On Some New Common Fixed Point Results for Finite Number of Mappings in Fuzzy Metric Spaces

Abstract

We essentially suggest the concept of mutual sequences and Cauchy mutual sequence and utilize the same to prove the existence and uniqueness of common fixed point results for finite number of self- and non-self-mappings using fuzzy -contractive mappings in fuzzy metric spaces. Our main result was obtained under generalized contractive condition in the fuzzy metric spaces. We provide examples to vindicate the claims and usefulness of such investigations. In this way, the present results generalize and enrich the several existing literature of the fuzzy metric spaces.

1. Introduction

In 1975, Kramosil and Michalek [1] introduced the notion of fuzzy metric spaces using the theory of fuzzy sets, which generalizes the metric spaces. Later on, many authors have introduced the notion of fuzzy metric spaces in different ways (see [2–5]). The widely accepted definition is given by George and Veeramani [6]. They presented slight modification on the definition of fuzzy metric spaces initiated by the respective authors by obtaining Hausdorff topology on the same setting. Utilizing the notion of the fuzzy metric, many authors proved various interesting common fixed point result for self- and non-self-mappings using different contraction in this setting. In 1984, Hadžić [7] proved some common fixed point theorems for family of mapping. After that, Bari and Vetro [8] also proved theorems for family of mappings in fuzzy metric spaces. In 1994, Subrahmanyam [9] generalized Jungck’s theorem [10] in the setting of fuzzy metric spaces introduced by Kramosil and Michalek [1]. Vasuki [11] proved common fixed point theorems in the same setting. In 2002, Rhoades [12] proved common fixed point theorems for non-self-mappings using quasicontraction.

Jungck and Rhoades [13] introduced the concept of weak compatibility in metric spaces, which was further studied by Singh and Jain [14] in the fuzzy metric settings. Sedghi et al. [15] proved common fixed point theorems for four weakly compatible mappings. For the common fixed point using the notion of common limit range property, we refer common fixed point theorems by Chauhan et al. [16]. In this continuation, Imdad et al. [17] proved common fixed point theorems in fuzzy metric spaces using common property (E.A) and Prasad et al. [18] presented some coincidence point theorems via contractive mappings.

Recently, Roldan and Sintunavarat [19] introduced an important concept of fuzzy metric spaces on the product space which is induced by a simple fuzzy metric structure and compare the convergence, Cauchy, and completeness between these two structures. They also proved common fixed point results using CLRg property in the same metric setting. On the other hand, Shukla et al. [20] unify classes of different fuzzy contractive mappings presented in [21–24] and introduced a new class of fuzzy -contractive mapping and notions of properties and to prove fixed point results in the fuzzy metric spaces.

In this paper, firstly we define the mutual sequences and Cauchy mutual sequences. The idea behind defining Cauchy mutual sequence is to collect those Cauchy sequences which are converging to the same limit. After that, we utilize this idea to find common fixed points. Indeed, Cauchy mutual sequences in a fuzzy metric space are the special type of Cauchy sequences in fuzzy metric space which converge to the same limit , if they are convergent. We also generalize the -contraction for finite number of mappings; using these contractive mappings, we will prove some unique common fixed point theorems in fuzzy metric spaces. The main aim of this paper is to prove unique common fixed point theorems using -contraction (which is the extension of -contraction for finite number of mappings) in fuzzy metric spaces for self- and non-self-mappings with the help of mutual sequences. We also give examples for validity of our claims. In this way, our results generalize and improve several existing results.

2. Preliminaries

Throughout this paper, are natural numbers, ; will denote Cartesian product of -copies of and is any nonempty set. In the sequel, sometimes will be denoted by .

Definition 1 (see [6]). An ordered triple is called a fuzzy metric space if is a (nonempty) set, is a fuzzy set on , and is a continuous -norm satisfying the following conditions, for all and : (1)(2), if and only if (3)(4)(5) is continuous

Definition 2 (see [6]). (i)Let be a fuzzy metric space. A sequence is said to be converge to in if and only if for all , i.e., for each and , there exists such that for all (ii)A sequence in a fuzzy metric space is a Cauchy sequence if and only if for each , there exists such that for all . On the other hand, is called a Cauchy sequence if for all and (iii)A fuzzy metric space is said to be complete if every Cauchy sequence in is convergent to some

Lemma 3 (see [19]). Let be a fuzzy metric space and , where and define a fuzzy set on such that Then, the following hold: (i) is also a fuzzy metric space(ii)Let be a sequence on and ; then, sequence converges to on if and only if all sequences converge to on , for all (iii)Let be a sequence on ; then, is Cauchy sequence on if and only if is Cauchy sequence on , for all (iv) is complete if and only if is complete

Definition 4 (see [20]). Let denote the family of all functions satisfying the following condition: .

Definition 5 (see [20]). Let be a self-mapping and a fuzzy metric space. If there exists such that for all , , , then is called a fuzzy -contractive mapping with respect to the function .

Definition 6 (see [20]). Let be any self-mapping in , and a fuzzy metric space then quadruplet has the property (), if there exists such that , for all and for all and , for all implies that

3. Proposed Results

For brevity, we observe that Definitions 4 and 6 can be unified as follows.

Definition 7. Let denote the set of all functions to satisfying the following conditions: (i), for all~(ii)Let and be two sequences in such that , for all and and then

Example 1. Define such that and , , we observe that as .

Definition 8. Let be a fuzzy metric space and are -mapping on satisfying following condition: for all , , , where and is a fuzzy metric space induced by . Then, are said to be fuzzy -contractive mappings.
Observe that for “,” Definition 5 is a particular case of Definition 8.

Definition 9. A sequence is said to be a mutual sequence in a fuzzy metric space .

Definition 10. Let be a mutual sequence in a fuzzy metric space and all sequences , converge, then the sequence is said to be convergent mutual sequence. If the sequence , for all converge to the unique limit , then the mutual sequence is said to be coconvergent mutual sequence and limit is said to be the mutual limit.

Definition 11. A mutual sequence is said to be a Cauchy mutual sequence in a fuzzy metric space , if for each , there exists such that for all , we have for all , i.e., a mutual sequence is said to be a Cauchy mutual sequence if as , for all , where is fuzzy metric spaces induced by .
Now, we are considering the following lemmas to prove the existence and uniqueness of common fixed points.

Lemma 12. Every Cauchy mutual sequence in is a Cauchy sequence in .

Proof. Let be a mutual sequence. By Definition 1 and Lemma 3, we have Now, we have sequence a Cauchy mutual sequence. So, as , we get Hence, is a Cauchy sequence in .
The following example shows that the converse of Lemma 12 may not be true.

Example 2. Let define a fuzzy metric space , where and be a continuous -norm defined as . Consider a mutual sequence on fuzzy metric space , then mutual sequence is a Cauchy sequence in but not Cauchy mutual sequence because as , we have for all .

Lemma 13. In a fuzzy metric space, every convergent Cauchy mutual sequence is coconvergent.

Proof. Let be a convergent mutual sequence which converges to , where , for . Since is a convergent Cauchy mutual sequence, as , we have for all , which implies that for all . Hence, .

Theorem 14. Let be a complete fuzzy metric space; are -self mappings on satisfying (a)fuzzy -contraction(b), for all , Then, have unique common fixed point.

Proof. Let and , for all and . We get as a mutual sequence on and according to Lemma 3, is also a fuzzy metric space.
If , for all and for any , then , i.e., is the fixed point of , for every and a fixed . Suppose that , for fixed . From, Definitions 8 and 7for all . So, is a common fixed point of .
Now, assume that no consecutive terms of the sequence are the same and for some , i.e., for some and for all . From Definition 8 and 7, we have for all . Similarly, we get for all , which is a contradiction in light of the inequality (14). Therefore, , for some .
Now consider, , for all . Then, from Definitions 8 and 7, we have for all and . Taking infimum (over ) in the above inequality, we have for all . Therefore, is a monotonic and bounded sequence, for all . So, there exist some such that for all .
Denote for all .
Now, our claim is , for every . Letting on contrary that , for some . In light of (3), we have and by condition (b), . Applying Definition 7, we get From (2), we have By (22) and as , we get which is a contradiction. We conclude that for all . Hence, the mutual sequence is a Cauchy mutual sequence. Completeness of and Lemma 12 ensure that there exists such that for all . From Lemma 13, sequence is coconvergent sequence, i.e., . Now, we have to prove that is a common fixed point of . Suppose that , . Without loss of generality, let us assume that and , for all . So, there exists such that , and for all . Then, we have as ; from (5) and Lemma 13, we get which is a contradiction. Hence, for all . Hence, is the common fixed point of , for all .
Now, we have to prove the uniqueness of the common fixed point of . Assume on contrary that be two distinct common fixed points of , for all and there exists such that . Then, from Definitions 8 and 7, we get implying thereby , for all . Hence, .

Remark 15. On putting , in Theorem 14, it reduces to Theorem 3.19 presented in [20].

Example 3. Let and be a fuzzy metric space in which is a fuzzy set defined on such that for all and , is a continuous -norm defined as , then is a complete metric space. Now, let us define 5-maps as and a function such that for all .

Let and for all , we get as a mutual sequence. We can easily observe that conditions and of Theorem 14 are satisfied. Hence, is the unique common fixed point of .

Now, we present the fixed point theorem for non-self-mappings.

Theorem 16. Let be a fuzzy metric space, are subset of . Let and are mappings satisfying the following conditions: (i) are complete subspace of (ii) for all , , , , where and is fuzzy metric spaces induced by (iii) for all , , Then, have unique common fixed point.

Proof. Let and , ,..., and , for all . We get as a mutual on .
If , for all and for any , then , , ..., and . Now, from Lemma 3, Definition 7, and condition (ii), we have for all , a contradiction, which implies that , i.e., is a common fixed point of ’s.
From Lemma 3, Definition 7, and condition (ii), for all , we have In the light of inequality (6), we observe that the behavior of mutual sequence in the proof of Theorem 14 and mutual sequence as above is alike. The proof of sequence to be a Cauchy mutual sequence is immediate from Theorem 14.
From (i) and Lemma 12, there exist such that converges to , and from Lemma 13, sequence is coconvergent to some point .
Now, we have to prove that is a common fixed point of ’s. Without loss of generality, let us assume that and So, there exists such that , for all . Then, we have as , and from (5), we get , a contradiction. Hence, , for all . Therefore, is the common fixed point of , for all .
The proof of uniqueness of common fixed point runs similar to the proof of Theorem 14. Hence, we are through.

Example 4. Let and be a fuzzy metric space where is continuous -norm defined as . Let be subset of ; we define and such that Now, define function similar to Example 3. We can easily verify that all conditions of Theorem 16 are satisfied. If and , ,..., and for all . We get as mutual sequence, as unique common fixed point of , and .

4. Conclusion

In this paper, the concept of mutual sequences in is given, and with the help of induced fuzzy metric structure , we define Cauchy mutual sequences in simple fuzzy metric structure . For brevity, Definitions 2.6 and 2.4 (presented in [20]) are unified as Definition 7. We also present contraction, which is an extension of -contraction for finite number of mappings. With the help of mutual sequences, we proved unique common fixed point theorems for finite numbers of mappings using contraction. We also provide many examples to show that our results are meaningful and to support our theorems. The given results generalize and extend several results in the existing literature. As perspectives, it would be interesting that the results presented in this paper proved for other contractive conditions and extend to other nonclassical metric structure, like bipolar fuzzy metric spaces [5] and relational fuzzy metric spaces [25].

Data Availability

No data were used to support this study.

Conflicts of Interest

All the authors declare that they have no conflict of interest.

Authors’ Contributions

Each author contributed equally and significantly to every part of this article. All authors read and approved the final version of the paper.