361 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License Generalization of Fixed Point Theorem for Contraction Mappings in the fuzzy Metric Space Rusul Abdul Kadhim Mohammed 1* and Zeana Zakai Jamil 2 1,2 Department of Mathematics, Collage of Science, University of Baghdad, Baghdad, Iraq. *Corresponding Author Received: 18 February 2024 Accepted: 14 August 2024 Published: 20 July 2025 doi.org/10.30526/38.3.3943 Abstract Let be a fuzzy - metric space, where is a non-empty set, is a fuzzy set on to [0,1], and is a continuous -norm, and let a function [ ] [ ] satisfies the following conditions: The function is strictly decreasing and continuous, If and only if equals 1 and ( ) ( ), where and in . Which is called - function and use it to define - Contraction mappings of type and . In this research, we will complete the study of many authors about fixed point theory on fuzzy -metric spaces as and generalize some results on fixed points theory on fuzzy metric spaces to fuzzy - metric spaces with simplify different proofs. It was established many results on compact fuzzy metric spaces and complete fuzzy metric spaces. we generalize results of other results to fuzzy -metric spaces by using – Contraction mappings of type and in both complete and compact fuzzy -metric spaces to show existence of fixed points for this type of self-mapping. Keywords : Complete fuzzy - metric space, Compact fuzzy - metric space, Fixed point. 1.Introduction The definition of a fuzzy metric space was defined by (1) to (5) defined the completeness of the fuzzy metric space, Bakhtin in 1989 present the concept of - metric spaces (6). After that, (7) generalized a - metric spaces and a fuzzy metric spaces to fuzzy - metric spaces in 2016. (8) used the continuous - norm to modify the idea of fuzzy metric spaces, and showed in the completeness definition of Grabiec fails to make complete, then they introduced another definition completeness for which , with the standard fuzzy metric induced by the Euclidean metric. In 1922, (9) presented the Banach fixed point theorem, which is known an essential concept in fixed point theory. From that time , many results fixed points on different spaces and types of Contraction mappings have been proved, for more detail see (10-28). One of them, (3) established many results on the compact or complete fuzzy metric spaces. In our work, we prove the existence a uniqueness of fixed point of a mapping on complete or compact fuzzy - metric space where is a non-empty set, is a fuzzy set on https://orcid.org/0009-0003-3781-6548 mailto:Rusul.Abd2203m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-9123-5849 mailto:zina.z@sc.uobaghdad.edu.iq https://orcid.org/0009-0003-3781-6548 mailto:Rusul.Abd2203m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-9123-5849 mailto:zina.z@sc.uobaghdad.edu.iq https://orcid.org/0009-0003-3781-6548 mailto:Rusul.Abd2203m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-9123-5849 mailto:zina.z@sc.uobaghdad.edu.iq https://orcid.org/0009-0003-3781-6548 mailto:Rusul.Abd2203m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-9123-5849 mailto:zina.z@sc.uobaghdad.edu.iq https://orcid.org/0009-0003-3781-6548 mailto:Rusul.Abd2203m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-9123-5849 mailto:zina.z@sc.uobaghdad.edu.iq https://orcid.org/0009-0003-3781-6548 mailto:Rusul.Abd2203m@sc.uobaghdad.edu.iq https://orcid.org/0000-0001-9123-5849 mailto:zina.z@sc.uobaghdad.edu.iq IHJPAS. 2025, 38(3) 362 to [0,1], and is a continuous -norm under - Contraction mapping of types and . 2. Preliminaries In this section we will cover some basic definitions and results. Definition 2.1 (29): A t- norm is a binary operation :[0,1]×[0,1]→[0,1] for all [ ], which satisfies the conditions: 1- Commutatively : ; 2- Associativity: ; 3- for each 4- for all [ ] such that and , 5- is continuous . The commutatively of (1), and conditions (2) , for all -norm, , and : (29) e.g of a -norm are , { } (29) . Definition 2.2 (30): Let be a -norm, and let [ ] [ ] , be defined as [ ] Remark 2.3 (29): Each t-norm T can be extended through associativity to an n-ary operation that takes the values of a ( ) [ ] as , . We say that a -norm is of -type if the family { } is equicontinuous at . Remark 2.4 (29): In -norm can take any sequence ( form [0,1] can extend to countable infinite operation . exists since the sequence is non-increasing and bounded from below. The concept a fuzzy -metric space is obtainable in (7). Definition 2.5 (7): A 3-tuple is called a fuzzy - metric space (Fb-MS) if X is an arbitrary (nonempty) set, and is a continues - norm, is a fuzzy set on × × [ satisfying the following conditions for all , 0 : - 0 , - 1 if and only if , - , - ( ( )) , - [ [ ] is continuous . Another definition of Fb-MS is introduced in (1). Definition 2.6 (1): A 3-tuple is called a (Fb-MS) if X is an arbitrary (nonempty) set, and is a continues - norm, is a fuzzy set on × × [ satisfying the following conditions for all , 0 : - 0, - 2 - 1 if and only if , - , - - [ [ ] is continuous. Let us give a simple application of a Fb-MS. Example 2.7 [1, P.31]: Suppose , a function [ define by IHJPAS. 2025, 38(3) 363 =exp .Then is a Fb-MS. We now start with certain fundamental concepts. Definition 2.8 (1): A sequence in is a converges to if ( , →1 as → ∞ for each > 0. We can write = . Definition 2.9 (1): If, for each 0 and 0 1, there are 0 such that , , 1– for all , ≥ o , then is called a Cauchy sequence in . Proposition 2.10(1):A sequence in said to be a Cauchy sequence if , , =1 for each . Definition 2.11 (1): If every Cauchy sequence in a Fb-MS is convergent ,then Fb- MS is said to be complete. In [3], give a concept of a compact fuzzy metric space. We will generalized to Fb-MS. Definition 2.12: If there exists a convergent subsequence for each sequence in , then a Fb- MS is called compact. In [2] proved the following propositions, which are main to in our results. Proposition 2.13 (2): Suppose be a sequence in a Fb-MS , and let be of - type. If there is λ such that , , ( ) , and there are , and such that ( ) 1 ; Then the sequence is a Cauchy. Proposition 2.14 (2): Let be a Fb-MS. If for some and , ( ) , then . 3. Results In this section. We establish several fixed point theorems for a class of self -mappings in complete fuzzy -metric spaces and compact fuzzy - metric spaces by using the function. First we need the following Definitions. Definition 3.1: Suppose be a Fb-MS, a function [ ] [ ] satisfies: - is strictly decreasing , - is continuous, - if and only if , - ( ) ( ) Is called -function. We use -function to define two types of Contraction mappings Definition 3.2: Let be a Fb-MS, be a self –mapping on , and [ ] [ ] be function and be a mapping then 1- is called – contraction of type ) if there is such that, for all ( ( )) (1) 2- is called – contraction of type ) if ( ) ( ) (2) For all and . Theorem 3.3: Let be a complete Fb-MS and let be of -type and be a – IHJPAS. 2025, 38(3) 364 contraction mapping of type ( . If for there is and such that ( ) 1 ; . Then there is a unique fixed point for . Proof : Let . Define for each { } and To prove has a fixed point we have two cases Case 1 if [ ]: Thus there is then i.e , then is a fixed point of . Case 2 if [ ]: hence for all . From Equation (1) we get ( ) ( ( )) ( ( )) . (3) Claim: ( ) for all and . Assume ( ), since is strictly decreasing, hence ( ( )) ( ) which is contradiction with Equation (3), so ( ) through Proposition (2.13) we hence is a Cauchy sequence. As is a complete Fb-MS, thus . Let and 2=1– 1. By the Definition ((2.5), part 4)). ( ( ) ( )) Since is strictly decreasing and by Definition ((3.1) , part (4)) implies that ( ( ( ) ( ))) ( ( ( )) ( ( ))) By Equation (1) we get ( ( ( ) ( )) ( ( ))) As , thanks to the continuity of , and by definition ((3.1) , part (3)) ( ) ( ( ( ) ) ) We have ( ) implies that , hence Assume that has another fixed point say Assume ( ), since is strictly decreasing, then ( ( )) , which is contradiction with equation (1), we get ( ). There for by Proposition (2.14) In the following finding, we established the presence of a fixed point in a compact Fb-MS. Theorem 3.4: Suppose be a compact Fb-MS and be continuous - contraction of type ( . Then there is a fixed point that is unique to in . Proof : Assume . Define with { } IHJPAS. 2025, 38(3) 365 Now, There are two cases: Case 1 [If for some ] , so be a fixed point of . Case 2: [If for every ]: Since be a compact Fb-MS, then there is a subsequence { } of { } such that convergent to as . According to continuity of , we have and As is continuous, we get But Equation (4) contradiction with equation (2), as ( ) ( ) ( ) Furthermore, Assume that has two fixed point say and This lead to a contradiction. Therefore Remark :The above theorem needs only condition(2) in the Definition (3.1). 4. Conclusion Based on the results, we will complete the study of many authors about fixed point theory on fuzzy - metric spaces, by using – Contraction mappings of type and to show existence of fixed points for this type of self-mapping. Acknowledgment The authors are greatly appreciated the referees for their valuable comment and suggestions for improving the paper. Conflict of Interest The authors declare that they have no conflicts of interest. 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