Fuzzy Meir-Keeler’s Contraction and Characterization

Abstract

In this study, a fuzzy Meir-Keeler’s contraction theorem for complete FMS based on George and Veeramani idea is established. Then, we characterize fuzzy Meir-Keeler’s contractions as contractive types induced by functions called fuzzy -function. Moreover, we show that the converse of it is true. Finally, we bring some examples and corollaries certify our results and new improvement.

1. Introduction

Fixed point theory and related topics are an active research field with a wide range of applications in mathematics, engineering, chemistry, physics, economics, and computer sciences. Many authors have been studied this theory in hyperstructure spaces alongside the classical metric spaces and normed spaces. Among them, it could be cited probabilistic (and fuzzy) metric spaces. The phrase of fuzzy metric space (FMS), introduced by Kramosil and Michalek [1], then George and Veeramani [2], modified this idea which has applications in quantum particle physics [3] and in the two-slit experiment [4, 5]. Also, the theory of FMS is, in this framework, very disparate from the usual theory of metric best approximation and completion, e.g., see [6] and [7–9], respectively. Grabiec [10] developed and extended fixed point theory to probabilistic metric space. Later on, several authors have participated to the growth of this theory (see [10–17]).

In 2006, Lim [18] introduced -functions (LF) and characterized Meir-Keeler’s contractive as a self-map on that satisfies for some LF (see also [19]). This characterization prepares it easy to compare such maps with those satisfying Boyd-Wong’s condition (see [20]). Then, Meir-Keeler [21] developed Boyd-Wong’s result as follows

In 2005, Razani ([22], Theorem 2.2) introduced a contraction theorem in FMS. Our main result in this paper is to extend this Theorem to fuzzy Meir-Keeler’s contraction. We assert that if is a FMS and on be a fuzzy Meir-Keeler’s contractive self-mapping, then, has a unique fixed point in . Our works are an extension of some recent results that we notice them. Then, we characterize fuzzy Meir-Keeler’s contractive map as a map so that where is a fuzzy LF and is the minimum -norm, i.e., .

2. Preliminaries

In what follows, we mention some reported results, definitions, and examples related to the theory of FMS which are needed. More details and explanations can be followed in [2, 6–9, 23, 24].

Definition 1 (see [24]). A -norm is a function such that is continuous, commutative, associative, , and , where and , and .

In the study of probabilistic and FM spaces, the presentation of -norms was raised by the requirement to assign the triangle inequality (condition (iv) below) in the setting of metric spaces to that of fuzzy metric spaces. Numerous examples of this concept have been cited by various researchers, for instance, one may be found in [14, 15].

Definition 2 (George and Veeramani [2]). is said FMS where is a set, is a continuous -norm, and on is a function with (1), for all (2), for all iff (3)(4),(5) be a continuous fuzzy setwhere and .Now, we bring two theorems and definitions that play as key roles in this paper, and we continue the next sections based on these concepts to reach our aims.

Theorem 3 (see [2]). In a FMS , (i.e., converges to ) iff, as .

Definition 4 (see [2]). is said a Cauchy sequence in if for all and , so that whenever . Also, we call it complete if every Cauchy sequence is convergent.

Theorem 5 (see [9, 17, 23]). In a FMS , is a continuous function.

Definition 6 (see [22]). In a FMS , on is said a fuzzy contractive self-mapping (FCM), if

3. Main Results

In this section, we discuss concerning fuzzy Meir-Keeler’s contractive self-mapping. We give a proof of Meir-Keeler’s fixed point theorem in FMS. Here, we consider fuzzy Meir-Keeler’s contraction (FMK) to state our main results.

Definition 7. In a FMS , on is a (FMK), if for all , such that for all ,

Remark 8. Each FMK is a FCM but the inverse is not necessarily true. To prove this claim, we bring the flowing example.

Example 1. It is precise that FMK implies fuzzy contraction but note the inverse is not true because assume that and defined as Example 2.9 of [2] with . If and then and . Thus, . Therefore, we get , for all , i.e., is a FCM. But if , , and , then, there exists so that holds. Hence, if be a FMK, we easily have that for all . It means that , which is a contradiction. Thus, does not satisfy FMK, and this proves our claim.

Theorem 9. If is a complete FMS, where -norm is defined as . Suppose self-mapping on is a FMK. Then, has a unique fixed point.

Proof. For all and , we get . Since if there are and so that . For simplicity, we put and . Hence, for each Then, by (4), , which is a contradiction. Let . Set and suppose that , since otherwise has a fixed point. Assume that be arbitrary and fixed after choosing. Now, let . is a nonincreasing sequence. So, converges to . Thus, such that, , for all . We claim that . If , there is such that, for all and , where , we have . Select such that . We have thus so . It means that . Hence by 2, , which is a contradiction, since , for all .
Now, we prove that is a Cauchy sequence. If is not, then there is , so that, for any , there are such that . For each , there is such that for any and , if then . There exists such that for all and , . Take such that Then, we get It presents that . Thus, by (4), . Also, In other words, . Thus, by (4), . By induction, . By (11), , i.e., , which is a contradiction. Thus, is convergent to . We get Then, . So, .
For proving the uniqueness of , suppose there is a with , it follows that This is a contradiction, then, the uniqueness is proved.
Note that Theorem 9 is a generalization of ([22], Theorem 2.2); when we consider -norm , this shows one of the most reason of improvement of [22].☐

4. Characterization of Fuzzy Meir-Keeler’s Contractions

In this part of the paper, we characterize FMK maps. In Theorem 11, we provided a sufficient and necessary condition for FMK maps by tools of fuzzy -function. Also, this generalizes Theorem 1 of [21]. More precisely, we show that if a self-mapping on be a FMK then there is a fuzzy -function from into itself such that, for all , . Also, we show that the converse of it is true. In some sense, our work is very close to Suzuki [19], and Lim [18].

Definition 10 (see [18]). Function from into itself is said to be a fuzzy LF if and so that .
In the following, we bring a theorem to show the condition of reaching a self-map on in a FMS to FMK.

Theorem 11. Suppose is a FMS, and the -norm is defined as . Then, a self-map on is FMK iff there exists a (nondecreasing) fuzzy LF as (2) is satisfied.

Proof. Assume that is a FMK. By Definition 7, let a function from into is defined such that for . With such , let present a nondecreasing function from into by for any . Since , we get for . Suppose that the function is defined by It is obvious that . for . Let be fixed. If , where , one can put . Otherwise, with . But , then, we get , and if then Thus, we get . This is a contradiction. Thus, it is concluded that Now, we shall select with , and let . Consider . Since we obtain . Therefore Hence, is a fuzzy LF. If we consider with and fixed. The definition of implies that there is in which . Thus, where . Therefore, holds. Therefore, satisfies (2). We define function as for any , we get Hence, also satisfies (2). Easily can be verified that is a nondecreasing fuzzy LF. This completes the proof.☐

Considering the following example, we briefly explain the Theorems 9 and 11.

Example 2. Let with usual distance on and on be defined as follows: Let we set as Example 2.9 of [2] and then, Therefore, satisfies the conditions of Theorem 9. Hence, has a (unique) fixed point in . Also, if we consider Thus, is a fuzzy LF and In this step, we can easily reach to the following corollaries.

Corollary 12. Suppose is a FMS, where and are a self-mapping on . If there exists a fuzzy -function where (2) is satisfied, then, has a unique fixed point.

Proof. Let has given. By Definition 10 there is such that for all , . It means that if then by (2) It means that (4) holds. Thus, has a (unique) fixed point.☐

Based on theorems in Section 3 and 4, we have the following result:

Corollary 13. Suppose that be a FMS where and is a self-mapping on . The followings statements are equivalent: (1) is a FMK(2)There is a fuzzy LF such that (2) satisfies

Proof. By Theorem 11 and Corollary 12, we easily obtain the desired results.☐

5. Conclusion

In this paper motivated by the results of Razani [22], a new class of FMK contractions in a complete FMS was introduced by reducing the contractive condition of the so-called Meir-Keeler’s contractive maps. In Theorems 9, we established a fixed point theorem; and in Theorem 11, we provided a sufficient and necessary condition for fuzzy FMK maps. Our work generalizes Theorem 1.1 of [22] and Theorem 1 of [21].

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. All authors read and approved the final manuscript.