Fuzzy Fixed Point Results of Fuzzy Mappings on -Metric Spaces via -Contractions

Abstract

In this manuscript, we establish some fixed point results for fuzzy mappings via -contractions. For validation of the proved results, some nontrivial examples are presented. Few interesting consequences are also stated which authenticate that our results generalize many existing ones in the literature.

1. Introduction

Fixed point theory has attained an important and significant role in analysis. The literature of the last four decades ornamented with results which discover fixed points of self and non-self nonlinear operators in a metric space. This branch of mathematics provides a strong tool for finding out the solution of integral, differential, and eigenvalue equations. Since 1922, the Banach contraction principle (BCP) has become the center of the attention for researchers working in different areas. The eminent Banach contraction principle (BCP) [1] states that for a complete metric space , each self-mapping on satisfying has a unique fixed point.

A lot of literature can be found on generalizations and extensions of the famous BCP by changing either the space under consideration or the condition on the mapping. In recent years, various authors presented interesting generalizations of a metric space, for example, uniform space [2, 3], -metric space [4], -algebra valued metric space [5, 6], -metric space [7], and -metric space [8].

A very interesting generalization of metric space is cone metric space. Gupta and Chauhan [9] presented an analogous to the BCP on cone -metric spaces in 2021. In contrast, Wardowski [10] extended the Banach contraction to a more generalized form, known as -contractions, and established a fixed point theorem in complete metric spaces. Furthermore, many mathematicians used -contractions for the existence of a fixed point, see [11–13]. Nadler’s theorem [14] states that for a complete metric space , each non-self-mapping satisfying has a fixed point. Here, denotes the Hausdorff metric defined on , the set of bounded and closed subsets of . Throughout the article, a metric space and a fuzzy fixed point are denoted by a bms and a ffp, respectively. Some important definitions are presented before constructing the main results. These definitions are inevitable for next discussion.

Definition 1. Let be a nonempty set. A function is called a -metric if, for all and , the following assertions are satisfied: (i)(ii)(iii)

The pair is called a bms. if we consider then above definition coincides with the definition of a metric space.

Example 2. Let . Define by Then, it is easy to show that is a bms with

Definition 3. Let be a bms. (i)A sequence is said to be a Cauchy sequence if, given there exists such that, for all , we have or (ii)A sequence converges to if

Among numerous advances of fuzzy sets theory, a significant development is made to find the fuzzy analogues of fixed point results of the classical fixed point theorems. The concept of fuzzy sets along with its related notions is given by Zadeh in [15]. In 1975, the idea of fuzzy metric space is introduced. This interesting concepts is investigated by many researchers with different contractions, see for example [16, 17].

Furthermore, Weiss [18] and Butnariu [19] used the notions of fuzzy maps to established various results. Heilpern [20] proved a fixed point theorem for fuzzy maps which is the analogue of Nadler’s multivalued result [14] in metric spaces. Notion of fuzzy maps is investigated by many mathematician in different directions. In this context, in [21], the existence of common fuzzy fixed points under a rational contractive condition has been established which is further generalized by Shoaib et al. [22] in dislocated complete metric spaces. Motivated by these research ideas, we formulate some results for fuzzy contractions using the Hausdorff metric. Namely, the present article provides certain fixed point results for fuzzy mappings via contractions. A mapping is said to be an -mapping [10], if the following conditions are satisfied:

(F1): is a strictly increasing function, that is, for , with then

(F2): for each sequence of the positive real numbers ,

(F3): There is a real number such as

Afterward, Altun et al. [13] altered the above definition by adding another condition given below:

(F4):

The set of all functions satisfying to is denoted by Let be a metric space and be a self mapping on . is said to be an -contraction [10] if there exists and satisfying - such that

Definition 4. (see [22]). Let be a nonempty subset of a bms , and let . An element is called the best approximation in if where The set is called a proximinal set if every has at least one best approximation in The set of all proximinal subsets of is denoted by .

Definition 5. (see [22]). The function, , defined by is called a Hausdorff -metric on

A function with domain and values in is called a fuzzy set. We will denote the collection of all fuzzy sets in by . If and is a fuzzy set, then is called the grade of membership of in . The set is called an -level set of a fuzzy set , and it is given by

If is a mapping from into then is called a fuzzy mapping. A fuzzy mapping is a fuzzy subset on with membership function . The function is the grade of membership of in .

Definition 6. (see [21]). Let be a fuzzy mapping. A point is called an fuzzy fixed point (ffp) if there exists such that

Lemma 7. (see [22]). Let and be nonempty proximal subsets of a bms .
If then .

Lemma 8. (see [22]). Let be a bms and suppose that is a Hausdorff bms. If, for all and for each , there exists satisfying , then .

2. Main Results

Definition 9. Consider a bms . A mapping is said to be a fuzzy -contractive mapping if for a function , there exists such that condition (F1) holds, and the following conditions are satisfied:
(Fb2): for each sequence of positive numbers, if then
(Fb3): such that
(Fb4): there exists for which for all .

Theorem 10. Let be a complete bms with . Let be a fuzzy -contractive mapping. Then, has an ffp if following conditions are satisfied: (i) is an -admissible mapping(ii)There exists (iii)For any sequence

Proof. Let be an arbitrary point of Choose If , then is a fixed point, and there is nothing to prove; so, . Therefore, Now, by Lemma 8, there exists such that Then, Consider Thus, Since is increasing, one writes We have That is, because Consequently, Continuing in the same manner, we can define a sequence such that and Now, let By (2.2), and hence By property (Fb2), we get that as Then, by condition (Fb3), there exists such that Therefore, we have Applying limit , we have Therefore, there exists such that Using Lemma 7 given in [23], is a Cauchy sequence in . Since is complete, there exists such that . We claim that has an ffp. Consider Taking limit we get So, we get , which implies that Hence, is an ffp of .

Letting in Theorem 10, we obtain the following:

Corollary 11. Let be a complete metric space. Let be a fuzzy -contractive mapping, and then has an ffp if following conditions are satisfied: (i) is an -admissible mapping(ii)There exists and such that (iii)For any sequence which converges to with we have

If we consider and for all , then we have the following result proved by Ahmed et al. [11].

Corollary 12. Let be a complete metric space, and let . Suppose for each and , there exists and such that . Assume there exists such that then has an ffp.

Example 13. Let and define Then, is a complete bms with Define a fuzzy mapping by and for all , there exists such that Let , , , , and and then we have where Therefore, where for all It follows that all the conditions of Theorem 10 are satisfied, and is an ffp of

Definition 14. Let be a bms. Let and be a pair of fuzzy mappings from into The pair is said to be -admissible if it satisfies the following conditions: (i)For each , there exists such that for all (ii)For each , there exists such that for all

Definition 15. Consider a complete bms . A pair of fuzzy mappings is said to be an -contraction if there exists so that the condition (F1) holds and the following conditions are satisfied:
(Fb2): for each sequence of positive numbers, if then
(Fb3): there exists such that
(Fb4): there exists for which with

Theorem 16. Let be a complete bms with . Let be a pair of fuzzy contraction mappings, and then and have a common ffp if following conditions are satisfied: (i)Both and are -admissible mappings(ii)There exists and such that (iii)For any sequence which converges to with we have

Proof. Let be an arbitrary point of Choose ; then by Lemma 8, there exists such that Now, by Lemma 8, By using assumption (ii), Since is increasing, one writes That is, So, we have By using symmetric property of a bm, we have Then, Consider Thus, Since is increasing, we have That is, Therefore, we obtain Add (35) and (42) to conclude By using the given condition, we have Consequently, Continuing in the same manner, we can define a sequence such that and Now, let By (2.10), and hence By property (Fb2), we obtain as Then, by condition (Fb3), there exists such that From (47), Applying limit , we have . Therefore, there exists such that Using Lemma 7 given in [23], is a Cauchy sequence in which is complete; so, there exists such that . We shall prove that has a fixed point. Consider Since is increasing and one writes Using (52), we obtain Taking limit , we obtain Therefore, This implies that is an ffp of By the same procedure, one can show that Hence, is a common ffp of and

If we consider and for all in then we have the following result proved by Ahmed et al. [11].

Corollary 17. Let be a complete metric space, and let . Suppose for each and , there exists and such that . Assume there exists such that then and have a common ffp.

Example 18. Let and be a bm (with ) defined by It is clear that it is a bms with , but it is not a metric space.
Take as fuzzy mappings defined by Now, we define and we have Therefore, Let , , and , and then we have and Further, it can also be seen that for , where and Further, it is easy to verify that all the conditions of Theorem 10 are satisfied with for all . Hence, there exists which is a common fixed point.

3. Conclusion

In this paper, the notion of fuzzy contractive mappings has been introduced, and few results have been established for the existence of fuzzy fixed points of an -contraction and a pair of -contractions. To elaborate the main results, examples have also been presented. Few corollaries have been established to show that our results generalize and extend many existing classical results available in the literature.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article.