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]).

Let . A -norm is said to be of H-type if the family of functions is equicontinuous at , where

(2.1)

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.

Let be a fuzzy metric space. For , the open ball with a center and a radius is defined by

(2.2)

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

(1)a sequence in is said to be convergent to (denoted by ) if

(2.3)

for all ;

(2)a sequence in is said to be a Cauchy sequence if for any , there exists , such that

(2.4)

for all and ;

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

Remark 1 (see [25]).

  1. (1)

    For all , is nondecreasing.

  2. (2)

    It is easy to prove that if , , , then

    (2.5)
  1. (3)

    In a fuzzy metric space , whenever for in , , , we can find a , such that .

  2. (4)

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

Definition 5 (see [6]).

Let be a fuzzy metric space. is said to satisfy the -property on if

(2.6)

whenever , and .

Lemma 1.

Let be a fuzzy metric space and satisfies the -property; then

(2.7)

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.

Let be an ordinary metric space, for all , and be defined as following:

(2.8)

where . Then is continuous and increasing in , and . Let

(2.9)

then is a fuzzy metric space and

(2.10)

But for any , , , ,

(2.11)

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]).

Let be a fuzzy metric space, where is a continuous -norm of H-type. If there exists such that if

(2.12)

for all , then .

Definition 6 (see [5]).

An element is called a coupled fixed point of the mapping if

(2.13)

Definition 7 (see [5]).

An element is called a coupled coincidence point of the mappings and if

(2.14)

Definition 8 (see [7]).

An element is called a common coupled fixed point of the mappings and if

(2.15)

Definition 9 (see [7]).

An element is called a common fixed point of the mappings and if

(2.16)

Definition 10 (see [7]).

The mappings and are said to be compatible if

(2.17)

for all whenever and are sequences in , such that

(2.18)

for all are satisfied.

Definition 11 (see [7]).

The mappings and are called commutative if

(2.19)

for all .

Remark 3.

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

3. Main Results

For convenience, we denote

(3.1)

for all .

Theorem 1.

Let be a complete FM-space, where is a continuous -norm of H-type satisfying (2.7). Let and be two mappings and there exists such that

(3.2)

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.

Let be two arbitrary points in . Since , we can choose such that and . Continuing in this way we can construct two sequences and in such that

(3.3)

The proof is divided into 4 steps.

Step 1.

Prove that and are Cauchy sequences.

Since is a -norm of H-type, for any , there exists a such that

(3.4)

for all .

Since is continuous and for all , there exists such that

(3.5)

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

(3.6)

From condition (3.2), we have

(3.7)

Similarly, we can also get

(3.8)

Continuing in the same way we can get

(3.9)

So, from (3.5) and (3.6), for , we have

(3.10)

which implies that

(3.11)

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.

Since complete, there exist such that

(3.12)

Since and are compatible, we have by (3.12),

(3.13)

for all . Next we prove that and .

For all , by condition (3.2), we have

(3.14)

for all . Let , since and are compatible, with the continuity of , we get

(3.15)

which implies that . Similarly, we can get .

Step 3.

Prove that and .

Since is a -norm of H-type, for any , there exists an such that

(3.16)

for all .

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

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

(3.17)

letting , we get

(3.18)

Similarly, we can get

(3.19)

From (3.18) and (3.19) we have

(3.20)

By this way, we can get for all ,

(3.21)

Then, we have

(3.22)

So for any we have

(3.23)

for all . We can get that and .

Step 4.

Prove that .

Since is a -norm of H-type, for any , there exists an such that

(3.24)

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 .

Since for ,

(3.25)

Letting yields

(3.26)

Thus we have

(3.27)

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.

Let be a complete FM-space, where is a continuous -norm of H-type satisfying (2.7). Let and there exists such that

(3.28)

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]).

Let for all and be a complete fuzzy metric space such that has -property. Let and be two functions such that

(3.29)

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.

Let , for all . is defined as (2.8). Let

(3.30)

for all . Then is a complete FM-space.

Let , and be defined as

(3.31)

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

In fact, it is easy to see that ,

(3.32)

For all and . (3.28) is equivalent to

(3.33)

Since , we can get

(3.34)

From (3.33), we only need to verify the following:

(3.35)

that is,

(3.36)

We consider the following cases.

Case 1 ().

Then (3.36) is equivalent to

(3.37)

it is easy to verified.

Case 2 ().

Then (3.36) is equivalent to

(3.38)

which is

(3.39)

since

(3.40)

that is

(3.41)

holds for all . So (3.36) holds for .

Case 3 ().

Then (3.36) is equivalent to

(3.42)

Let , we only need to verify

(3.43)

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 .