Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 34 https://internationalpubls.com Coincidence Point Results for a New Class of Fuzzy Contractive Mappings Prapoorna Manthena1, K Santosh Reddy2, Ch Shashi Kumar3, V Lakshmi Narayana4, Divvela Surendra5, N Veerraju6 1Department of Mathematics, TSWRDCW LB Nagar, Hyderabad, Telangana, India Email: m.prapoorna@gmail.com 2 Assistant Professor Department of Mathematics, Vardhaman College of Engineering, Hyderabad, Telangana, India Email: santureddyk@gmail.com 3Assistant Professor, Department of Mathematics, VNR Vignana Jyothi Institute of Engineering and Technology, Hyderabad -500090, Telangana,India.Mailid: skch17@gmail.com. 4Assistant Professor, Department of Mathematics, Vardhaman College of Engineering, Hyderabad. Telangana, India. Email.ID: vlnarayana@vardhaman.org, 5Assistant Professor, Department of English, Koneru Lakshmaiah Education Foundation (KL Deemed to be University) Vaddeswaram, Guntur, Andhra Pradesh, India, Pin code: 522502 Email: dsurendra@kluniversity.in 6Assistant Professor, Department of Engineering Mathematics, SRKR Engineering College, Bhimavaram, West Godavari District, Andhra Pradesh, India, Pincode: 534204 Email: veerrajunalla@gmail.com Article History: Received: 22-01-2024 Revised: 28-03-2024 Accepted: 20-04-2024 Abstract: This work is mainly aimed to enhance the concept of fuzzy Z -contractive mapping [11] by establish a different contraction called fuzzy ( ), gZ. - contraction. we obtain some sufficient conditions for the existence and uniqueness of point of coincidence o f fuzzy ( ), gZ. - contraction mappings in the context of fuzzy metric spaces and prove some intriguing results. Keywords: Fixed point, M -completeness, Fuzzy metric space, Fuzzy Z-contractive mapping. 2010 M a t h e m a t i c s Subject Classification: 54H25; 47H10 1. Introduction In light of its numerous uses across a variety of fields, the analysis of fixed points of mappings following contraction conditions has become the focus of numerous research initiatives. Regarding this, the classical findings of Banach [1] and Edelstien [2] have served as the inspiration for several writers working in the field of metric fixed-point theory. Grabiec [4], who pioneered the introduction of fixed-point theory in fuzzy metric spaces, extended the discoveries of Banach [1] and Edelstein [2] in the framework of fuzzy metric spaces in the design of Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 35 https://internationalpubls.com Kramosil and Michlek [8]. The fixed-point conclusion proposed by Grabiec in his work [4] was based on the notion of the -completeness of fuzzy metric spaces, which George and Veeramani [3] weakened by introducing. Many authors explored and examined various contractions in fuzzy metric spaces, and they came up with insightful results. (See [5], [6], [9], [12], [13]). In order to integrate various types of fuzzy contractive mappings, Shukla et al. [11] recently introduced fuzzy Z − contractive mappings. They demonstrated that this new type is more thorough than the well- known earlier ones and properly incorporates the classes that were proposed by numerous researchers ([5], [9], [12], and [13]) in fuzzy metric spaces in the sense of George and Veeramani [3]. By expanding on the work of Shukla et al. [11], we introduce the concept of fuzzy ( ), g −Z. contraction in this paper. In the context of fuzzy metric spaces, we also develop certain discoveries regarding the presence and uniqueness of coincidence points for such contractions. 2. Preliminaries Let us recall some prefaces here: Definition 2.1. [10] A continuous t -norm is a binary operation      : 0,1 0,1 0,1  → which satisfies the following conditions  , , , 0,1l p r t  (1) 1l l = (2) l p p l =  (3) l p r t   whenever &l r p t  (4) ( ) ( )l p r l p r  =   Definition 2.2. [3] A fuzzy metric space is a 3-tuple ( ), ,X M where X is an arbitrary set, is a continuous t -norm and M is a fuzzy set in ( )0,X X   , satisfying the following conditions: (1) ( ), , 0a b t M , (2) ( ), , 1, 0a b t t a b=    =M , (3) ( ) ( ), , , ,a b t b a t=M M , (4) ( ) ( ) ( ), , , , , ,a c t r a b t b c r+  M M M , (5) ( ) ( ) ( , , : 0, 0,1a b   →M is a continuous function, , , & , 0a b c X t r   . Lemma 2.3. [4] ( ), ,a b M is nondecreasing ,a b X  . Definition 2.4. [3] Let ( ), ,X M be a fuzzy metric space. A sequence  qs in X is a M -Cauchy sequence, if ( )0,1 & 0,t   there exists 0q N such that ( ) 0, , 1 ,p qs s t p q −  M . A sequence  qs in X is convergent to x X if ( ) l1 m , ,i q q s x t → =M for each 0t  . If every −M Cauchy sequence in X is convergent, then such fuzzy metric space X is said to be −M complete.  Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 36 https://internationalpubls.com Definition 2.5. [7] On a fuzzy metric space ( ), ,X M , two self-mappings &  are said to be compatible if ( ) ll ,im ,q q qx x t  → =M for each 0t  , whenever  qx is a sequence in X .1im 1imq q q q x x x X  → → = =  Definition 2.6. [7] Two self-maps &  of a fuzzy metric space ( ), ,X M are said to be weakly compatible if they commute at their coincidence points; x x = for some x X implies that x x = . Remark 2.7. Let &  be weakly compatible s e l f -maps of a set X . If &  have a unique point of coincidence ,w x x = = then w is the unique common fixed point of &  . Definition 2.8. [11] Let the family of all functions (  ( : 0, l 0, l  → satisfying, ( ) ( ), , , 0, lt r r t r    be denoted by Z . Remark 2 . 9 . [11] From the above definition, we can see that , , 0, l .( ) ( )t t t t    Example 2.10. [11] Consider the following functions (  ( : 0, l 0, l  → by: (1) ( ) ( ), ,p q q = where (  ( : 0, l   0, l → is a function such that ( ) ,q q ,0( ),1q  (2) ( ) l  ,       ,p q p q p  = + + (3) ( ),  . q p q p  = Then, in all the cases   . Z Definition 2.11. [11] Let : X X → be a self - mapping on a fuzzy metric space ( ), , .X M Suppose,    Z such that, ( ) ( ) ( )( ), , , , , , ,    x y t x y t x y t    M M M (2.1) for all , , , 0. x y X x y t  =  Then  is called a fuzzy −Z contractive mapping with respect to . Z Remark 2.12. Even in the −M complete fuzzy metric space, a fuzzy −Z contractive mapping may not have a fixed point. (See example 3.10. of [11]) S. Shukla et al. [11] considered a space with the additional characteristic listed below to guarantee the existence of the fixed point of a fuzzy −Z contractive mapping. Definition 2.13. [11] Let  :       X X → be a mapping in a fuzzy metric space ( ), ,  X M and   . Z Then , we say that the quadruple ( ), , ,X  M has the property ( )S , if for any Picard sequence { }qs with initial value , . .,   ,q qx X i e s x q N =   such that l 1 0( ) ( )inf , , inf , ,     ,,q p q p q q s s t s s t q N t+ +     M M p p implies that 1 1 .1 ), ,im ( ( ), , l), 0(q q p q p p q inf s s t s s t t + + →  =  M M (2.2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 37 https://internationalpubls.com Theorem 2.14. [11] Let :      X X → be a fuzzy −Z contraction on an −M complete fuzzy metric space , , . ( )X M If the quadruple ( ), , ,X  M has the property ( ) ,S then f has a unique fixed point .z X 3. Main Results Definition 3.1. Let &  be two self-mappings on a fuzzy metric space , , . ( )X M Then  is called a fuzzy ( ), −Z contraction if  Z such that ( ) ( ) ( )( ), , , , , , ,x y t x y t x y t      M M M (3.1) for all , , , 0.x y X x y t    Remark 3.2. If   I= (identity mapping) in (3.1), then we get fuzzy −Z contraction (Definition 2.11). Now, we state our result for the notion of fuzzy ( ), −Z contraction. Theorem 3.3. Let ( , , )X M be a fuzzy metric space and , :  X X  → be self-mappings. Let  be a fuzzy ( ), −Z contraction and the quadruple ( ), , ,X  M has the property ( )S . Also assume that, at least one of the following conditions hold: (i) ( ) ( )   X or X  is complete. (ii) X is complete,  is continuous and ,  are commuting. (iii) X is complete,  is continuous and ,  are compatible. Then &  have unique point of coincidence. Proof. Firstly, we shall show that the point of coincidence of &  , if exists is unique. Let us suppose that 1 2 &  z z are distinct points of coincidence of &  which follows that there exist two points ( )1 2 1 2 &    w w w w= such that 1 1 1 w w z = = and 2 2 2.w w z = = In view of (3.1) and the property of , we obtain ( ) ( ) ( ) ( )( )1 2 1 2 1 2 1 2, , , , , , , , ,z z t w w t w w t w w t      = M M M M 1 2 1 2( ) ( ), , , ,w w t z z t  =M M (3.2) which is a contradiction. Let us consider a sequence {  }qy such that 1q q qy x x  += = where { }0 .q N If 0 0 1    q qy y += for any 0 {    }  0 ,q N  then 0 0 0 01 l  1     q q q qx y y x + += = + = which shows that &  have a point of coincidence. Thus, we assume that 1  q qy y += for all     0 .{ }q N  Consider ( ) ( ) ( ) ( )( )l 2 l 2 l 2 1 2, , , , , , , , ,q q q q q q q qy y t x x t x x t x x t      + + + + + + + += M M M M 1 2 ),( ,q qx x t + + M ( )l, ,  q qy y t+= M (3.3) which gives ( ) ( )l l 2, , , , 0.q q q qy y t y y t t+ + +  M M If  q py y= for some ,qp then we have l l l l      .q q p py x x y + + + += = = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 38 https://internationalpubls.com Now using (3.1) and (3.3), one can get ( ) ( )( )l l l, , , , , , , 0.( )q q p p q qy y t y y t y y t t+ + + =  M M M M which is a contradiction. Thus, we assume that  q py y= for all distinct , .q Np Now we prove { }qy is a Cauchy sequence: For 0,t  let us suppose that ( ) ,( ).,q p p p q a t inf y y t  = M Consider ( )l l l l( ), , , ,q p q py y t x x t + + + +=M M ( ) ( )( )l l l l, , , , ,q p q px x t x x t    + + + + M M ( )l l ,, , , , 0.( )q p q px x t y y t t + + =  M M (3.4) Thus, ( ) ( )l l , ., , , ,q p q py y t y y qt p+ +  M M Taking infimum over all ( )qp in the above inequality, we get ( ) l l )inf , ,  inf , ,(q p q p p q p q y y t y y t+ +   M M ( ) ( )l. .,      .,q qi e a t a t q N+   Thus, ( ) qa t is monotonic and bounded for each 0.t  We prove ( ) 0lim l,q q a t t → =   . On the contrary, let us suppose that  some 0t , where 00 t t  such that ( ) ( )0 0lim l.q q a t a t → =  For such 0t , using the fact that the quadruple ( ), , ,X  M have the property ( )S ( )( )Since  ,q qy q x= = we obtain ( ) ( )( )l l 0 0liminf , , , , , l q p q p q p q y y t y y t + + →  =M M ( )  3.5 ( ) ( ) ( )( ) ( )l l 0 l l 0 l l 0 0inf , ,  inf ,  ,  , , , , ,q p q p q p p q q p p q p q y y t x x t x x t y y t     + + + + + +     MM M M ( ) ( ) ( )( ) ( )l l 0 l l 0 0 0  inf , ,   inf , , , , ,  inf , ,q p q p q p q p p q p q p q y y t y y t y y t y y t+ + + +      MMM M ( ) ( )( ) ( )l 0 l l 0 0 0. .,   inf , , , ,)  ( ,q q p q p q p q i e a t y y t y y t a t+ + +   MM Letting    q→ in the above inequality and using (3.5), we get ( )0 la t = which is a contradiction to our assumption. Thus, 1iminf , , l, 0.( )q p q p q y y t t →  =  M Hence from the definition of ,qa we get , 1im , , l, 0. ( )q p p q y y t t → =  M (3.6) which shows {  }qy is an −M Cauchy sequence. Suppose that (i) holds: ( ). .,  i e X is complete. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 39 https://internationalpubls.com Then since {  }qy is in ( ) , X it has a limit in ( ).X Let     . .,    q qy u i e x u  → → as  q→ for . u X Now, we shall prove that u is the coincidence point of and . ., .i e u u   = It is clear that, we can suppose ,qy u u  for all }0{n N . In view of (3.1) and the property of , we have ( ) ( ) ( )( ), , , , , , , , ,( )q q q qx u t x u t x u t x u t        M MM M Letting  q→ in the above inequality and considering the extremities gives ( ) ( ) ( ), , 1  im , , l 1im , , .n q q q u u t x u t x u t      → →   M M M Thus, as  qx u q → → which implies as q .qy ku→ → Since limit is unique, we obtain .u u = Hence, u u = is a (unique) point of coincidence of &   . Similarly, we can show that u u = is a (unique) point of coincidence of &   when ( )X is complete. Suppose that (ii) holds: Since X is complete, there exists    u X such that          as q   .q qy u x u→  → → Since  is continuous, we have 2 as .qx q → → In view of (3.1) and the property of , we have ( )( ) ( )( ) ( )( )( ) ( )( ), , , , , , , , ,q q q qx u t x u t x u t x u t            M M M M Letting  q→ in the above inequality and considering the extremities gives ( ) ( ) ( ) ( ), , 1  im , , 1im , ,  l 1im , , .q q n q q q u u t x u t x u t x u t        → → →  =  M MM M Thus, as q . qx u → → But from the definition of ,qy we get 2 l l q q qx x x  + += = from which the result follows . .,i e u u = is a (unique) point of coincidence of & .  Suppose that (iii) holds: Since X is complete, there exists  u X such that           as   . q qy u x u q→  → → Since  is continuous, we have as q .qx u → → In view of (3.1) and the property of  and 2 l l  ,q q qx x x  + += = we have ( ) ( )( ) ( )( ) ( )( )( ) ( )( )l , , , , , , , , , , ,q q q q qx u t x u t x u t x u t x u t            − =  M M M M M Letting  q→ in the above inequality and considering the extremities gives ( ) ( ) ( ), , 1  im , ,  l 1im , , .q q q q u u t x u t x u t      → →   M M M Thus as q .qx u → → Since ,  are compatible, we have ( )1im , , l, 0.q q q x x t t  → = M Consider ( ), , , ,      , ,      , ,    3 3 3 q n q q t t t u u t u x x x x u                            M M M M Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 40 https://internationalpubls.com Letting    q→ in the above inequality gives u u = thus showing that u u = is a (unique) point of coincidence of & .  Hence, the result is proved for all three cases (i), (ii) and (iii), i.e., the mappings &  have a unique point of coincidence. Theorem 3.4. Besides the hypotheses of Theorem 3.3, if &  are weakly compatible, then they have a unique common fixed point in .X Proof. Using Theorem 3.3, &  have unique point of coincidence. Further, if &  are weakly compatible, then according to Remark 2.7, they have a unique common fixed point in .X Example 3.5. Let l[ ]0,X = and M be the fuzzy set on ( )0,X X   defined by ( ), ,      t x y t t x y = + − M and  is minimum norm. Then, ( ), ,X M is an complete fuzzy metric space. Let { }qx be a sequence given by   .q N  Consider the function (  ( : 0, l 0, l    → by ( ), s s t t  = ( ,   0, l .t s  Let ,  be given by l 0 0, 4 7 1 l , 20 4 2 4 l , l 25 2 if x x if x if x             =              and l 0, 3 4 2 1 l , 5 4 2 2 l , l 3 2 x if x x if x if x             =              Then  Z and the quadruple ( ), , ,X  M has the property ( ). S Furthermore, the mapping  is a fuzzy ( ), −Z contractive mapping with respect to the function . Thus, all the conditions of Theorem (3.4) are satisfied i.e., the mappings and  have a coincidence point 0.x = On the other word, they have a unique common fixed point i.e., at 0.x = References [1] Banach, S. (1922), Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae, 3(1), 133 - 181. [2] Edelstein, M. (1962), On fixed and periodic points under contractive mappings, J.Lond. Math. 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