Abstract and Applied Analysis

Volume 2013 (2013), Article ID 135202, 6 pages

http://dx.doi.org/10.1155/2013/135202

## On Convergence of Fixed Points in Fuzzy Metric Spaces

^{1}School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China^{2}School of Mathematics, Beijing Institute of Technology, Beijing 100081, China^{3}College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{4}School of Information, Capital University of Economics and Business, Beijing 100070, China

Received 7 November 2012; Accepted 9 July 2013

Academic Editor: Fasma Diele

Copyright © 2013 Yonghong Shen 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

We mainly focus on the convergence of the sequence of fixed points for some different sequences of contraction mappings or fuzzy metrics in fuzzy metric spaces. Our results provide a novel research direction for fixed point theory in fuzzy metric spaces as well as a substantial extension of several important results from classical metric spaces.

#### 1. Introduction

Fixed point theory of classical metric spaces plays an important role in general topology. In 1988, Grabiec [1] first extended fixed point theorems of Banach and Edelstein to fuzzy metric spaces in the sense of Kramosil and Michalek. Since then, many authors had dedicated themselves to the study of fixed point theory in fuzzy metric spaces [2–18]. Besides, some authors extended fixed point theory to other types of fuzzy metric spaces in recent years. For instance, Alaca et al. [19] extended the well-known fixed point theorems of Banach and Edelstein to intuitionistic fuzzy metric spaces with the help of Grabiec's work. Simultaneously, Mohamad [20] and Razani [21] proved the existence of fixed point for a nonexpansive mapping of intuitionistic fuzzy metric spaces and the intuitionistic Banach fixed point theorem in complete intuitionistic fuzzy metric spaces, respectively. Later, Ćirić et al. [22] investigated the existence of fixed points for a class of asymptotically nonexpansive mappings in an arbitrary intuitionistic fuzzy metric space. On the other hand, Adibi et al. [23] extended a common fixed point theorem to -fuzzy metric spaces and proved a coincidence point theorem and a fixed point theorem for compatible mappings of type () in these spaces. In 2008, Ješić and Babačev [24] further studied some common fixed point theorems for a pair of -weakly commuting mappings with nonlinear contractive condition in intuitionistic fuzzy metric spaces and -fuzzy metric spaces. In the same year, Park et al. [25] extended some common fixed point theorems for five mappings to -fuzzy metric spaces. Up to now, one can see that the majority of papers mainly focus on the existence of fixed points for different mappings in different fuzzy metric spaces. However, the aim of this paper is to show that the convergence of the sequence of fixed points to some sequences of contraction mappings or fuzzy metrics satisfies certain conditions in fuzzy metric spaces.

#### 2. Preliminaries

Now, we begin with some basic concepts and lemmas. Let denote the set of all positive integers.

*Definition 1 (Schweizer and Sklar [26]). *A binary operation is called a *continuous triangular norm* (shortly, continuous -norm) if it satisfies the following conditions:(TN-1) is commutative and associative;(TN-2) is continuous;(TN-3) for every ;(TN-4) whenever , , and .

In particular, a *t*-norm is said to be *positive* [27] if whenever .

We redefine the notion of a fuzzy metric space by appending the following condition (FM-6) based on the one in the sense of George and Veeramani [2].

*Definition 2. *A *fuzzy metric space* is an ordered triple such that is a (nonempty) set, is a continuous *t*-norm, and is a fuzzy set on satisfying the following conditions, for all , : (FM-1) ; (FM-2) if and only if ; (FM-3) ; (FM-4) ; (FM-5) is continuous; (FM-6) .

*Definition 3 (Grabiec [1] and Vasuki and Veeramani [17]). *Let be a fuzzy metric space. Then(a)a sequence is said to *converge to * in , denoted by , if and only if for all ; that is, for each and , there exists an such that for all ;(b)a sequence in is a *G-Cauchy sequence* if and only if for any and ;(c)the fuzzy metric space is called *G-complete* if every G-Cauchy sequence is convergent.

*Definition 4 (Grabiec [1]). *Let be a fuzzy metric space. A mapping is called a *contraction mapping* if there exists such that
for every and .

According to fuzzy Banach contraction theorem of complete fuzzy metric space in the sense of Grabiec [1], we can obtain the following lemma.

Lemma 5. *Let be a G-complete fuzzy metric space. If is a contraction mapping, then has a unique fixed point. *

*Definition 6. *Let be a fuzzy metric space and let be a sequence of self-mappings on . is a given mapping. The sequence is said to *converge pointwise to * if for each and , there exists an such that
for all and .

*Definition 7. *Let be a fuzzy metric space and let be a sequence of self-mappings on . is a given mapping. The sequence is said to *converge uniformly to * if for each and , there exists an such that
for all and .

*Definition 8. *Let be a fuzzy metric space. A sequence of self-mappings is *uniformly equicontinuous* if for each , there exists an such that implies for every , , and .

*Definition 9 (George and Veeramani [2]). *Let be a fuzzy metric space. The *open ball * and *closed ball * with center and radius , , , respectively, are defined as follows:

Lemma 10 (George and Veeramani [2]). *Every open (closed) ball is an open (a closed) set.*

*Definition 11 (Gregori and Romaguera [3]). *A fuzzy metric space is a *compact space* if is a compact topological space, where is a topology induced by the fuzzy metric .

Based on the corresponding conclusions stated in [2], we can easily obtain the following lemma.

Lemma 12. *Every closed subset of a compact fuzzy metric space is compact. *

Lemma 13. *Let be a fuzzy metric space and let be a sequence of self-mappings on . is a contraction mapping of ; that is, there exists such that for every . is a compact subset of . If converges pointwise to in and it is a uniformly equicontinuous sequence, then the sequence converges uniformly to in . *

*Proof. *For each , we may choose an appropriate such that . Since is uniformly equicontinuous, there exists such that for every , , and . For the foregoing , we fix . Define . By Lemma 10, is a family of open sets of . Obviously, constitutes an open covering of ; that is, . Since is compact, there exist such that . For every (), since converges pointwise to in , for , there exist () such that for all . Set . Clearly, depends only on . For every , there is an such that . Then we have for all . Thus, for all ,
Hence, the sequence converges uniformly to in .

*Definition 14. *A fuzzy metric space in which every point has a compact neighborhood is called *locally compact*.

*Definition 15. *Let be a fuzzy metric space and let be a sequence of fuzzy metrics on . The sequence is said to *upper semiconverge uniformly to * if for each and , there exists an such that and for all , .

#### 3. Main Results

Theorem 16. *Let be a G-complete fuzzy metric space and let be a sequence of self-mappings on where t-norm . is a contraction mapping of ; that is, there exists such that for all , , and satisfying . If there exists at least a fixed point for each () and the sequence converges uniformly to , then . *

*Proof. *Suppose that ; namely, there exist and such that for any there is a satisfying . Fix a number . According to the condition (FM-6) of Definition 2, for , we can find an appropriate such that for any . Since the sequence converges uniformly to , we can make sufficiently large such that for all , . Now for , we have
This leads to a contradiction. Hence, .

Theorem 17. *Let be a G-complete fuzzy metric space where t-norm is positive. If is a self-mapping of and is a contraction mapping for a certain positive integer , then has a unique fixed point. *

*Proof. *First of all, if , the theorem is evident. In addition, if , according to Lemma 5, we need only to prove that is a contraction mapping. Since is a contraction mapping, there is such that for every , and . Define another fuzzy metric on using as follows:
Actually, it is easy to verify that the foregoing two fuzzy metrics are equivalent. Meantime, we claim that is a contraction mapping with respect to the fuzzy metric , since

Corollary 18. *Let be a G-complete fuzzy metric space and let be a sequence of self-mappings on where t-norm . is a self-mapping of , and is a contraction mapping for a certain positive integer . If there exists at least a fixed point for each and the sequence converges uniformly to , then . *

*Proof. *It follows from Theorems 16 and 17.

Theorem 19. *Let be a locally compact fuzzy metric space and let be a sequence of self-mappings on . is a contraction mapping; that is, there exists a such that for all , . If the following conditions are satisfied:*(i)* is a contraction mapping for a certain ,*(ii)* converges pointwise to and is a uniformly equicontinuous sequence,*(iii)*, ,**then the sequence converges to ; that is, . *

*Proof. *For each , we choose such that . If given , we may assume that is sufficiently small such that is a compact subset of . By Lemma 13, since is uniformly equicontinuous and pointwise convergent on , we know that converges uniformly to on the compact subset . Then, for the foregoing , there exists such that for all , , and . In addition, since is a contraction mapping, we have for all . Thus, for all and , we can obtain
Therefore, for all , is an invariant set for . Since is a contraction mapping for a certain positive integer , it follows that the fixed point of is contained in the set , when . By the definition of , we have for all . In fact, although should satisfy the foregoing condition, it may be sufficiently small. Hence, we can obtain .

In addition, if *t*-norm , then we can obtain the following some important conclusions.

Lemma 20. *Let be a G-complete fuzzy metric space and let be a compact subset of where t-norm . and are a sequence of fuzzy metrics and a sequence of self-mappings on , respectively. If they satisfy the following conditions:*(i)

*upper semiconverges uniformly to ,*(ii)

*is a contraction mapping for the fuzzy metric , ,*(iii)

*converges pointwise to ,*

*then converges uniformly to in with regard to the fuzzy metric .*

*Proof. *For each , choose such that . Since upper semiconverges uniformly to , there exists such that and for all , . Choose in such that for each . Then, for all , we have
Therefore, the sequence is uniformly equicontinuous in with regard to the fuzzy metric . Since is pointwise convergent and is a compact subset of , according to Lemma 13, it follows that the subsequence () converges uniformly to in . Hence, converges uniformly to in .

Theorem 21. *Let be a locally compact fuzzy metric space where t-norm . If and satisfy the following conditions:*(i)

*upper semiconverges uniformly to ,*(ii)

*is a contraction mapping for the fuzzy metric , ,*(iii)

*converges pointwise to ,*(iv)

*, ,*

*then the sequence of fixed points of converges to the fixed point of ; that is, .*

*Proof. *For each , choose such that . Meantime, for , we may make sufficiently small such that is compact in for each . By Lemma 20, we know that converges uniformly to in with respect to the fuzzy metric . Then, for every , there exists an such that for all , . Thus, when , for all , we have
Therefore, is an invariant set in with regard to . Since is still a contraction mapping restricted to concerning on , one can see that the fixed point is also included in . Apparently, for all , we can obtain . Since is sufficiently small, it can easily be shown that converges to ; that is, . This completes the proof.

Theorem 22. *Let be a compact fuzzy metric space where t-norm . The sequences and satisfy the following conditions:*(i)

*upper semiconverges uniformly to ;*(ii)

*is a contraction mapping for the fuzzy metric , ;*(iii)

*converges pointwise to .*

*If every mapping () has a fixed point and there is a subsequence of which converges to , then .*

*Proof. *Let denote the closure of the set . By Lemma 12, we can easily know that is a compact set. According to Lemma 20, it follows that the subsequence converges uniformly to in with regard to . Obviously, converges to . Hence, .

Theorem 23. *Let be a fuzzy metric space. is a sequence of contraction mappings and satisfying (). is a contraction mapping. If is a pointwise convergent sequence with respect to and the subsequence of converges to , then has a fixed point . *

*Proof. *For each , choose such that . Since is a convergent subsequence and is a pointwise convergent sequence, for a given , we may choose such that and for all , . For every , we denote by () the contraction constant of . Thus, for all , we have
Therefore, the subsequence converges to . According to the uniqueness of limit, it follows that ; that is, is a fixed point of .

#### Acknowledgments

This work was supported by “Qing Lan” Talent Engineering Funds by Tianshui Normal University. Dong Qiu acknowledges the support of the National Natural Science Foundation of China (Grant no. 11201512) and the Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001). Wei Chen acknowledges the support of the Beijing Municipal Education Commission Foundation of China (no. KM201210038001) and the Humanity and Social Science Youth Foundation of Ministry of Education of China (no. 13YJC630012).

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