On Some Fixed Point Results in Fuzzy Metric Spaces

Abstract

In the existing literature, Banach contraction theorem as well as Meir-Keeler fixed point theorem were extended to fuzzy metric spaces. However, the existing extensions require strong additional assumptions. The purpose of this paper is to determine a class of fuzzy metric spaces in which both theorems remain true without the need of any additional condition. We demonstrate the wide validity of the new class.

1. Introduction

The well-known Banach’s contraction principle (BCP) is a classic method in nonlinear analysis and is one of the most important and heavily researched fixed point theorems. It states that if is a contraction on the complete metric space , then  has a unique fixed point in , and  for all .

There exist many different concepts of a fuzzy metric space (cf. [1–5]). However, the definition of George and Veeramani [4] is reverently cited in most references in this area. The authors of [4] updated Kramosil and Michalek’s definition of fuzzy metric space and obtained a Hausdorff topology for this kind of fuzzy metric space. The topology induced by a fuzzy metric space in the sense of George and Veeramani has recently been demonstrated to be metrizable in [6]. It would be very useful if the BCP remains true in fuzzy metric spaces. This has been discussed in [7, 8] and then in numerous other directions over the years (see, for example, [9–11]). In [12], the generalized altering distance function has been defined and the Banach contraction principal in complete fuzzy metric spaces using altering distance has been extended.

In [7], in order to give the fuzzy version of the BCP, Grabiec introduced a new definition of a Cauchy sequence in fuzzy metric spaces as follows.

Definition 1. A sequence in a fuzzy metric space is Cauchy if for each and .
It can be seen easily that this definition of a Cauchy sequence is incorrect, for more details see [13–15], whereas, in [8], Gregori and Sapena extended the Banach fixed point theorem to fuzzy version stating the following theorem.

Theorem 1. Let be a complete fuzzy metric space in which fuzzy contractive sequences are Cauchy. Let be a fuzzy contractive mapping with serving as the contractive constant. Then, has a unique fixed point.

Here, authors added a fairly strong assumption, that is, “every contractive sequence is Cauchy.”

In this paper, we prove that the Banach contraction theorem as well as the Meir-Keeler fixed point theorem remain true in fuzzy metric spaces with only a slight modification in the definition of fuzzy spaces given by George and Veeramani. Last, in this paper, we give some results to illustrate the broad validity of our results.

Before stating the main results, we need the following definitions.

Definition 2 (Schweizer and Sklar [16]). A binary operation is called a continuous t-norm if it satisfies the following assertions: (T1) is commutative and associative; (T2) is continuous; (T3) for all ; (T4) when and , with .Here is the definition of a fuzzy metric space given by George and Veeramani:

Definition 3 (George and Veeramani [4]). A fuzzy metric space is an ordered triple in which is a nonempty set, is a continuous t-norm, and is a fuzzy set on such that (T5) ; (T6) if and only if ; (T7) ; (T8) ; (T9) is left continuous,for all and .
In this paper, we will consider the following class of fuzzy metric spaces.

Definition 4 (fuzzy metric space). Let denote a nonempty set, refers to a continuous t-norm, and serves as a fuzzy set on such that (F1) ; (F2) if and only if ; (F3) ; (F4) ; (F5) is left continuous. (F6) For some , the family is uniformly equicontinuous,for all , and . Then, the triple is called an fuzzy metric space.

Remark 1. Obviously, all fuzzy metric space is a fuzzy metric space. So, all properties in fuzzy metric spaces remain true in fuzzy metric spaces.

Definition 5 (George and Veeramani [4]). Let be a fuzzy metric space. Then,(i)A sequence converges to if and only if as for all ;(ii)A sequence in is a Cauchy sequence if and only if for all and , there exists such that for all ;(iii)The fuzzy metric space is complete if every Cauchy sequence converges to some .In the sequel, we use the following essential technical lemma.

Lemma 1. Let be an fuzzy metric space, be the continuous extension of up to , and be a sequence in such that , for all . Then,

Proof. For all , function is positive, continuous, and nondecreasing on , so is well defined. Let be a monotonically decreasing sequence of positive numbers, converging to 0, and be a sequence in such that , for all , i.e., .
From which it follows that ,Therefore, ,On the other hand, by the fact that and assumption (F6), we deduce that , such thatfor all and . Hence, by relations (3)–(5), it yields , and this meanswhich achieves the proof of the lemma.

2. Main Results

Now, we will present our key finding.

Theorem 2. Let be a complete fuzzy metric space. Let be a fuzzy contractive mapping with the contractive constant , i.e., there exists such thatfor all in and for all . Then, has a unique fixed point . Furthermore, for all , the sequence converges to .

Proof. Let in and . Let and . By inequality (7), we obtainfor all and for all in , which deduce thatfor all . Now, to prove that is a Cauchy sequence, we assume to the contrary. Since is a nondecreasing function, so thatfor all . Let . By virtue of limit (9) and the last relation, we can write that :Taking into account the continuity of the function and the fact that , we can choose such thatBy virtue of assumptions (T4) and (F4) and relations (11) and (12), it follows thatSo, according to assumptions (T2)-(T3), limit (9), and Lemma 1, one hasSuppose that for all , there exists such that means, having in mind relations (7) and (14), that the sequence has two subsequences and verifying(for the sake of simplicity, we have saved the same notation for the subsequence).
Now, we suppose that there exists such that for all . We claim that . Suppose not, i.e., there exists and two subsequences and verifyingfor all .
Having satisfying , we obtainas .
This is a contradiction. Then,Relations (14), (15), and (18) drive to a clear contradiction with condition (7). So, is a Cauchy sequence in the complete fuzzy metric space and we deduce that there exists such thatfor all , and by relation (7), we obtainfor all and for all . Passing to the limit, having in mind the limit in (19), it follows that , which, with assumption (F2) and relation (7), means that is the unique fixed point of mapping . This achieves the proof.

Theorem 3. (fuzzy Meir-Keeler fixed point theorem). Let be a complete fuzzy metric space. Let be a fuzzy Meir-Keeler type mapping, i.e., for all , there exists such thatfor all in and for all . Then, has a unique fixed point . Furthermore, for all , the sequence converges to .

Proof. Let and and . Obviously, we havefor all , and due to relation (21), we obtain . Recursively, we obtain a sequence in verifyingfor all in . It is a bounded increasing sequence. Then, there exists a function such thatfor all . We claim that , for all . Suppose not, i.e., there exists such that . By the limit in (24), for all , there exists such thatfor all , which, with condition (21), implies that . This is a clear contradiction with (24). Therefore,for all . Now, we follow, exactly, the same lines as in the proof of Theorem 2 to deduce that is a Cauchy sequence in the complete fuzzy metric space , which deduce that there exists such thatOn the other hand, for all and all , we haveCondition (21) assures thatwhich, with the limit in (27), gives , and finallyFor the uniqueness, we assume that there exists such that . It is clear that .
Hence, by (21), or , a contradiction, and this achieves the proof.
Now, we give the following corollary.

Corollary 1. Let be a complete metric space, and a Meir-Keeler mapping on , i.e., for each , there exists such that for all ,

Let be a function on defined by

Then,(1) is an fuzzy metric space, where · is the product t-norm.(2)For all , there exists such thatfor all in and for all .

Proof. is a fuzzy metric space (see [4]) and is a set of functions with common Lipschitz constant “1.” So, it is uniformly equicontinuous. This means that is an fuzzy metric space. For the second assumption, it suffices to see that
For all and all , we haveLet and such thatNow, we choose in (34) such that . Therefore, using relations (34) and (35), it follows thatand this achieves the proof.

3. Application

The purpose of this section is to give an example of the existence of a solution for an integral equation, where we can apply Theorem 2 to get its solution. For such integral equations, we refer the reader to [17] where the authors provide a common solution for a system of two integral equations.

Consider the integral equation,and Banach space of all continuous functions defined on equipped with supremum normwith induced metric

Now, consider the fuzzy metric space with product norm as

According to George and Veeramani, standard fuzzy metric space and the corresponding metric space have same topologies. So, fuzzy metric space defined in (40) is complete.

Theorem 4. Consider the integral operator on as

Suppose that there exists such that and suppose that satisfies the following condition:for all and for all where

Then, the integral equation (37) has a unique solution.

Proof. Let and considerSo,Using (40), we can writeSince all the conditions of Theorem 2 hold, (37) has a unique solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.