- Research Article
- Open Access

# Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces

- Xin-Qi Hu
^{1}Email author

**2011**:363716

https://doi.org/10.1155/2011/363716

© Xin-Qi Hu. 2011

**Received: **23 November 2010

**Accepted: **27 January 2011

**Published: **23 February 2011

## Abstract

## Keywords

- Point Theorem
- Fixed Point Theorem
- Open Ball
- Cauchy Sequence
- Contractive Mapping

## 1. Introduction

Since Zadeh [1] introduced the concept of fuzzy sets, many authors have extensively developed the theory of fuzzy sets and applications. George and Veeramani [2, 3] gave the concept of fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space which have very important applications in quantum particle physics particularly in connection with both string and -infinity theory.

Bhaskar and Lakshmikantham [4], Lakshmikantham and Ćirić [5] discussed the mixed monotone mappings and gave some coupled fixed point theorems which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi et al. [6] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang [7] gave some common fixed point theorems under -contractions for compatible and weakly compatible mappings in Menger probabilistic metric spaces. Many authors [8–23] have proved fixed point theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces.

In this paper, using similar proof as in [7], we give a new common fixed point theorem under weaker conditions than in [6] and give an example which shows that the result is a genuine generalization of the corresponding result in [6].

## 2. Preliminaries

First we give some definitions.

Definition 1 (see [2]).

A binary operation is continuous -norm if is satisfying the following conditions:

(1) is commutative and associative;

Definition 2 (see [24]).

The -norm is an example of -norm of H-type, but there are some other -norms of H-type [24].

Obviously, is a H-type norm if and only if for any , there exists such that for all , when .

Definition 3 (see [2]).

A 3-tuple is said to be a fuzzy metric space if is an arbitrary nonempty set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions, for each and :

A subset is called open if, for each , there exist and such that . Let denote the family of all open subsets of . Then is called the topology on induced by the fuzzy metric . This topology is Hausdorff and first countable.

Example 1.

Let be a metric space. Define -norm and for all and , . Then is a fuzzy metric space. We call this fuzzy metric induced by the metric the standard fuzzy metric.

Definition 4 (see [2]).

Let be a fuzzy metric space, then

(3)a fuzzy metric space is said to be complete if and only if every Cauchy sequence in is convergent.

- (1)
- (2)

- (3)
- (4)

Definition 5 (see [6]).

Lemma 1.

Proof.

If not, since is nondecreasing and , there exists such that , then for , when as and we get , which is a contraction.

Remark 2.

Condition (2.7) cannot guarantee the -property. See the following example.

Example 2.

Define , where and each satisfies the following conditions:

(-2) is upper semicontinuous from the right;

It is easy to prove that, if , then for all .

Lemma 2 (see [7]).

Definition 6 (see [5]).

Definition 7 (see [5]).

Definition 8 (see [7]).

Definition 9 (see [7]).

Definition 10 (see [7]).

Definition 11 (see [7]).

Remark 3.

It is easy to prove that, if and are commutative, then they are compatible.

## 3. Main Results

Theorem 1.

Suppose that , and is continuous, and are compatible. Then there exist such that , that is, and have a unique common fixed point in .

Proof.

The proof is divided into 4 steps.

Step 1.

Prove that and are Cauchy sequences.

for all with and . So is a Cauchy sequence.

Similarly, we can get that is also a Cauchy sequence.

Step 2.

Prove that and have a coupled coincidence point.

for all . Next we prove that and .

which implies that . Similarly, we can get .

Step 3.

Since is continuous and for all , there exists such that and .

for all . We can get that and .

Step 4.

Since is continuous and , there exists such that .

On the other hand, since , by condition we have . Then for any , there exists such that .

Thus we have proved that and have a unique common fixed point in .

This completes the proof of the Theorem 1.

Taking (the identity mapping) in Theorem 1, we get the following consequence.

Corollary 1.

Then there exist such that , that is, admits a unique fixed point in .

Let , where , the following by Lemma 1, we get the following.

Corollary 2 (see [6]).

for all , where , and is continuous and commutes with . Then there exists a unique such that .

Next we give an example to demonstrate Theorem 1.

Example 3.

for all . Then is a complete FM-space.

Then satisfies all the condition of Theorem 1, and there exists a point which is the unique common fixed point of and .

We consider the following cases.

it is easy to verified.

holds for all . So (3.36) holds for .

for all that . We can verify it holds.

Thus it is verified that the functions , , satisfy all the conditions of Theorem 1; is the common fixed point of and in .

## Declarations

### Acknowledgment

The author is grateful to the referees for their valuable comments and suggestions.

## Authors’ Affiliations

## References

- Zadeh LA:
**Fuzzy sets.***Information and Computation*1965,**8:**338–353.MATHMathSciNetGoogle Scholar - George A, Veeramani P:
**On some results in fuzzy metric spaces.***Fuzzy Sets and Systems*1994,**64**(3):395–399. 10.1016/0165-0114(94)90162-7MATHMathSciNetView ArticleGoogle Scholar - George A, Veeramani P:
**On some results of analysis for fuzzy metric spaces.***Fuzzy Sets and Systems*1997,**90**(3):365–368. 10.1016/S0165-0114(96)00207-2MATHMathSciNetView ArticleGoogle Scholar - Bhaskar TG, Lakshmikantham V:
**Fixed point theorems in partially ordered metric spaces and applications.***Nonlinear Analysis. Theory, Methods & Applications*2006,**65**(7):1379–1393. 10.1016/j.na.2005.10.017MATHMathSciNetView ArticleGoogle Scholar - Lakshmikantham V, Ćirić L:
**Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces.***Nonlinear Analysis. Theory, Methods & Applications*2009,**70**(12):4341–4349. 10.1016/j.na.2008.09.020MATHMathSciNetView ArticleGoogle Scholar - Sedghi S, Altun I, Shobe N:
**Coupled fixed point theorems for contractions in fuzzy metric spaces.***Nonlinear Analysis. Theory, Methods & Applications*2010,**72**(3–4):1298–1304. 10.1016/j.na.2009.08.018MATHMathSciNetView ArticleGoogle Scholar - Fang J-X:
**Common fixed point theorems of compatible and weakly compatible maps in Menger spaces.***Nonlinear Analysis. Theory, Methods & Applications*2009,**71**(5–6):1833–1843. 10.1016/j.na.2009.01.018MATHView ArticleMathSciNetGoogle Scholar - Ćirić LB, Miheţ D, Saadati R:
**Monotone generalized contractions in partially ordered probabilistic metric spaces.***Topology and its Applications*2009,**156**(17):2838–2844. 10.1016/j.topol.2009.08.029MATHMathSciNetView ArticleGoogle Scholar - O'Regan D, Saadati R:
**Nonlinear contraction theorems in probabilistic spaces.***Applied Mathematics and Computation*2008,**195**(1):86–93. 10.1016/j.amc.2007.04.070MATHMathSciNetView ArticleGoogle Scholar - Jain S, Jain S, Bahadur Jain L:
**Compatibility of type (P) in modified intuitionistic fuzzy metric space.***Journal of Nonlinear Science and its Applications*2010,**3**(2):96–109.MATHMathSciNetGoogle Scholar - Ćirić LB, Ješić SN, Ume JS:
**The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy metric spaces.***Chaos, Solitons and Fractals*2008,**37**(3):781–791. 10.1016/j.chaos.2006.09.093MATHMathSciNetView ArticleGoogle Scholar - \'Cirić L, Lakshmikantham V:
**Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces.***Stochastic Analysis and Applications*2009,**27**(6):1246–1259. 10.1080/07362990903259967MATHMathSciNetView ArticleGoogle Scholar - Ćirić L, Cakić N, Rajović M, Ume JS:
**Monotone generalized nonlinear contractions in partially ordered metric spaces.***Fixed Point Theory and Applications*2008,**2008:**-11.Google Scholar - Aliouche A, Merghadi F, Djoudi A:
**A related fixed point theorem in two fuzzy metric spaces.***Journal of Nonlinear Science and its Applications*2009,**2**(1):19–24.MATHMathSciNetGoogle Scholar - Ćirić L:
**Common fixed point theorems for a family of non-self mappings in convex metric spaces.***Nonlinear Analysis. Theory, Methods & Applications*2009,**71**(5–6):1662–1669. 10.1016/j.na.2009.01.002MATHView ArticleMathSciNetGoogle Scholar - Rao KPR, Aliouche A, Babu GR:
**Related fixed point theorems in fuzzy metric spaces.***Journal of Nonlinear Science and its Applications*2008,**1**(3):194–202.MATHMathSciNetGoogle Scholar - Ćirić L, Cakić N:
**On common fixed point theorems for non-self hybrid mappings in convex metric spaces.***Applied Mathematics and Computation*2009,**208**(1):90–97. 10.1016/j.amc.2008.11.012MATHMathSciNetView ArticleGoogle Scholar - Ćirić L:
**Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces.***Chaos, Solitons and Fractals*2009,**42**(1):146–154. 10.1016/j.chaos.2008.11.010MATHMathSciNetView ArticleGoogle Scholar - Shakeri S, Ćirić LJB, Saadati R:
**Common fixed point theorem in partially ordered -fuzzy metric spaces.***Fixed Point Theory and Applications*2010,**2010:**-13.Google Scholar - Ćirić L, Samet B, Vetro C:
**Common fixed point theorems for families of occasionally weakly compatible mappings.***Mathematical and Computer Modelling*2011,**53**(5–6):631–636. 10.1016/j.mcm.2010.09.015MATHMathSciNetView ArticleGoogle Scholar - Ćirić L, Abbas M, Saadati R, Hussain N:
**Common fixed points of almost generalized contractive mappings in ordered metric spaces.***Applied Mathematics and Computation*2011,**217**(12):5784–5789. 10.1016/j.amc.2010.12.060MATHMathSciNetView ArticleGoogle Scholar - Ćirić L, Abbas M, Damjanović B, Saadati R:
**Common fuzzy fixed point theorems in ordered metric spaces.***Mathematical and Computer Modelling*2011,**53**(9–10):1737–1741. 10.1016/j.mcm.2010.12.050MATHMathSciNetView ArticleGoogle Scholar - Kamran T, Cakić N:
**Hybrid tangential property and coincidence point theorems.***Fixed Point Theory*2008,**9**(2):487–496.MATHMathSciNetGoogle Scholar - Hadžić O, Pap E:
*Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and its Applications*.*Volume 536*. Kluwer Academic, Dordrecht, The Netherlands; 2001:x+273.Google Scholar - Grabiec M:
**Fixed points in fuzzy metric spaces.***Fuzzy Sets and Systems*1988,**27**(3):385–389. 10.1016/0165-0114(88)90064-4MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.