# 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

## 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

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