞, ∑∞ n=1ν (2) n < ∞, ∑∞ n=1ν (3) n < ∞,ii. ‖x−t1(pt1)n−1y‖≤‖sn1x−t1(pt1)n−1y‖,‖x−t2(pt2)n−1y‖≤‖sn2x−t2(pt2)n−1y‖, ‖x −t3(pt3)n−1y‖≤‖sn3x −t3(pt3) n−1y‖iii. there exists a constant m1,m2 > 0 such that ψ(t) ≤ m1t, φ(t) ≤ m2t, t ≥ 0. then, limn∞‖xn −sixn‖ = 0 and limn∞‖xn −tixn‖ = 0, for i = 1, 2, , 3. abbreviations usednot applicable declaration: availability of data and materialnot applicable competing lnterestthe authors declare that there is no conflict of interest. fundingno specific funding received for this work authors contributionika and ncu wrote the paper while dii suggested the idea and did the analysis. the three authorsread and approved the final manuscript. acknowledgementthe authors thank the anonymous 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https://doi.org/10.1016/j.jmaa.2006.09.023 https://doi.org/10.1016/s0362-546x(99)00200-x https://doi.org/10.1016/s0362-546x(99)00200-x https://doi.org/10.1090/s0002-9939-1972-0298500-3 https://doi.org/10.1016/j.aml.2011.06.022 https://doi.org/10.1186/1687-1812-2012-224 https://doi.org/10.1016/s0895-7177(00)00199-0 https://doi.org/10.1016/s0895-7177(00)00199-0 https://doi.org/10.1016/j.jmaa.2005.10.062 1. introduction 2. preliminary 3. main results references ©2023 ada academica https://adac.eeeur. j. math. anal. 3 (2023) 10doi: 10.28924/ada/ma.3.10 complex oscillation of solutions and their arbitrary-order derivatives of linear differential equations with analytic coefficients of [p,q]-order in the unit disc benharrat belaïdi∗, meriem belmiloud department of mathematics, laboratory of pure and applied mathematics, university of mostaganem (umab), b. p. 227 mostaganem, algeria benharrat.belaidi@univ-mosta.dz, meriem.belmiloud27@gmail.com ∗correspondence: benharrat.belaidi@univ-mosta.dz abstract. throughout this article, we investigate the growth and fixed points of solutions of complexhigher order linear differential equations in which the coefficients are analytic functions of [p,q]−orderin the unit disc. this work improves some results of belaïdi [3–5], which is a generalization of recentresults from chen et al. [9]. 1. introduction and main results consider for k ≥ 2 the following complex linear differential equations f (k) + ak−1 (z) f (k−1) + · · · + a1 (z) f ′ + a0 (z) f = 0, (1.1) ak (z) f (k) + ak−1 (z) f (k−1) + · · · + a1 (z) f ′ + a0 (z) f = 0, (1.2)where ai 6≡ 0 (i = 0, 1, ...,k) are analytic functions in the unit disc d = {z ∈ c : |z| < 1}. it iswell-known that the solutions of (1.1) are analytic in d too and that there are exactly k linearlyindependent solutions of (1.1), see [13]. bernal [6] was the first to use the concept of iteratedorder to study the growth of fast growing solutions of equation (1.1) . after that, the iterated orderof solutions of higher order equations was investigated by cao in [8], he extended the results ofchen and yang [10], belaïdi [2] on c. in addition, cao [8] obtained some results concerning thefixed points of homogeneous linear differential equations (1.1) and (1.2) . in [15, 16], juneja and hisco-authors have investigated some properties of entire functions of [p,q]-order, and obtained someresults of their growth. in [20], by using the concept of [p,q]-order liu, tu and shi have consideredthe equation (1.1) with entire coefficients and obtained different results concerning the growth of itssolutions in the complex plane. in [3], the [p,q]−order was introduced in the unit disc d, and manyresults on [p,q]−order of solutions of (1.1) have been found by different researchers [3–5,14,18,22]in d. recently, chen et al. in [9] gave some results about the growth and fixed points of solutions received: 30 sep 2022. key words and phrases. linear differential equations; analytic function; [p,q]−order; fixed points.1 https://adac.ee https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 2 of higher-order linear differential equations in the unit disc, they studied and estimated the fixedpoints of solutions of (1.1) and (1.2), and also extended the coefficient conditions to a type ofone-constant-control coefficient comparison and obtained the same estimates of iterated order ofsolutions. the aim of this paper is to contrast coefficients by producing better estimates of thegrowth of solutions by using the concept of [p,q]−order, and optimizing the coefficients’s conditionswith less control constants of the coefficients’s modulus or characteristic functions and we will obtainresults which improve and generalize those of chen et al., belaïdi, cao, tu and xuan. throughout this paper, we shall assume that the reader is familiar with the fundamental resultsand the standard notations of the nevanlinna’s theory in the unit disc d = {z ∈ c : |z| < 1}(see, [12, 13, 17, 21]). now, we give the definitions of iterated order and growth index to classify generally thefunctions of fast growth in d as those in c (see, [6]). let us define inductively, for r ∈r, exp1 r := erand expp+1 r := exp (expp r) , p ∈ n. we also define for all r sufficiently large in (0, +∞) , log1 r := log r and logp+1 r := log (logp r) , p ∈n. moreover, we denote by exp0 r := r, log0 r := r, log−1 r := exp1 r and exp−1 r := log1 r. definition 1.1 (see [7]) let f be a meromorphic function in d. then the iterated n−order of f is defined by σn (f ) = lim sup r→1− log+n t (r, f ) log ( 1 1−r ) (n ≥ 1 is an integer ) , where log+1 x = log +x = max{log x, 0} , log+n+1x = log + ( log+n x ) . for n = 1, this notation is called order (σ1 (f ) = σ (f )) and for n = 2 hyper-order ([19]). if f is an analytic in d, then the iterated n−order of f is defined by σm,n (f ) = lim sup r→1− log+n+1m (r, f ) log ( 1 1−r ) (n ≥ 1 is an integer ) . for n = 1, σm,1 (f ) = σm (f ) . now, we introduce the concept of [p,q]-order of meromorphic and analytic functions in theunit disc. definition 1.2 ([3]) let p ≥ q ≥ 1 be integers and f be a meromorphic function in d. then, the [p,q]-order of f is defined by σ[p,q] (f ) = lim sup r→1− log+p t (r, f ) logq ( 1 1−r ) . https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 3 for an analytic function f in d, we also define σm,[p,q] (f ) = lim sup r→1− log+p+1m (r, f ) logq ( 1 1−r ) . remark 1.1 it is easy to see that 0 ≤ σ[p,q] (f ) ≤ ∞ (0 ≤ σm,[p,q] (f ) ≤ ∞), for any p ≥ q ≥ 1. by definition 1.2, we have that σ[1,1] = σ (f ) (σm,[1,1] = σm (f )) and σ[2,1] = σ2 (f )( σm,[2,1] = σm,2 (f ) ). proposition 1.1 ([3]) let p ≥ q ≥ 1 be integers, and let f be an analytic function in d of [p,q]-order. the following two statements hold: (i) if p = q, then σ[p,q] (f ) ≤ σm,[p,q] (f ) ≤ σ[p,q] (f ) + 1. (ii) if p > q, then σ[p,q] (f ) = σm,[p,q] (f ) . definition 1.3 ([4]) let p ≥ q ≥ 1 be integers and f be a meromorphic function in d. then, the [p,q]-exponent of convergence of the sequence of zeros of f is defined by λ[p,q] (f ) = lim sup r→1− log+p n ( r, 1 f ) logq ( 1 1−r ) , where n ( r, 1 f ) is the integrated counting function of zeros of f in {z : |z| ≤ r}. similarly, the [p,q]-exponent of convergence of the sequence of distinct zeros of f is defined by λ[p,q] (f ) = lim sup r→1− log+p n ( r, 1 f ) logq 1 1−r , where n ( r, 1 f ) is the integrated counting function of distinct zeros of f in {z : |z| ≤ r}. definition 1.4 let p ≥ q ≥ 1 be integers and f be a meromorphic function in d. then, the [p,q]-exponent of convergence of the sequence of fixed points of f is defined by λ[p,q] (f −z) = lim sup r→1− log+p n ( r, 1 f−z ) logq ( 1 1−r ) . similarly, the [p,q]-exponent of convergence of the sequence of distinct fixed points of f is defined by λ̄[p,q] (f −z) = lim sup r→1− log+p n̄ ( r, 1 f−z ) logq ( 1 1−r ) . recall that for a measurable set e ⊂ [0, 1) , the upper and lower densities of e are defined by densde = lim sup r→1− m (e ∩ [0, r)) m ([0, r)) and densde = lim inf r→1− m (e ∩ [0, r)) m ([0, r)) , respectively, where m (f ) = ∫ f dt 1−t for f ⊂ [0, 1). it is clear that 0 ≤ densde ≤ densde ≤ 1for any measurable set e ⊂ [0, 1) . https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 4 proposition 1.2 if a set e satisfies densde > 0, then m (e) = ∫ e dt 1−t = +∞. proof. suppose that m (e) = ∫ e dt 1−t = δ < ∞. we have m ([0, r)) = − log (1 − r) . since m (e ∩ [0, r)) ≤ m (e) , then densde = lim sup r→1− m (e ∩ [0, r)) m ([0, r)) ≤ lim sup r→1− δ − log (1 − r) = 0. so densde = 0. hence densde > 0 =⇒ m (e) = ∫ e dt 1 − t = +∞. in 2012, belaïdi in [4] and [5] treated the growth of solutions of homogeneous linear differentialequations in which the coefficients are analytic functions of [p,q]−order in d. as for the equation (1.1), he got the following results. theorem a (see [4]) let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0 (z) , ...,ak−1 (z) be analytic functions in the unit disc d such that for real constants α,β, where 0 ≤ β < α, we have |a0 (z)| ≥ expp+1 { α logq ( 1 1 −|z| )} and |ai (z)| ≤ expp+1 { β logq ( 1 1 −|z| )} (i = 1, ...,k − 1) as |z| → 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ α. theorem b (see [5]) let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0 (z) , ...,ak−1 (z) be analytic functions in the unit disc d such that for real constants α,β, where 0 ≤ β < α, we have t (r,a0) ≥ expp { α logq ( 1 1 −|z| )} and t (r,ai ) ≤ expp { β logq ( 1 1 −|z| )} (i = 1, ...,k − 1) as |z| = r → 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ α. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 5 after that in 2021, chen et al. [9] investigated the growth of solutions of equations (1.1) and (1.2) in d by using the iterated order, and they got the following results. theorem c (see [9]) let n ≥ 1 be an integer. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0,a1, ...,ak−1 be analytic functions in the unit disc d such that max{σm,n (ai ) : i = 1, 2, ...,k − 1}≤ σm,n (a0) = µ (0 < µ < ∞) , and for a constant α ≥ 0, we have lim inf |z|→1−,z∈h ( (1 −|z|)µ logn |a0 (z)| ) > α and |ai (z)| ≤ expn { α ( 1 1 −|z| )µ} , (i = 1, 2, ...,k − 1) as |z|→ 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σn (f ) = σm,n (f ) = ∞ and σn+1 (f ) = σm,n (a0) = µ. theorem d (see [9]) let n ≥ 1 be an integer. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0,a1, ...,ak be analytic functions in the unit disc d, and for some constants α ≥ 0 and µ > 0, we have lim inf |z|→1−,z∈h ( (1 −|z|)µ logn−1t (r,a0) ) > α and t (r,ai ) ≤ expn−1 { α ( 1 1 −|z| )µ} , (i = 1, 2, ...,k) as |z| = r → 1− for z ∈ h. then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σn (f ) = ∞ and σn+1 (f ) ≥ µ. theorem e (see [9]) assume that the assumptions of theorem c hold. then every solution f 6≡ 0 of equation (1.1) satisfies λ̄n ( f (j) −z ) = λ̄n (f −z) = σn (f ) = ∞, λ̄n+1 ( f (j) −z ) = λ̄n+1 (f −z) = σn+1 (f ) = µ, (j = 1, 2, ...) .in this paper, we improve and generalize the recent results of chen et al. [9] by using theconcept of [p,q]−order instead of the iterated order with less control constant. at the same time,our work improve some results of belaïdi in [4] and [5]. to be specific, we will decrease the controlconstants of the coefficients’ modulus or characteristic functions and obtain the same results ofbelaïdi, tu and xuan. here, we study the problem and get the following results. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 6 theorem 1.1 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0, ...,ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai ) : i = 1, 2, ...,k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and for a constant α ≥ 0, we have lim inf |z|→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z| ))µ > α (1.3) and |ai (z)| ≤ expp { α ( logq−1 ( 1 1−|z| ))µ} (i = 1, ...,k − 1) (1.4) as |z| → 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. by theorem 1.1, we easily obtain the following corollary. corollary 1.1 ([22]) let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0, ...,ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai ) : i = 1, 2, ...,k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and for some real constants α,β where 0 ≤ β < α, we have |a0 (z)| ≥ expp { α ( logq−1 ( 1 1 −|z| ))µ} and |ai (z)| ≤ expp { β ( logq−1 ( 1 1 −|z| ))µ} , i = 1, ...,k − 1 as |z| → 1− for z ∈ h. then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. theorem 1.2 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0, ...,ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai ) : i = 1, 2, ...,k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and lim sup |z|→1−,z∈h logp |ai (z)|( logq−1 ( 1 1−|z| ))µ < lim inf |z|→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z| ))µ (i = 1, ...,k − 1) (1.5) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 7 as |z| → 1− for z ∈ h.then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. theorem 1.3 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0, ...,ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai ) : i = 1, 2, ...,k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and for a constant α ≥ 0, if p ≥ q ≥ 2 we have lim inf |z|→1−,z∈h logp−1t (r,a0)( logq−1 ( 1 1−|z| ))µ > α (1.6) and t (r,ai ) ≤ expp−1 { α ( logq−1 ( 1 1−|z| ))µ} , (i = 1, ...,k − 1) (1.7) as |z| = r → 1− for z ∈ h, then every solution f 6≡ 0 of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. if p = q = 1, we have lim inf |z|→1−,z∈h t (r,a0)( 1 1−|z| )µ > (k − 1) α (1.8) and t (r,ai ) ≤ α ( 1 1−|z| )µ , (i = 1, ...,k − 1) (1.9) as |z| = r → 1− for z ∈ h, then every nontrivial solution f of equation (1.1) satisfies σ(f ) = σm(f ) = ∞ and σ2 (f ) = σm,2 (f ) = µ. by theorem 1.3, we easily obtain the following corollary. corollary 1.2 ([22]) let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0, ...,ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai ) : i = 1, 2, ...,k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and for some real constants α,β, where 0 ≤ β < α, we have t (r,a0) ≥ expp−1 { α ( logq−1 ( 1 1 −|z| ))µ} and t (r,ai ) ≤ expp−1 { β ( logq−1 ( 1 1−|z| ))µ} (i = 1, ...,k − 1) as |z| = r → 1− for z ∈ h. then the following statements hold: https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 8 (i) if p = q = 1 and 0 ≤ (k − 1) β < α, then every nontrivial solution f of equation (1.1) satisfies σ(f ) = σm(f ) = ∞ and σ2 (f ) = σm,2 (f ) = µ. (ii) if p ≥ q ≥ 2 and 0 ≤ β < α, then every nontrivial solution f of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. theorem 1.4 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0, ...,ak−1 be analytic functions in the unit disc d such that max { σm,[p,q] (ai ) : i = 1, 2, ...,k − 1 } ≤ σm,[p,q] (a0) = µ (0 < µ < +∞) and if p ≥ q ≥ 2, we have lim sup |z|→1−,z∈h logp−1t (r,ai )( logq−1 ( 1 1−|z| ))µ < lim inf |z|→1−,z∈h logp−1t (r,a0)( logq−1 ( 1 1−|z| ))µ , (i = 1, ...,k − 1) (1.10) as |z| = r → 1− for z ∈ h, then every nontrivial solution f of equation (1.1) satisfies σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. if p = q = 1, we have lim sup |z|→1−,z∈h (k − 1) t (r,ai )( 1 1−|z| )µ < lim inf |z|→1−,z∈h t (r,a0)( 1 1−|z| )µ , (i = 1, ...,k − 1) (1.11) as |z| = r → 1− for z ∈ h, then every nontrivial solution f of equation (1.1) satisfies σ(f ) = σm(f ) = ∞ and σ2 (f ) = σm,2 (f ) = µ. theorem 1.5 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0, ...,ak be analytic functions in the unit disc d such that for some constants α ≥ 0 and µ > 0, we have (1.3) and |ai (z)| ≤ expp { α ( logq−1 ( 1 1−|z| ))µ} , (i = 1, ...,k) as |z| → 1− for z ∈ h. then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ[p,q] (f ) = ∞ and σ[p+1,q] (f ) ≥ µ. theorem 1.6 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 9 and let a0, ...,ak be analytic functions in the unit disc d such that for a constant µ > 0, we have lim sup |z|→1−,z∈h logp |ai (z)|( logq−1 ( 1 1−|z| ))µ < lim inf |z|→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z| ))µ , (i = 1, ...,k) as |z| → 1− for z ∈ h.then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ[p,q] (f ) = ∞ and σ[p+1,q] (f ) ≥ µ. theorem 1.7 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0, ...,ak be analytic functions in the unit disc d such that for some constants α ≥ 0 and µ > 0, if p ≥ q ≥ 2 we have lim inf |z|→1−,z∈h logp−1t (r,a0)( logq−1 ( 1 1−|z| ))µ > α (1.12) and t (r,ai ) ≤ expp−1 { α ( logq−1 ( 1 1−|z| ))µ} , (i = 1, ...,k) (1.13) as |z| = r → 1− for z ∈ h, then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ[p,q] (f ) = ∞ and σ[p+1,q] (f ) ≥ µ. if p = q = 1, we have lim inf |z|→1−,z∈h t (r,a0)( 1 1−|z| )µ > kα (1.14) and t (r,ai ) ≤ α ( 1 1−|z| )µ , (i = 1, ...,k) (1.15) as |z| = r → 1− for z ∈ h, then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ (f ) = ∞ and σ2 (f ) ≥ µ. theorem 1.8 let p ≥ q ≥ 1 be integers. let h be a set of complex numbers satisfying densd {|z| : z ∈ h ⊆ d} > 0, and let a0 (z) , ...,ak (z) be analytic functions in the unit disc d such that for a constant µ > 0, if p ≥ q ≥ 2 we have lim sup |z|→1−,z∈h logp−1t (r,ai )( logq−1 ( 1 1−|z| ))µ < lim inf |z|→1−,z∈h logp−1t (r,a0)( logq−1 ( 1 1−|z| ))µ , (i = 1, ...,k) (1.16) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 10 as |z| = r → 1− for z ∈ h, then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ[p,q] (f ) = ∞ and σ[p+1,q] (f ) ≥ µ. if p = q = 1, we have lim sup |z|→1−,z∈h kt (|z| ,ai )( 1 1−|z| )µ < lim inf |z|→1−,z∈h t (|z| ,a0)( 1 1−|z| )µ , (i = 1, ...,k) (1.17) as |z| = r → 1− for z ∈ h, then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies σ(f ) = ∞ and σ2 (f ) ≥ µ. remark 1.2 for equation (1.1) , we can easily conclude that theorems a-c are generalized totheorems 1.1-1.4. in the same paper, chen et al. [9] obtained some results of the fixed points of solutions andtheir arbitrary order derivatives of equations (1.1) and (1.2). here, we generalize these results,and we obtain our theorems as following. theorem 1.9 assume that the assumptions of theorem 1.1 or theorem 1.2 hold. then every solution f 6≡ 0 of equation (1.1) satisfies λ̄[p,q] ( f (j) −z ) = λ[p,q] (f −z) = σ[p,q] (f ) = ∞, λ̄[p+1,q] ( f (j) −z ) = λ̄[p+1,q] (f −z) = σ[p+1,q] (f ) = µ, (j = 1, 2, ...) . theorem 1.10 assume that the assumptions of theorem 1.3 or theorem 1.4 hold. then every solution f 6≡ 0 of equation (1.1) satisfies λ̄[p,q] ( f (j) −z ) = λ[p,q] (f −z) = σ[p,q] (f ) = ∞, λ̄[p+1,q] ( f (j) −z ) = λ̄[p+1,q] (f −z) = σ[p+1,q] (f ) = µ, (j = 1, 2, ...) . theorem 1.11 assume that the assumptions of one of theorem 1.5 to theorem 1.8 hold. then every meromorphic (or analytic) solution f 6≡ 0 of equation (1.2) satisfies λ̄[p,q] ( f (j) −z ) = λ[p,q] (f −z) = σ[p,q] (f ) = ∞, λ̄[p+1,q] ( f (j) −z ) = λ̄[p+1,q] (f −z) = σ[p+1,q] (f ) ≥ µ, (j = 1, 2, ...) . 2. some lemmas in this section we give some lemmas which are used in the proofs of our theorems. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 11 lemma 2.1 ([11], theorem 3.1) let k and j be integers satisfying k > j ≥ 0, and let ε > 0 and d ∈ (0, 1). if f is a meromorphic function in d such that f (j) does not vanish identically, then for |z| /∈ e1 ∣∣∣∣∣f (k) (z)f (j) (z) ∣∣∣∣∣ ≤ [( 1 1 −|z| )2+ε max { log ( 1 1 −|z| ) ; t (s (|z|) , f ) }]k−j , where e1 ⊂ [0, 1) is a set with ∫ e1 dr 1−r < ∞ and s (|z|) = 1 −d (1 −|z|) . lemma 2.2 ([13]) let f be a meromorphic function in the unit disc d, and let k ≥ 1 be an integer. then m ( r, f (k) f ) = s (r, f ) , where s(r, f ) = o ( log+t (r, f ) + log ( 1 1−r )), possibly outside a set e2 ⊂ [0, 1) with ∫e2 dr1−r < ∞. lemma 2.3 ([1]) let g : (0, 1) → r and h : (0, 1) → r be monotone increasing functions such that g (r) ≤ h (r) holds outside of an exceptional set e3 ⊂ [0, 1) for which ∫ e3 dr 1−r < ∞. then there exists a constant d ∈ (0, 1) such that if s (r) = 1 −d (1 − r) , then g (r) ≤ h (s (r)) for all r ∈ [0, 1). lemma 2.4 ([3]) let p ≥ q ≥ 1 be integers. if a0 (z) , ...,ak−1 (z) are analytic functions of [p,q]−order in the unit disc d, then every solution f 6≡ 0 of (1.1) satisfies σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≤ max { σm,[p,q] ( aj ) : j = 0, 1, ...,k − 1 } . lemma 2.5 ([4, 18]) let p ≥ q ≥ 1 be integers. if f and g are non-constant meromorphic functions of [p,q]−order in d, then we have (i) σ[p,q] (f ) = σ[p,q](1f ) , σ[p,q] (af ) = σ[p,q] (f ) and σ[p,q] (f + a) = σ[p,q] (f ) (a ∈c∗) , (ii) σ[p,q] (f ′) = σ[p,q] (f ) , (iii) σ[p,q] (f + g) ≤ max {σ[p,q] (f ) ,σ[p,q] (g)} , (iv) σ[p,q] (f g) ≤ max {σ[p,q] (f ) ,σ[p,q] (g)} , if σ[p,q] (f ) > σ[p,q] (g) , then we obtain σ[p,q] (f + g) = σ[p,q] (f g) = σ[p,q] (f ) . lemma 2.6 ([4]) let p ≥ q ≥ 1 be integers. let a0, ...,ak−1 and f 6≡ 0 be finite [p,q]−order analytic functions in the unit disc d. if f is a solution with σ[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σ < ∞ of equation f (k) + ak−1 (z) f (k−1) + · · · + a1 (z) f ′ + a0 (z) f = f, (2.1) then λ̄[p,q] (f ) = λ[p,q] (f ) = σ[p,q] (f ) = ∞, λ̄[p+1,q] (f ) = λ[p+1,q] (f ) = σ[p+1,q] (f ) = σ. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 12 by using the same arguments of the proof of lemma 3.5 in the paper [14, p. 4], we obtain thefollowing lemma in the case when σ[p,q] (f ) = σ = ∞. lemma 2.7 let p ≥ q ≥ 1 be integers. let aj (j = 0, ...,k − 1) , f 6≡ 0 be meromorphic functions in d, and let f be a solution of the differential equation (2.1) satisfying max { σ[p,q] ( aj ) (j = 0, ...,k − 1) ,σ[p,q] (f ) } < σ[p,q] (f ) = σ ≤∞. then we have λ[p,q] (f ) = λ[p,q] (f ) = σ[p,q] (f ) and λ[p+1,q] (f ) = λ[p+1,q] (f ) = σ[p+1,q] (f ) . 3. proofs of theorems 1.1 to 1.8 proof of theorem 1.1. suppose that every solution f of equation (1.1) not being identically equalto 0. from the conditions of theorem 1.1, there exists a set h of complex numbers satisfying densdh1 > 0, where h1 = {r = |z| : z ∈ h ⊆ d} . then h1 is a set with ∫h1 dr1−r = +∞, suchthat for z ∈ h we have (1.3) and (1.4) as |z|→ 1−. by lemma 2.1, there exists a set e1 ⊂ [0, 1)with ∫ e1 dr 1−r < ∞ such that for |z| /∈ e1, we have for j = 1, ...,k∣∣∣∣∣f (j) (z)f (z) ∣∣∣∣∣ ≤ [( 1 1 −|z| )2+ε max { log ( 1 1 −|z| ) ,t (s (|z|) , f ) }]j , (3.1) where s (|z|) = 1 −d (1 −|z|) , d ∈ (0, 1). from (1.1) , we get |a0 (z)| ≤ ∣∣∣∣∣f (k)f ∣∣∣∣∣ + |ak−1 (z)| ∣∣∣∣∣f (k−1)f ∣∣∣∣∣ + · · · + |a1 (z)| ∣∣∣∣f ′f ∣∣∣∣ . (3.2) by (1.3), we know that ∃γ ∈r : lim inf |z|→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z| ))µ > γ > α. obviously logp |a0 (z)|( logq−1 ( 1 1−|z| ))µ > γ > α ≥ 0 (3.3) as |z|→ 1− for z ∈ h. by (1.4) and (3.3) , we obtain |a0 (z)| > expp { γ ( logq−1 ( 1 1 −|z| ))µ} > expp { α ( logq−1 ( 1 1 −|z| ))µ} ≥ |ai (z)| (i = 1, 2, ...,k − 1) (3.4) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 13 as |z|→ 1− for z ∈ h. applying (3.1) and (3.4) into (3.2) , we have expp { γ ( logq−1 ( 1 1 −|z| ))µ} ≤ |a0 (z)| ≤ k [( 1 1 −|z| )2+ε max { log ( 1 1 −|z| ) ,t (s (|z|) , f ) }]k ×expp { α ( logq−1 ( 1 1 −|z| ))µ} holds for all z satisfying |z| ∈ h1\e1 as |z|→ 1−. noting that γ > α, by the last inequality, weobtain exp ( (1 −o (1)) expp−1 { γ ( logq−1 ( 1 1 −|z| ))µ}) ≤ k ( 1 1 −|z| )k(2+ε) tk (s (|z|) , f ) (3.5) for all z satisfying |z| ∈ h1\e1 as |z| → 1−. then, by (3.5) and combining with lemma 2.3, weget for all r = |z| ∈ h1 exp ( (1 −o(1)) expp−1 { γ ( logq−1 ( 1 1 − r ))µ}) ≤ k ( 1 1 − s (r) )k(2+ε) tk (s1 (r) , f ) , (3.6) where s1 (r) = 1 − d2 (1 − r) with d ∈ (0, 1). therefore, from (3.6) we obtain σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = lim sup s1(r)→1− log+p+1t (s1 (r) , f ) logq ( 1 1−s1(r) ) ≥ µ. (3.7) by lemma 2.4, we get σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≤ max { σm,[p,q] (ai ) : i = 0, 1, ...,k − 1 } = σm,[p,q] (a0) = µ. (3.8)therefore, by (3.7) and (3.8) , we obtain σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = σm,[p,q] (a0) = µ. proof of theorem 1.2. set α0 = lim inf |z|→1−,z∈h logp |a0 (z)|( logq−1 ( 1 1−|z| ))µ , αi = lim sup |z|→1−,z∈h logp |ai (z)|( logq−1 ( 1 1−|z| ))µ , (i = 1, 2, ...,k − 1) . https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 14 by (1.5), there exist real numbers α,γ such that αi < α < γ < α0, i = 1, 2, ...,k − 1. it yields logp |ai (z)|( logq−1 ( 1 1−|z| ))µ < α < γ < logp |a0 (z)|( logq−1 ( 1 1−|z| ))µ as |z| → 1− for z ∈ h. hence, we have (3.4) as |z| → 1− for z ∈ h. then, by using the sameproof of theorem 1.1, we get σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ µ and by lemma 2.4 we obtain the conclusion of theorem 1.2. proof of theorem 1.3. suppose that every solution f of equation (1.1) not being identically equalto 0. by (1.1) , we can write −a0 (z) = f (k)(z) f (z) + ak−1 (z) f (k−1)(z) f (z) + · · · + a1 (z) f ′(z) f (z) . (3.9) from (3.9) , we obtain t (r,a0) = m(r,a0) ≤ k−1∑ i=1 m(r,ai ) + k∑ i=1 m ( r, f (i) f ) + o(1) = k−1∑ i=1 t (r,ai ) + k∑ i=1 m ( r, f (i) f ) + o(1). (3.10) if p ≥ q ≥ 2, then by (1.6), we know that ∃γ ∈r : lim inf |z|→1−,z∈h logp−1t (r,a0)( logq−1 ( 1 1−|z| ))µ > γ > α. obviously logp−1t (r,a0)( logq−1 ( 1 1−|z| ))µ > γ > α ≥ 0 (3.11) as |z|→ 1− for z ∈ h. by (1.7) and (3.11) , we obtain t (r,a0) > expp−1 { γ ( logq−1 ( 1 1 −|z| ))µ} > expp−1 { α ( logq−1 ( 1 1 −|z| ))µ} ≥ t (r,ai ) , (i = 1, 2, ...,k − 1) (3.12) as |z|→ 1− for z ∈ h. by applying lemma 2.2 and substituting (3.12) into (3.10) , we get expp−1 { γ ( logq−1 ( 1 1 − r ))µ} ≤ (k − 1) expp−1 { α ( logq−1 ( 1 1 − r ))µ} +o ( log+t (r, f ) + log ( 1 1 − r )) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 15 for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. noting that γ > α, by the last inequality,we have exp { (1 −o (1)) expp−2 { γ ( logq−1 ( 1 1 − r ))µ}} ≤ o ( log+t (r, f ) + log ( 1 1 − r )) (3.13) for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. therefore, from (3.13) we obtain σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ µ. (3.14) by lemma 2.4, we get σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≤ max { σm,[p,q] (ai ) : i = 0, 1, ...,k − 1 } = σm,[p,q] (a0) = µ. (3.15)therefore, by (3.14) and (3.15) , we obtain σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) = µ. if p = q = 1, then by (1.8), we know that ∃γ ∈r : lim inf |z|→1−,z∈h t (r,a0)( 1 1−|z| )µ > γ > (k − 1) α. obviously t (r,a0)( 1 1−|z| )µ > γ > (k − 1) α ≥ 0 (3.16) as |z|→ 1− for z ∈ h. by (1.9) and (3.16) , we obtain t (r,a0) > γ ( 1 1 −|z| )µ > (k − 1) α ( 1 1 −|z| )µ ≥ α ( 1 1 −|z| )µ ≥ t (r,ai ) , (i = 1, 2, ...,k − 1) (3.17) as |z|→ 1− for z ∈ h. by applying lemma 2.2 and substituting (3.17) into (3.10) , we get γ ( 1 1 − r )µ ≤ (k − 1) α ( 1 1 − r )µ +o ( log+t (r, f ) + log ( 1 1 − r )) for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. noting that γ > (k − 1) α, by the lastinequality, we have (γ − (k − 1) α) ( 1 1 − r )µ ≤ o ( log+t (r, f ) + log ( 1 1 − r )) (3.18) for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. therefore, from (3.18) we obtain σ (f ) = σm (f ) = ∞ and σ2 (f ) = σm,2 (f ) ≥ µ. (3.19) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 16 by lemma 2.4, we get σ2 (f ) = σm,2 (f ) ≤ max{σm (ai ) : i = 0, 1, ...,k − 1} = σm (a0) = µ. (3.20) therefore, by (3.19) and (3.20) , we obtain σ (f ) = σm (f ) = ∞ and σ2 (f ) = σm,2 (f ) = µ. proof of theorem 1.4. if p ≥ q ≥ 2, we set α0 = lim inf |z|→1−,z∈h logp−1t (r,a0)( logq−1 ( 1 1−|z| ))µ , αi = lim sup |z|→1−,z∈h logp−1t (r,ai )( logq−1 ( 1 1−|z| ))µ , (i = 1, 2, ...,k − 1) . by (1.10), there exist real numbers α,γ such that αi < α < γ < α0, i = 1, 2, ...,k − 1. it yields logp−1t (r,ai )( logq−1 ( 1 1−|z| ))µ < α < γ < logp−1t (r,a0)( logq−1 ( 1 1−|z| ))µ as |z|→ 1− for z ∈ h. hence, we have t (r,a0) > expp−1 { γ ( logq−1 ( 1 1 −|z| ))µ} > expp−1 { α ( logq−1 ( 1 1 −|z| ))µ} ≥ t (r,ai ) , (i = 1, 2, ...,k − 1) as |z|→ 1− for z ∈ h. then, by using the same proof of theorem 1.3, we get σ[p,q] (f ) = σm,[p,q] (f ) = ∞ and σ[p+1,q] (f ) = σm,[p+1,q] (f ) ≥ µ, and by lemma 2.4 we obtain the conclusion of theorem 1.4.if p = q = 1, we set α0 = lim inf |z|→1−,z∈h t (r,a0)( 1 1−|z| )µ , αi = lim sup |z|→1−,z∈h (k − 1) t (r,ai )( 1 1−|z| )µ , (i = 1, 2, ...,k − 1) . by (1.11), there exist real numbers α,γ such that αi < α < γ < α0, i = 1, 2, ...,k − 1. it yields (k − 1) t (r,ai )( 1 1−|z| )µ < α < γ < t (r,a0)( 1 1−|z| )µ (3.21) as |z|→ 1− for z ∈ h. by (3.21) , we obtain t (r,a0) > γ ( 1 1 −|z| )µ > α ( 1 1 −|z| )µ ≥ (k − 1) t (r,ai ) , (i = 1, 2, ...,k − 1) (3.22) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 17 as |z|→ 1− for z ∈ h. by applying lemma 2.2 and substituting (3.22) into (3.10) , we get γ ( 1 1 − r )µ ≤ α ( 1 1 − r )µ + o ( log+t (r, f ) + log ( 1 1 − r )) for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. noting that γ > α, by the last inequality,we have (γ −α) ( 1 1 − r )µ ≤ o ( log+t (r, f ) + log ( 1 1 − r )) (3.23) for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. therefore, from (3.23) we obtain σ (f ) = σm (f ) = ∞ and σ2 (f ) = σm,2 (f ) ≥ µ. (3.24) by lemma 2.4, we get σ2 (f ) = σm,2 (f ) ≤ max{σm (ai ) : i = 0, 1, ...,k − 1} = σm (a0) = µ. (3.25) therefore, by (3.24) and (3.25) , we obtain σ (f ) = σm (f ) = ∞ and σ2 (f ) = σm,2 (f ) = µ. proof of theorems 1.5 and 1.6. suppose that every meromorphic (or analytic) solution f ofequation (1.2) not being identically equal to 0. from (1.2) , we get a0 (z) ≤ |ak (z)| ∣∣∣∣∣f (k)f ∣∣∣∣∣ + |ak−1 (z)| ∣∣∣∣∣f (k−1)f ∣∣∣∣∣ + · · · + |a1 (z)| ∣∣∣∣f ′f ∣∣∣∣ . (3.26) by using a similar proof as in theorem 1.1 or theorem 1.2, we obtain |a0 (z)| > expp { γ ( logq−1 ( 1 1 −|z| ))µ} > expp { α ( logq−1 ( 1 1 −|z| ))µ} ≥ |ai (z)| (i = 1, 2, ...,k) (3.27) for |z| ∈ h1\e1 as |z|→ 1−. applying (3.1) and (3.27) into (3.26) , we get expp { γ ( logq−1 ( 1 1 −|z| ))µ} ≤ |a0 (z)| ≤ k [( 1 1 −|z| )2+ε max { log ( 1 1 −|z| ) ,t (s (|z|) , f ) }]k ×expp { α ( logq−1 ( 1 1 −|z| ))µ} for all z satisfying |z| ∈ h1\e1 as |z|→ 1−. noting that γ > α, by the last inequality, we have exp ( (1 −o (1)) expp−1 { γ ( logq−1 ( 1 1 −|z| ))µ}) ≤ k ( 1 1 −|z| )k(2+ε) tk (s (|z|) , f ) (3.28) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 18 for all z satisfying |z| ∈ h1\e1 as |z| → 1−. then, by (3.28) and combining with lemma 2.3, weget for all r = |z| ∈ h1 exp ( (1 −o(1)) expp−1 { γ ( logq−1 ( 1 1 − r ))µ}) ≤ k ( 1 1 − s (r) )k(2+ε) tk (s1 (r) , f ) , (3.29) where s1 (r) = 1−d2 (1 − r) with d ∈ (0, 1). therefore, from (3.29) we obtain σ[p,q] (f ) = ∞ and σ[p+1,q] (f ) = lim sup s1(r)→1− log+p+1t (s1 (r) , f ) logq ( 1 1−s1(r) ) ≥ µ. proof of theorems 1.7 and 1.8. suppose that every meromorphic (or analytic) solution f ofequation (1.2) not being identically equal to 0. by (1.2) , we can write −a0 (z) = ak (z) f (k)(z) f (z) + ak−1 (z) f (k−1)(z) f (z) + · · · + a1 (z) f ′(z) f (z) . (3.30) from (3.30) , we have t (r,a0) = m(r,a0) ≤ k∑ i=1 m(r,ai ) + k∑ i=1 m ( r, f (i) f ) + o(1) = k∑ i=1 t (r,ai ) + k∑ i=1 m ( r, f (i) f ) + o(1). (3.31) if p ≥ q ≥ 2, then by using a similar proof as in theorem 1.3 or theorem 1.4, we obtain t (r,a0) > expp−1 { γ ( logq−1 ( 1 1 −|z| ))µ} > expp−1 { α ( logq−1 ( 1 1 −|z| ))µ} ≥ t (r,ai ) , (i = 1, 2, ...,k) (3.32) as |z|→ 1− for z ∈ h. by applying lemma 2.2 and substituting (3.32) into (3.31) , we get expp−1 { γ ( logq−1 ( 1 1 − r ))µ} ≤ k expp−1 { α ( logq−1 ( 1 1 − r ))µ} +o ( log+t (r, f ) + log ( 1 1 − r )) for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. noting that γ > α, by the last inequality,we have exp { (1 −o (1)) expp−2 { γ ( logq−1 ( 1 1 − r ))µ}} ≤ o ( log+t (r, f ) + log ( 1 1 − r )) (3.33) for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. therefore, from (3.33) we obtain σ[p,q] (f ) = ∞ and σ[p+1,q] (f ) ≥ µ. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 19 if p = q = 1, then by using a similar proof as in theorem 1.3 or theorem 1.4, we get t (r,a0) > γ ( 1 1 −|z| )µ > kα ( 1 1 −|z| )µ > α ( 1 1 −|z| )µ ≥ t (r,ai ) , (i = 1, 2, ...,k) (3.34) as |z|→ 1− for z ∈ h. by applying lemma 2.2 and substituting (3.34) into (3.31) , we obtain γ ( 1 1 − r )µ ≤ kα ( 1 1 − r )µ + o ( log+t (r, f ) + log ( 1 1 − r )) for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. noting that γ > kα, by the last inequality,we have (γ −kα) ( 1 1 − r )µ ≤ o ( log+t (r, f ) + log ( 1 1 − r )) (3.35) for all z satisfying |z| = r ∈ h1\e2 as |z| = r → 1−. therefore, from (3.35) we obtain σ (f ) = σm (f ) = ∞ and σ2 (f ) = σm,2 (f ) ≥ µ. 4. proof of theorem 1.9 suppose that every solution f of equation (1.1) not being identically equal to 0. first step. we consider the fixed points of f . define the function g by setting g (z) := f (z) −z, z ∈ d. it follows from (1.1) that g(k) + ak−1g (k−1) + · · · + a1g′ + a0g = −a1 −za0 (4.1) and by theorem 1.1 or theorem 1.2, we get σ[p,q] (g) = σ[p,q] (f ) = ∞, σ[p+1,q] (g) = σ[p+1,q] (f ) = µ, λ̄[p+1,q] (g) = λ̄[p+1,q] (f −z) . (4.2) now, we prove that −a1 − za0 6≡ 0. assume that −a1 − za0 ≡ 0. clearly a0 6≡ 0. then lim |z|→1−,z∈h ∣∣∣a1a0∣∣∣ = 1 and by (3.4), we have ∣∣∣∣a1 (z)a0 (z) ∣∣∣∣ < expp { α ( logq−1 ( 1 1−|z| ))µ} expp { γ ( logq−1 ( 1 1−|z| ))µ} = 1 exp { (1 −o (1)) expp−1 { γ ( logq−1 ( 1 1−|z| ))µ}} → 0 https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 20 as |z| → 1− for z ∈ h. then lim |z|→1−,z∈h ∣∣∣a1a0∣∣∣ = 0. it is easy to see the contradiction. hence, −a1 −za0 6≡ 0. next by lemma 2.5, we get max { σ[p,q] (ai ) (i = 0, 1, ...,k − 1) ,σ[p,q] (−a1 −za0) } < ∞. we deduce, by using (4.1) , (4.2) and lemma 2.6 that λ̄[p,q] (g) = σ[p,q] (g) = ∞, λ̄[p+1,q] (g) = σ[p+1,q] (g) = µ. therefore, we obtain λ̄[p,q] (f −z) =λ̄[p,q] (g) = σ[p,q] (g) = σ[p,q] (f ) = ∞, λ̄[p+1,q] (f −z) = λ̄[p+1,q] (g) = σ[p+1,q] (g) = σ[p+1,q] (f ) = µ. second step. for the following proof, we use the principle of mathematical induction. set ak (z) ≡ 1, then |ak (z)| ≤ expp { α ( logq−1 ( 1 1 −|z| ))µ} and equation (1.1) becomes (1.2) . we consider the fixed points of f (j) (z) (j = 1, 2, ...). definethe function g1 by setting g1 (z) := f ′ (z) −z, z ∈ d. then, by lemma 2.5 and (4.2) , we have σ[p,q] (g1) = σ[p,q] (f ′) = ∞, σ[p+1,q] (g1) = σ[p+1,q] (f ′) = µ, λ̄[p+1,q] (g1) = λ̄[p+1,q] (f ′ −z) . (4.3) dividing both sides of (1.2) by a0, we obtain ak a0 f (k) + ak−1 a0 f (k−1) + · · · + a1 a0 f ′ + f = 0. (4.4) it follows, by differentiating both sides of equation (4.4) that ak a0 f (k+1) + (( ak a0 )′ + ak−1 a0 ) f (k) + · · · + (( a2 a0 )′ + a1 a0 ) f ′′ + (( a1 a0 )′ + 1 ) f ′ = 0. (4.5) multiplying (4.5) by a0, we have ak,1f (k+1) + ak−1,1f (k) + · · · + a1,1f ′′ + a0,1f ′ = 0. (4.6) substituing f ′ = g1 + z into (4.6) , we obtain ak,1g (k) 1 + ak−1,1g (k−1) 1 + · · · + a1,1g ′ 1 + a0,1g1 = f1, (4.7) where ak,1 = ak = 1, ai,1 = a0 (( ai+1 a0 )′ + ai a0 ) (i = 1, 2, ...,k − 1) , (4.8) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 21 a0,1 = a0 (( a1 a0 )′ + 1 ) , (4.9) f1 = −(a1,1 + za0,1) . (4.10) next, we prove that a0,1 6≡ 0 and f1 6≡ 0. assume that a0,1 ≡ 0, then a1a0 = −z + c0, where c0 isan arbitrary constant. hence, we have a1 + (z −c0) a0 = 0. then, f0 = z −c0 is a solution of (1.1) and σ[p,q] (f0) < ∞. this contradicts (4.2) . now, assume that f1 ≡ 0. by (4.6) and (4.10) ,we know that the function f1 such that f ′1 = z is a solution of equation (4.6) and σ[p,q] (f1) < ∞.this contradicts (4.2) . therefore, a0,1 6≡ 0 and f1 6≡ 0. it follows by (4.8) − (4.10) and lemma2.5 that max { σ[p,q] (ai,1) (i = 0, 1, ...,k) ,σ[p,q] (f1) } < ∞. we deduce by using (4.3) , (4.7) and lemma 2.6 that λ̄[p,q] (g1) = σ[p,q] (g1) = ∞, λ̄[p+1,q] (g1) = σ[p+1,q] (g1) = µ. therefore, we obtain λ̄[p,q] ( f ′ −z ) = λ̄[p,q] (g1) = σ[p,q] (g1) = σ[p,q] (f ) = ∞, λ̄[p+1,q] ( f ′ −z ) = λ̄[p+1,q] (g1) = σ[p+1,q] (g1) = σ[p+1,q] (f ) = µ.set g2 (z) = f ′′ (z) − z, z ∈ d. then, by using a similar discussion as in the case of the function g1, we can get ak,2f (k+2) + ak−1,2f (k+1) + · · · + a1,2f (3) + a0,2f ′′ = 0 and ak,2g (k) 2 + ak−1,2g (k−1) 2 + · · · + a1,2g ′ 2 + a0,2g2 = f2,where ak,2 = 1, ai,2 = a0,1 (( ai+1,1 a0,1 )′ + ai,1 a0,1 ) (i = 1, 2, ...,k − 1) , a0,2 = a0,1 (( a1,1 a0,1 )′ + 1 ) , f2 = −(a1,2 + za0,2) . therefore, by the same procedure as for g1, we obtain λ̄[p,q] ( f ′′ −z ) = λ̄[p,q] (g2) = σ[p,q] (g2) = σ[p,q] (f ) = ∞, λ̄[p+1,q] ( f ′′ −z ) = λ̄[p+1,q] (g2) = σ[p+1,q] (g2) = σ[p+1,q] (f ) = µ.now, assume that a0,s 6≡ 0, λ̄[p,q] ( f (s) −z ) = σ[p,q] (f ) = ∞, λ̄[p+1,q] ( f (s) −z ) = σ[p+1,q] (f ) = µ (4.11) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 22 for all s = 0, 1, ..., j−1, and we prove that for s = j we have (4.11) holds. set gj (z) = f (j) (z)−z, z ∈ d. then, by using (4.2) , we obtain σ[p,q] ( gj ) = σ[p,q] ( f (j) ) = ∞, σ[p+1,q] ( gj ) = σ[p+1,q] ( f (j) ) = µ, λ̄[p+1,q] ( gj ) = λ̄[p+1,q] ( f (j) −z ) . (4.12) by following the same procedure as before, we have ak,jf (k+j) + ak−1,jf (k+j−1) + · · · + a1,jf (j+1) + a0,jf (j) = 0 (4.13) and ak,jg (k) j + ak−1,jg (k−1) j + · · · + a1,jg′j + a0,jgj = fj, (4.14)where ak,j = 1, ai,j = a0,j−1 (( ai+1,j−1 a0,j−1 )′ + ai,j−1 a0,j−1 ) (i = 1, 2, ...,k − 1) , a0,j = a0,j−1 (( a1,j−1 a0,j−1 )′ + 1 ) 6≡ 0 (a0,0 = a0, a1,0 = a1) , fj = − ( a1,j + za0,j ) 6≡ 0. we deduce by applying lemma 2.6 in (4.14) that λ̄[p,q] ( f (j) −z ) = λ̄[p,q] ( gj ) = σ[p,q] ( gj ) = σ[p,q] ( f (j) ) = ∞, λ̄[p+1,q] ( f (j) −z ) = λ̄[p+1,q] ( gj ) = σ[p+1,q] ( gj ) = σ[p+1,q] ( f (j) ) = µ (j = 1, 2, ...) . therefore, we obtain λ̄[p,q] ( f (j) −z ) = λ̄[p,q] (f −z) = σ[p,q] (f ) = ∞, λ̄[p+1,q] ( f (j) −z ) = λ̄[p+1,q] (f −z) = σ[p+1,q] (f ) = µ (j = 1, 2, ...) . 5. proofs of theorem 1.10 and 1.11 proof of theorem 1.10. suppose that every solution f of equation (1.1) not being identicallyequal to 0. by applying theorem 1.3 or theorem 1.4, we get σ[p,q] (f ) = ∞, σ[p+1,q] (f ) = µ. now, we prove that −a1 −za0 6≡ 0. assume that −a1 −za0 ≡ 0, then we can easily obtain t (r,a1) = t (r,−za0) ≤ t (r,a0) + t (r,z) , t (r,a0) = t ( r, a1−z ) ≤ t (r,a1) + t (r,z) + o (1) . (5.1) it follows from (5.1) that 1 − t (r,z) + o (1) t (r,a0) ≤ t (r,a1) t (r,a0) ≤ 1 + t (r,z) t (r,a0) . (5.2) https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 23 by following the same reasoning as in the proof of theorem 1.3 or theorem 1.4, we have t (r,a0) > expp−1 { γ ( logq−1 ( 1 1 −|z| ))µ} > expp−1 { α ( logq−1 ( 1 1 −|z| ))µ} ≥ t (r,a1) (5.3) as r = |z|→ 1− for z ∈ h. by using (5.3) , we obtain t (r,z) t (r,a0) ≤ t (r,z) expp−1 { γ ( logq−1 ( 1 1−|z| ))µ} → 0 (5.4) as |z|→ 1− for z ∈ h. then, by (5.2) and (5.4) , we get lim |z|→1−,z∈h t (r,a1) t (r,a0) = 1. (5.5) on the other hand, we have for p = q = 1 t (r,a1) t (r,a0) < α γ < 1 (5.6) and for p ≥ q ≥ 2 t (r,a1) t (r,a0) < expp−1 { α ( logq−1 ( 1 1−|z| ))µ} expp−1 { γ ( logq−1 ( 1 1−|z| ))µ} → 0 (5.7) as |z|→ 1− for z ∈ h. it follows by (5.6) and (5.7) that lim |z|→1−,z∈h t (r,a1) t (r,a0) 6= 1. (5.8) obviously, (5.5) contradicts with (5.8). hence, −a1 −za0 6≡ 0. set ak (z) ≡ 1, then t (r,ak) ≤ expp−1 { α ( logq−1 ( 1 1−|z| ))µ} . clearly, a0 6≡ 0. we can get the conclusion of theorem 1.10, byreasoning in the same way as we did in the proof of theorem 1.9. proof of theorem 1.11. suppose that every meromorphic (or analytic) solution f of equation (1.2)not being identically equal to 0. by applying one of theorem 1.5 to theorem 1.8, we get σ[p,q] (f ) = ∞, σ[p+1,q] (f ) ≥ µ. then, we can get the conclusion of theorem 1.11, by reasoning in the same way as we did in theproof of theorem 1.9 and theorem 1.10 by using σ[p+1,q] (f ) ≥ µ instead of σ[p+1,q] (f ) = µ and σ[p+1,q] ( f (j) ) ≥ µ instead of σ[p+1,q](f (j)) = µ (j = 1, 2, ...) . https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 24 6. examples example 6.1 consider the following differential equation f ′′ + k1 (z) exp4 {( log2 ( 1 1 −z ))5} f ′ + k0 (z) exp4 { 3 ( log2 ( 1 1 −z ))5} f = 0, (6.1) where k0 and k1 are analytic functions in the unit disc d such that |k0| > 1, |k1| < 1 and max { σm,[4,3] (k0) ,σm,[4,3] (k1) } < 5. in the equation (6.1) , we have a0 (z) = k0 (z) exp4 { 3 ( log2 ( 1 1 −z ))5} , a1 (z) = k1 (z) exp4 {( log2 ( 1 1 −z ))5} . then max { σm,[4,3] (a0) ,σm,[4,3] (a1) } = 5. let h = {z ∈c : |z| = r < 1 and arg z = 0}⊂ d be a set of complex numbers satisfying densd {|z| : z ∈ h} = 1 > 0. then |a0 (z)| = |k0 (z)| ∣∣∣∣∣exp4 { 3 ( log2 ( 1 1 −z ))5}∣∣∣∣∣ > exp4 { 3 ( log2 ( 1 1 − r ))5} ⇒ log4 |a0 (z)|( log2 ( 1 1−r ))5 > 3 ⇒ lim inf r→1−,z∈h log4 |a0 (z)|( log2 ( 1 1−r ))5 ≥ 3 > 1, and |a1 (z)| = |k1 (z)| ∣∣∣∣∣exp4 {( log2 ( 1 1 −z ))5}∣∣∣∣∣ ≤ exp4 {( log2 ( 1 1 − r ))5} as r → 1− for z ∈ h. it is clear that the conditions of theorem 1.1 hold with α = 1,µ = 5, p = 4and q = 3 on the set h. by theorem 1.1, every solution f 6≡ 0 of equation (6.1) satisfies σ[4,3] (f ) = σm,[4,3] (f ) = ∞ and σ[5,3] (f ) = σm,[5,3] (f ) = σm,[5,3] (a0) = 5. https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 25 example 6.2 consider the following differential equation k2 (z) exp4 {( log2 ( 1 1−z ))7} f ′′+k1 (z) exp4 { 2 ( log2 ( 1 1−z ))7} f ′ + k0 (z) exp4 { 5 ( log2 ( 1 1−z ))7} f = 0, (6.2) where k0,k1 and k2 are analytic functions in the unit disc d such that |k0| > 1, |k1| < 1, |k2| < 1and max { σm,[4,3] (k0) ,σm,[4,3] (k1) ,σm,[4,3] (k2) } < 7. in the equation (6.2) , we have a0 (z) = k0 (z) exp4 { 5 ( log2 ( 1 1−z ))7} , a1 (z) = k1 (z) exp4 { 2 ( log2 ( 1 1−z ))7} , a2 (z) = k2 (z) exp4 {( log2 ( 1 1−z ))7} . then max { σm,[4,3] (a0) ,σm,[4,3] (a1) ,σm,[4,3] (a2) } = 7. let h = {z ∈c : |z| = r < 1 and arg z = 0}⊂ d be a set of complex numbers satisfying densd {|z| : z ∈ h} = 1 > 0. then |a0 (z)| = |k0 (z)| ∣∣∣exp4{5 (log2( 11−z))7}∣∣∣ > exp4 { 5 ( log2 ( 1 1−r ))7} ⇒ log4 |a0 (z)|( log2 ( 1 1−r ))7 > 5 ⇒ lim inf r→1−,z∈h log4 |a0 (z)|( log2 ( 1 1−r ))7 ≥ 5 > 2, and |a1 (z)| = |k1 (z)| ∣∣∣∣∣exp4 { 2 ( log2 ( 1 1 −z ))7}∣∣∣∣∣ ≤ exp4 { 2 ( log2 ( 1 1 − r ))7} |a2 (z)| = |k2 (z)| ∣∣∣∣∣exp4 {( log2 ( 1 1 −z ))7}∣∣∣∣∣ ≤ exp4 { 2 ( log2 ( 1 1 − r ))7} https://doi.org/10.28924/ada/ma.3.10 eur. j. math. anal. 10.28924/ada/ma.3.10 26 as r → 1− for z ∈ h. it is clear that the conditions of theorem 1.5 hold with α = 2, µ = 7, p = 4and q = 3 on the set h. by theorem 1.11, every meromorphic (or analytic) solution f 6≡ 0 ofequation (6.2) satisfies λ̄[4,3] ( f (j) −z ) = λ̄[4,3] (f −z) = σ[4,3] (f ) = ∞ and λ̄[5,3] ( f (j) −z ) = λ̄[5,3] (f −z) = σ[5,3] (f ) ≥ 7, (j = 1, 2, ...) . references [1] s. bank, general theorem concerning the growth of solutions of first-order algebraic differential equations. compos.math. 25 (1972) 61–70. http://www.numdam.org/item/cm_1972__25_1_61_0.[2] b. belaïdi, estimation of the hyper-order of entire solutions of complex linear ordinary differential equations whosecoefficients are entire functions. electron. j. qual. theory diff. equ. 2002 (2002) 5. 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