- 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

We prove a common fixed point theorem for mappings under -contractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of S.Sedghi et al. (2010)

## 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;

(2) is continuous;

(3) for all ;

(4) whenever and for all .

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 :

(FM-1) ;

(FM-2) if and only if ;

(FM-3) ;

(FM-4) ;

(FM-5) is continuous.

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

for all ;

for all and ;

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

- (1)
For all , is nondecreasing.

- (2)

- (3)
In a fuzzy metric space , whenever for in , , , we can find a , such that .

- (4)
For any , we can find an such that and for any we can find a such that ).

Definition 5 (see [6]).

whenever , and .

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:

(-1) is nondecreasing;

(-2) is upper semicontinuous from the right;

(-3) for all , where , .

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

Lemma 2 (see [7]).

for all , then .

Definition 6 (see [5]).

Definition 7 (see [5]).

Definition 8 (see [7]).

Definition 9 (see [7]).

Definition 10 (see [7]).

for all are satisfied.

Definition 11 (see [7]).

for all .

Remark 3.

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

## 3. Main Results

for all .

Theorem 1.

for all , .

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 .

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.

Prove that and .

for all .

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

for all . We can get that and .

Step 4.

Prove that .

for all .

Since is continuous and , there exists such that .

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

which implies 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.

for all , .

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.

Case 1 ( ).

it is easy to verified.

Case 2 ( ).

holds for all . So (3.36) holds for .

Case 3 ( ).

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

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