Journal of Mathematics
Volume 2021 (2021), Article ID 9196642, 6 pages
https://doi.org/10.1155/2021/9196642

Research Article

On Some Fixed Point Results in Fuzzy Metric Spaces

Rachid Mecheraoui,¹ Zoran D. Mitrović,² Vahid Parvaneh,³ Hassen Aydi,4,5,6 and Naeem Saleem⁷

¹Department of Mathematics and Informatics, Abbes Laghror University, Khenchela 40000, Algeria
²University of Banja Luka, Faculty of Electrical Engineering, Patre 5, Banja Luka 78000, Bosnia and Herzegovina
³Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran
⁴Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia
⁵China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
⁶Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa
⁷Department of Mathematics (SSC), University of Management and Technology, C-II, Johar Town, Lahore, Pakistan

Correspondence should be addressed to Vahid Parvaneh; zam.dalahoo@gmail.com and Hassen Aydi; hassen.aydi@isima.rnu.tn

Received 12 June 2021; Accepted 6 September 2021; Published 20 September 2021

Academic Editor: Ali Jaballah

Copyright © 2021 Rachid Mecheraoui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 that

for all and . Hence, by relations (3)–(5), it yields , and this means

which 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 that

for 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 obtain

for all and for all in , which deduce that

for all . Now, to prove that is a Cauchy sequence, we assume to the contrary. Since is a nondecreasing function, so that

for 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 that

By virtue of assumptions (T4) and (F4) and relations (11) and (12), it follows that

So, according to assumptions (T2)-(T3), limit (9), and Lemma 1, one has

Suppose 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 verifying

for all .
Having satisfying , we obtain

as .
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 that

for all , and by relation (7), we obtain

for 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 that

for all in and for all . Then, has a unique fixed point . Furthermore, for all , the sequence converges to .

Proof. Let and and . Obviously, we have

for all , and due to relation (21), we obtain . Recursively, we obtain a sequence in verifying

for all in . It is a bounded increasing sequence. Then, there exists a function such that

for all . We claim that , for all . Suppose not, i.e., there exists such that . By the limit in (24), for all , there exists such that

for 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 that

On the other hand, for all and all , we have

Condition (21) assures that

which, with the limit in (27), gives , and finally

For 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 that

for 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 have

Let and such that

Now, we choose in (34) such that . Therefore, using relations (34) and (35), it follows that

and 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 norm

with 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 consider

So,

Using (40), we can write

Since 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.

References

Z. Deng, “Fuzzy pseudo-metric spaces,” Journal of Mathematical Analysis and Applications, vol. 86, no. 1, pp. 74–95, 1982.

M. A. Erceg, “Metric spaces in fuzzy set theory,” Journal of Mathematical Analysis and Applications, vol. 69, no. 1, pp. 205–230, 1979.

O. Kaleva and S. Seikkala, “On fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 12, no. 3, pp. 215–229, 1984.

A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395–399, 1994.

I. Kramosil and J. Michalek, “Fuzzy metrics and statistical metric spaces,” Kybernetika, vol. 11, no. 6, pp. 336–344, 1975.

V. Gregori and S. Romaguera, “Some properties of fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 115, no. 3, pp. 485–489, 2000.

M. Grabiec, “Fixed points in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 27, no. 3, pp. 385–389, 1988.

V. Gregori and A. Sapena, “On fixed-point theorems in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 125, no. 2, pp. 245–252, 2002.

D. Gopal and C. Vetro, “Some new fixed point theorem in fuzzy mertric spaces,” Iranian Journal of Fuzzy Systems, vol. 11, no. 3, pp. 95–107, 2014.

D. Gopal, M. Imdad, C. Vetro, and M. Hasan, “Fixed point theory for cyclic weak -contraction in fuzzy metric space,” Journal of Nonlinear Analysis and Application, vol. 2012, 11 pages, 2012.

C. di Bari and C. Vetro, “A fixed point theorem for a family of mappings in a fuzzy metric space,” Rendiconti del Circolo Matematico di Palermo, vol. 52, no. 2, pp. 315–321, 2003.

V. Gupta, R. K. Saini, and M. Verma, “Fixed point theorem by altering distance technique in complete fuzzy metric spaces,” International Journal of Computer Aided Engineering and Technology, vol. 13, no. 4, pp. 437–447, 2020.

G. Song, “Comments on ‘A common fixed point theorem in a fuzzy metric space’,” Fuzzy Sets and Systems, vol. 135, no. 3, pp. 409–413, 2003.

D. Mihet, “A Banach contraction theorem in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 144, pp. 431–439, 2004.

R. Vasuki and P. Veeramani, “Fixed point theorems and Cauchy sequences in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 135, no. 3, pp. 415–417, 2003.

B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, no. 1, pp. 314–334, 1960.

P. Borisut, P. Kumam, V. Gupta, and N. Mani, “Generalized (ψ, α, β)-weak contractions for initial value problems,” Mathematics, vol. 7, no. 3, p. 266, 2019.