Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 227 https://internationalpubls.com Fixed Point Theorems in Revised Fuzzy Metric Space via 𝑹𝑭 −Contraction Parakath Nisha Bagam P1, Sandhya P2, Thangathamizh R3, Shanmugavel P4, Sarathbabu K5, Anusuya R6 1 PG & Research Department of Mathematics, Urumu Dhanalakshmi College, Bharathidasan University, Trichy 620019, India; parakath.nisha979@gmail.com 2 Department of Mathematics, SRM Trichy Arts & Science college, Trichy, India; sandhyaprasad2684@gmail.com 3 Department of Mathematics, K. Ramakrishnan College of Engineering, Samayapuram, Trichy 621112, India; thamizh1418@gmail.com 4 Department of Mathematics, Selvamm Arts & Science college, Periyar University, Namakkal, India; p.sham1988@gmail.com 5 Department of Mathematics, Selvamm Arts & Science college, Periyar University, Namakkal, India; babusarath234@gmail.com 6 PG & Research Department of Mathematics, Urumu Dhanalakshmi College, Bharathidasan University, Trichy 620019, India; anusuyar23@gmail.com Article History: Received: 14-04-2024 Revised: 29-05-2024 Accepted: 15-06-2024 Abstract: The aim of this paper is to introduce a new type of contraction called 𝑅𝐹 −contraction. As compared to the F-contraction in the existing literature, our 𝑅𝐹 −contraction is much simpler and more straightforward, since it contains only one condition that is, the function revised fuzzy is strictly non-increasing. Moreover, some fixed-point theorems for 𝑅𝐹 −contraction are presented. Further, some examples are given to illustrate its validity and superiority. In addition, by applying a very significant lemma, we show that our proofs of most fixed-point theorems are shorter and more elegant than ones in the literature. Keywords and phrases: T-conorm, revised fuzzy metric, 𝑅𝐹 −contraction, fixed point. 2020 Mathematical classification: 46N20, 46S40, 47H10, 37C25. 1. Introduction George and Veeramani [3] proposed axioms to fuzzy metric spaces [for short, FMS] based on Zadeh's theory of fuzzy sets [25]. The triangular norm (for short, t-norm), initially proposed by Schweizer and Sklar [21], is one of the most significant axioms in binary functions. In various domains, such as fuzzy sets, fuzzy logic, and its applications, this is a critical process. The fixed-point [for short, FP] theory constructed in FMSs, which was pioneered by Grabiec [4], where a FM version of the Banach contraction principle was introduced, is one of the most fascinating motives. Following that, Gregori and his coauthors presented a number of fuzzy contractive mappings [for short, FCM] in FMS (see [5]). Mihet [11], on the other hand, established a fixed-point theorem for weak Banach contraction in W-complete FMS, and expanded prior results including additional types of contractions such as Edelstein fuzzy contractive mappings, fuzzy y-contractive mappings, and so on (for details, see [11]). Wardowski [23] recently developed a novel idea of fuzzy H-contractive mapping and deduced some interesting FP theorems. Wardowski [24] also developed a FP theorem in metric spaces https://cran.r-project.org/web/classifications/MSC-2010.html#code:37C25 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 228 https://internationalpubls.com and introduced the F-contraction contraction. [7,11,22] recently proposed further contractions in FMS. Alexander Sostak [1] introduced the notion of Revised Fuzzy metric [for short, RFM] in the year 2018, which allows for the incremental assessment of the membership of components in a set. Muraliraj and Thangathamizh developed Revised Fuzzy contraction mappings [for short, RFCM] and established FP findings for them [11-14, 24-26]. Many generic topological ideas and conclusions were then applied to the revised fuzzy topological space. We intraduce a new contraction called RF-contraction throughout this research, which differs from [7,17,23] in that it incorporates a simpler criterion, namely that the mapping is strictly non- increasing. Furthermore, in the context of RFMS, we deal with FP theorems for 𝑅𝐹 −contraction. In particular, we prove a lemma in RFMS with regard to the Cauchy sequence. Second, we introduce the idea of 𝑅𝐹 −contraction, which requires just that the function be strictly non-increasing. Third, we get certain FP theorems for 𝑅𝐹 −contraction with shorter requirements and simple proofs using the preceding lemma. Fourth, we provide some instances to back up our findings. Our examples demonstrate that our findings are true generalizations in the literature. 2. Preliminaries We'll go over a few fundamental definitions and ideas in the sections that follow. Definition 2.1 ([21]). A binary operation     2 : 0,1  0,1 T → is called a triangular norm (for short, t-norm) if the following conditions hold: (i) ( ), 0T g g= , for each  0,1 g  ; (ii) ( ) ( ),  , T g d T l m , for any     ,     g l d m  and  ,  ,  ,  0,1 g d l m ; (iii) T is associative and commutative. Three basic examples of continuous t-conorms are as follows: ( )  ,      , maxT g d max g d= , ( ),     PT g d g d gd= + − and ( )  ,        ,1LT g d ming g d= + (maximum, product, and Lukasiewicz t-conorm, respectively). Definition 2.2 ([1]). A triple ( ), ,X W T is called a RFMS if X is a nonempty set, T is a continuous t-conorm, and ( )  2 :  0,   0,1 W X  + → be a RF satisfying the following conditions: (RF 1) ( ),  ,    1W q r t  for all , q r X and    0t ; (RF 2) ( ),  ,     0(    0)W q r t t=  if and only if    q r= ; (RF 3) ( ) ( ),  ,      ,  , W q r t W r q t= for all , q r X and    0t ; Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 229 https://internationalpubls.com (RF 4) ( ) ( ) ( )( ),  ,        ,  ,  , ,  , W q l t s T W q r t W r l s+  for all ,  ,q r l X and ,    0t s ; (RF 5) ( ) ( )  ,  ,   :  0,   0,1 W q r − + → is continuous for all , q r X . If (RF 4) is replaced by the following condition: (RF 4)’ ( ) ( ) ( )( ),  ,    ,  ,  , ,  , W q l t T W q r t W r l t for all ,  ,q r l X and 0t  ; then ( ), , X W T is called a strong RFMS. Moreover, if ( ), , X W T is a RFMS, then W is a continuous function on ( )2 0,X  + and ( ),  , W q r − is non-increasing for all , q r X . In the sequel, unless there is a special explanation, we always denote by N , the set of all positive integers; 0N , the set of all nonnegative integers; R , the set of all real numbers and +R , the set of all positive real numbers. Definition 2.3 ([12]). Let ( ), ,X W T be a RFMS and  n n q N be a sequence in X . Then, we say the following: (i)  n n q N converges to q X (say lim n n q q → = ), if ( )lim  , , 0n n W q q t → = for any 0t  ; (ii)  n n q N is a Cauchy sequence if, for any ( )0,1   and 0t  , there exists 0n N such that ( ),  , m nW q q t  for any 0, m n n ; (iii) ( ), , X W T is complete if every Cauchy sequence is convergent. Definition 2.4 Let ( ), , X W T be a RFMS and  :   f X X→ a mapping. Then f is called a revised fuzzy contraction if there exists ( )0,1 k  such that ( ) ( )( ) ( )( ),  ,  ,  , W f q f r t k W q r t (1) for all , q r X and 0t  . In this case, k is called the contractive constant of f . We say that the mapping :   T X X→ is called a Tirado contraction if there exists ( )0,1 k  such that ( ) ( )( ),  ,  ,  , W Tq Tr t k W q r t for all , q r X and 0t  . (2) 3. Methods Lemma 3.1. Let ( ), ,X W T be a RFMS and  nq be a sequence in X such that for each n N , Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 230 https://internationalpubls.com ( )1 0 lim ,  ,  1n n t W q q t + + →  , and for any    0t , (3) ( )1lim ,  ,  0n n t W q q t+ → = . (4) If  nq is not a Cauchy sequence in X , then there exist ( ) 0  0,1  ,  0t   , and two sequences of positive integers  kn ,  km ,   , k kn m k k N   , such that the following sequences ( )  ( )  ( ) 0 1 0 1 0, , , , , , , , ,mk nk mk nk mk nkW q q t W q q t W q q t+ − ( ) 1 1 0, ,mk nkW q q t− + , ( ) 1 1 0, ,mk nkW q q t+ + tend to  as k → . Proof. Let  nq be a sequence in X , which is not a Cauchy sequence. Then, by Definition 2.3, there exists ( ) 00,1  , 0 t   and sequence  kn ,  km ,   , k kn m k k N   , such that for any k N , we have ( )0, ,mk nkW q q t  . (5) and ( )0, ,mk nkW q q t  . (6) clearly, by (5), one has ( )0lim inf , ,mk nk k W q q t  →  . (7) Using Condition (RF 4), for any k N and ( )00, p t , it is not hard to verify that ( ) ( ) ( )( )0 1 1 0, , , , , , ,mk nk mk mk mk nkW q q t T W q q p W q q t p− − − . (8) Note that, by (3) and (4), it follows that ( )1 0 lim lim ,  ,  0n n n p W q q p + + → →   =    .. (9) If we take 0p +→ in (8), then by (9), (6) and the continuity of T , we obtain ( ) ( ) ( )( )0 1 1 0 0 lim , , lim lim , , , , ,mk nk mk mk mk nk k k p W q q t T W q q p W q q t p + − − → → →   −    . ( ) ( )1 1 0 0 0 lim lim , , , lim lim , ,mk mk mk nk k kp p T W q q p W q q t p + +− − → →→ →     = −          ( )( )1 00, lim , ,mk nk k T W q q t− → = ( )1 0lim , ,mk nk k W q q t− → =  . This inequality and (7) imply Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 231 https://internationalpubls.com ( )0lim , ,mk nk k W q q t  → = . (10) Let us prove that ( )1 0lim , ,mk nk k W q q t + → = . (11) ( ) ( ) ( )( )1 0 0 1 0 lim , , lim lim , , , , ,mk nk mk nk nk nk k k p W q q t T W q q t p W q q p ++ + → → →   −    . ( )( )0lim , , ,0mk nk k T W q q t → = ( )0lim , ,mk nk k W q q t → = = . (12) On the other hand, by (10) and (4), we have ( )0lim , ,mk nk k W q q t → = ( ) ( )( )1 0 1 0 lim lim , , , , ,mk nk nk nk k p T W q q t p W q q p + + + → →   −    ( )( ) ( )1 0 1 0lim , , ,0 lim , ,mk nk mk nk k k T W q q t W q q t+ + → →  = . (13) Then, by (12) and (13), we obtain (11). The left proofs are similar to the above argument, and therefore we omit them. Remark 3.2. Condition (3) in Lemma 3.1 can be omitted if ( ), , X W T is a strong RFMS. In this case, instead of (8), we have ( ) ( ) ( )( )0 1 1 0, , , , , , ,mk nk mk mk mk nkW q q t T W q q p W q q t p− − − . In the following, denote by F the class of all mappings   ( ) :  0,1    0,F → + satisfying the following condition: for all , q r X , q r implies ( ) ( )   F q F r . That is to say, F is strictly non-increasing on  0,1  . Definition 3.3. Let ( ), X d be a metric space and  : F R R+ → be a mapping, satisfying the following: (RF 1) F is strictly non increasing on R+ ; (RF 2) For each sequence  n n  N of positive numbers, ( )lim n k F  → = − if, and only if lim 0n k  → = ; (RF 3) There exists ( )0,1 k  such that ( ) 0 lim    0k k F a +→ = . The mapping    :   J X X→ is said to be an F−contraction if there exists    0t such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 232 https://internationalpubls.com ( ) ( )( )( ) ( )( )  ,  , F d J q J r F d q r +  for all ,q r X with ( ) ( )( ),  0d j q j r  . Definition 3.4. Let ( ), ,X W T be a RFMS and FF . The mapping  :   J X X→ is said to be a 𝑅𝐹 −contraction if there exists ( )0,1   such that ( ) ( )( )( ) ( )( )( ) 1 ,  , F d f q f r F d q r   for all ,    , q r X q r  , and 0t  . (14) Theorem 3.5. Let ( ), ,X W T be a complete RFMS such that ( ) 0 lim ,  ,  1 t W q r t +→  , for all ,   q r X . If  :   J X X→ is a continuous RF F-contraction, then J has a unique fixed-point in X . Proof. Choose 0q X and ( )1n nq J q+ = for all 0n N . Suppose that :   J X X→ is a 𝑅𝐹 −contractive mapping. If ( )1n n nq q j q+= = holds for some 0n N , then nq is a FP. Assume that 1n nq q + for any 0n N . By (14), for every n N and 0t  , one has ( )( ) ( )( )( ) ( )( )1 1 1 1 ,  ,    ,  ,  ,  , n n n n n nF W q q t F W q q t F W q q t  + + −  . Then, we get ( ) ( )1 1,  ,  ,  , n n n nW q q t W q q t− + . Thus, ( ) 1,  , n nW q q t+ ( 0)t  is a strictly non-increasing sequence bounded from above, so ( ) 1,  , n nW q q t+ ( 0)t  is convergent. In other words, there exists ( )  0,1 a t  such that for any    0t , one has ( ) ( )1lim ,  ,     n n n W q q t a t+ → = . (15) Clearly, for any    0t and n N , it follows that ( ) ( )1,  ,   n nW q q t a t+  . (16) Note that, by (15) and (16), for any    0t , we have ( )( ) ( )( )1lim  ,  ,    0n n n F W q q t F a t+ → = − . (17) Assume that ( )  1 a t  for some 0t  . By (14), it implies that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 233 https://internationalpubls.com ( )( ) ( )( )( ) ( )( )1 1 1 1 ,  ,    ,  ,  ,  , n n n n n nF W q q t F W q q t F W q q t  + + −  . (18) Taking the limit from both sides of (18) together with (17), we get ( )( ) ( )( )( ) ( )( ) 1  0     0 0 ,F a t F a t F a t  −  −  − which means that ( )( )0   0F a t − = . This is a contradiction with ( )( )0 0F a t −  . Therefore, we have ( )1lim  ,  ,  0n n n W q q t+ → = . (19) Further, we need to prove that  nq is a Cauchy sequence. Suppose that this claim is not true. Using Lemma 3.1 and noting that (19) is in fact Condition (4), then there exist ( ) 00,1  ,  0t   and sequences   kmq and   knq such that ( )0lim  ,  ,  k km n k W q q t  → = . By (14), we have ( ) ( )( )( ) ( ) ( )( )( )( ) ( )( )0 0 0 1   ,    ,    ,    ,    ,  ,  . k k k k k km n m n m nF W J q J q t F W J q J q t F W q q t    Letting k → from both sides of the above inequality, we have ( )( ) ( )( )( ) ( )( ) 1 0     0      0F F F    −  −  − , which establishes that ( )( ) 0 0F  − = . This is in contradiction with ( )( ) 0 0F  −  . Hence,  nq is a Cauchy sequence. Since ( ), ,X W T is complete, then there exists q X such that lim  n n q q → = . (20) Let us prove that q is a FP of J . As a matter of fact, it follows immediately from (20) and the continuity of J that ( ) ( )1lim    lim   n n n n q q J q f q+ → → = = = . Finally, we prove the uniqueness of the FP. Suppose that q and r are distinct FPs of J . Again, by using (14), we easily obtain that ( ) ( )( )( ) ( ) ( )( )( )( ) ( )( )1 ( ,  ,  ( ,  ,  ,  , F W J q J r t F W J q J r t F W q r t    . As a consequence, we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 234 https://internationalpubls.com ( ) ( )( ) ( ) ( ) ( )( )( ,  ,  ,  ,    ( ,  , W J q J r t W q r t W J q J r t = . This is a contradiction. Remark 3.6. Let ( ), ,X W T be a RFMS. (i) Define a strictly non- increasing function ( ) 1   1 F t t = + for any ( )0,1 t and let J be a RF-contraction. Then, the RF contraction (1) is obtained. Indeed, since J is RF-contractive, then there exists ( )0,1 t such that ( ) ( )( ) ( ) 1 1 1 1 ,  , 1 ,  ,  W q r tW J q J r t       ++   that is, ( ) ( )( )( ) ( ) 1 1 ,  ,  1 ,  , W J q J r t W q r t  +  + Therefore, ( ) ( )( ) ( )( ),  ,  ,  , W J q J r t W q r t holds for all , ,q r X and 0t  . (ii) let ( ) 1   1 t t = + , where ( )0,1 t , and suppose that J is a RF-contraction. Then we easily obtain the Tirado contraction. Corollary 3.7. Let ( ), X d be a complete RFMS, and : J X X→ be a function such that there exists ( )( )0,1      1  , 2, 3, 4iK i = and for all ,  ,   q r X q r  , and one of the followings: (1) ( ) ( )( ) ( )( ) 1 1 ,  ,      ,  , W J q J r t W q r t K  ; (2) ( ) ( )( )  ( ) ( )( )  ( )  ( )  , , , , , ,, , 2 1       11 W J q J r t W q r t W q r tW J q J r t ex e e K e p − − −−       ++   ; (3) ( ) ( )( )  ( ) ( )( )  ( ) ( )( )  ( ) ( )( )  , , , , , , , , 3 W J q J r t W J q J r t W J q J r t W J q J r t e e e e − − − + −  ( )  ( )  ( )  ( )  , , , , , , , , 1 W q r t W q r t W q r t W q r t e e e e − − − + − ; (4) ( ) ( )( ) , ,exp W J q J r t . Then, J has a unique FP in X . Proof. For Cases (1)–(4), put Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 235 https://internationalpubls.com ( ) ( )     ( )         ( )   1 2 3 4,  ,  ,  , 1 3 t t t t t t t exp exp exp F t t F t F t F t exp exp exp exp − − − − − = = = = + + − respectively. Using Theorem 3.5, we claim that J has a unique FP. Example 3.8. Let    X R= and define the usual metric ( ), d q r q r= − for all , q r X . Let T be a product t-conorm. Define a RFM as follows: ( ) ( ) ( ),  ,     1    1 ,  ,    1 d q r d q r t t W q r t exp exp        −    + +          = −     , Where , q r X , and    0t . Clearly, ( ),  , W q r t satisfies the conditions of (RF 1)–(RF 3) and (RF 5). Moreover, for all ,  ,q r l X and ,     0t s , it is clear that ( ) ( ) ( ),  ,   1  1 ,  ,    1 d q l d q l t s t s W q l t s e e        −    + + + +          + = −     ( ) ( )  ,  ,   .  ,  ,  ,W q r t W r l s= that is, Condition (RF 4) holds. Let ( ) ( ) ( ) 1 1 1   ,   (0      1  )    4   2 J x x x X F y y and ln y =  = −   = . Since ( ) ( )( )( ) ( )( ),  ,  ,  ,   F W f q f r t F W q r t= holds for all , q r X ,  q r and    0t , then Condition (14) is fulfilled. Hence, by Theorem 3.5, it follows that J has a unique FP. It is worth mentioning that this example is true for arbitrary function ( )   J q kq= , where 0      1 k  is a constant with    t k . Example 3.9. Let ( ), X d be a metric space and T a t-conorm. Then for all , q r X and    0t , ( ) ( ) ( ) ,  ,  ,      1      ,  d q r W q r t t d q r = + + defines a RFM. Define a function ( )   F x x= on  0,1  and let ( )0,1   be a constant. If Condition (14) is fulfilled, then ( ) ( )( ) ( ) ( )( )( ) ( )( )( ) ( ) ( ) ,  ,  ,      ,  ,  ,  ,  ,  ,    1      ,  d q r t W f q f r t F W f q f r t F W q r t W q r t d q r        =  = =    + +    Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 236 https://internationalpubls.com holds for all , q r X and 0t  . That is to say, we obtain the contractive condition (1) from [22]. Theorem 3.10. Let ( ), ,X W T be a complete RFMS and FF be a continuous mapping. If  :   J X X→ is a RFM, 𝐹 −contraction, then J has a unique fixed-point in X. Proof. Choose 0q X and define a sequenced  nq by ( )1n nq J q+ = for all 0n N . If 1n nq q += for some 0n N , then the proof is finished. Assume that 1n nq q + for any 0n N . From the definition of the F-contractive, we have ( )( ) ( )( )( ) ( )( )1 1 1 1 1 ,  ,    ,  ,  ,  , n m n m n mF W q q t F W q q t F W q q t  + + + +  . Then, ( ) ( )1 1,  ,  ,  , n m n mW q q t W q q t+ +  . For any n m . Let ( ) ( )inf ,  , m n m n m a t W q q t  = . Notice that ( ) ( )1 1inf ,  ,  inf ,  , n m n m n m n m W q q t W q q t+ +    Then ( ) ( )1m ma t a t+  , for any m N . Since ( ) ma t is bounded, then there exists ( )  0,1a t  such that ( ) ( )lim m n a t a t → = for all 0t  . let us prove that ( ) 0a t = for all    0t . Suppose the contrary, and there exists 0s  such that ( )0 1a s  . Then by (14)  0,1   such that for any    0t , one has ( )( ) ( )( )1 1 1 1 1 lim inf ,  ,    lim inf ,  , n m n m m n m m n m F W q q t F W q q t  + + + + →  →   ( )( )  lim inf ,  , n m m n m F W q q s →   Using the assumption that F is continuous, we have ( )( ) ( )( ) ( )( ) 1 F a s F a s F a s    , which means that ( )( )  1F a s = . This is in contraction with ( )( ) 1F a s  . Thus ( )( )lim inf ,  ,  0n m m n m W q q s →  = . For any 0t  . Consequently, ( ) , lim    ,  ,  0n m m n W q q s → = , for any 0t  . Thus  nq is a Cauchy sequence. Since ( ), ,W X T is complete, then there is q X such that lim n n n q q → = . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 237 https://internationalpubls.com Taking advantage of (14), we have ( )( )( ) ( )( )( ) ( )1 1 1 ,  ,    ,  ,  ,  ,n n nF W q J q t F W q J q t W q q t  + +  . For any 0t  . Consequently, ( ) , lim    ,  ,  0n m m n W q q s → = for any 0t  . Thus,  nq ( )( ) ( )( )( ) ( )( )1 1 1 1 ,  ,    ,  ,  ,  , n n n n n nF W q q t F W q q t F W q q t  + + −  for all n N . Letting n→ and using the assumption that F is continuous, we have ( )( )( ) ( )( )( ) ( ) 1 ,  ,     ,  ,    0F W q J q t F W q J q t F    . Thus, it leads to ( )( ),  ,    0W q f q t  . Therefore, ( ) q J q= . Suppose now that J has distinct FPs , q r X , then by (14), we obtain ( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( )( ) 1 ,  ,    ,  ,    ,  ,      ,  , F W q r t F W J q J r t F W J q J r t F W q r t  =   . This is a contradiction. Hence,    q r= . Theorem 3.10. Let ( ), ,X W T be a complete RFMS such that ( ) 0 lim ,  ,  1 t W q r t +→  for all , q r X . Let  : f X X→ be a mapping and FF . Suppose that for all , q r X ,  q r and    0t , there exists ( )0,1   such that ( ) ( )( )( ) ( ) ( )( ) ( )( ) ( )1 ,  ,    ,  ,  , ,  ,  , ,  , F W f q f r t F max W q r t W q J q t W r J r t   . (21) Then, J has a unique FP, provided that J or F is continuous. Proof. Choose 0q X and define a sequence  nq as follows: ( )( )1 0 n nq J q n N+ =  . By (21), we have ( )( ) ( ) ( )( )( )1 1,  ,      ,  , n n nF W q q t F W J q J q t+ −= ( ) ( )( )( )1 1   ,  , n nF W J q J q t  − ( ) ( ) ( ) ( )1 1 1  ,  ,  , ,  ,  , ,  , n n n n n nF max W q q t W q q t W q q t− − + ( ) ( ) ( )1 1  ,  ,  , ,  , n n n nF max W q q t W q q t− += , for all n N and 0t  . (22) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 238 https://internationalpubls.com If ( ) ( )  ( )1 1 1,  ,  , ,  ,  ,  , n n n n n nmax W q q t W q q t W q q t− + += , then by (22), we get ( ) ( )1 1,  ,  ,  , n n n nW q q t W q q t+ + , which is a contradiction. If ( ) ( )  ( )1 1 1,  ,  , ,  ,  ,  , n n n n n nmax W q q t W q q t W q q t− + −= , then by (22), we have ( ) ( )1 1,  ,  ,  , n n n nW q q t W q q t+ − . Following the proof of Theorem 3.5, we find q X such that lim  n n q q → = . Suppose first that J is continuous. Then, by the construction of sequence  nq it follows that J has a FP q . Suppose that F is continuous. Then, by (21), we have ( )( )( ) ( )( )( )1 1 1 ,  ,      ,  , n nF W q J q t F W q J q t  + + ( ) ( ) ( )( ) ( )1  ,  ,  , ,  ,  , ,  , n n nF max W q q t W q q t W q J q t+ , (23) for all n N and 0t  . If ( )J q q= , then taking n→ from both sides of (23), we have ( )( )( ) ( )( )( ) ( )( )  ( )( )1 ,  ,      ,  ,    ( 0,0, ,  ,     ( ,  ,F W q J q t F W q J q t F max W q J q t F W q J q t    = , which means that ( )( )( ,  , 1F W q J q t = . This is in contradiction with ( )( )( ,  , 1F W q J q t  . Finally, we prove the uniqueness of the FP. Assume that J has two distinct FPs, ,p q . Then, by (21), we have ( )( ) ( ) ( )( )( ) ( ) ( )( )( )1 ,  ,  ( ,  ,    ( ,  , F W p q t F W J p J q t F W J p J q t  =  . ( ) ( )( ) ( )( ) ( )  ,  ,  , ,  ,  , ,  , F max W p q t W p J p t W q J q t , ( ) ( ) ( ) ( ),  ,  , ,  ,  , ,  , F max W p q t W p p t W q q t= ( ) ( ),  ,  ,0,0F max W p q t= ( )( ),  , F W p q t= , This is a contradiction. Therefore, p q= . Refrences [1] Alexander Sostak “George-Veeramani Fuzzy Metrics Revised” Axioms 2018,7,60. 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