Some Coupled Fixed Point Theorems in Cone Metric Spaces

Abstract

We prove some coupled fixed point theorems for mappings satisfying different contractive conditions on complete cone metric spaces.

1. Introduction

Recently, Huang and Zhang in [1] generalized the concept of metric spaces by considering vector-valued metrics (cone metrics) with values in an ordered real Banach space. They proved some fixed point theorems in cone metric spaces showing that metric spaces really doesnot provide enough space for the fixed point theory. Indeed, they gave an example of a cone metric space and proved existence of a unique fixed point for a selfmap of which is contractive in the category of cone metric spaces but is not contractive in the category of metric spaces. After that, cone metric spaces have been studied by many other authors (see [1–9] and the references therein).

Regarding the concept of coupled fixed point, introduced by Bhaskar and Lakshmikantham [10], we consider the corresponding definition for the mappings on complete cone metric spaces and prove some coupled fixed point theorems in the next section. First, we recall some standard notations and definitions in cone metric spaces.

A cone is a subset of a real Banach space such that

(i) is closed, nonempty and ;

(ii)if are nonnegative real numbers and , then ;

(iii).

For a given cone , the partial ordering with respect to is defined by if and only if . The notation will stand for , where denotes the interior of . Also, we will use to indicate that and .

The cone is called normal if there exists a constant such that for every if then . The least positive number satisfying this inequality is called the normal constant of (see [1]). The cone is called regular if every increasing (decreasing) and bounded above (below) sequence is convergent in . It is known that every regular cone is normal (see [1], or [7, Lemma  1.1]).

Huang and Zhang defined the concept of a cone metric space in [1] as follows.

Definition 1.1 (see [1]).

Let be a nonempty set and let be a real Banach space equipped with the partial ordering with respect to the cone . Suppose that the mapping satisfies the following conditions:

(d₁) for all and if and only if ;

(d₂) for all ;

(d₃) for all .

Then is called a cone metric on , and is called a cone metric space.

Definition 1.2 (see [1]).

Let be a cone metric space, and be a sequence in . Then

(i) converges to , denoted by , if for every with there exists a natural number such that for all ;

(ii) is a Cauchy sequence if for every with there exists a natural number such that for all .

A cone metric space is said to be complete if every Cauchy sequence in is convergent in . If for any sequence in there exists a subsequence of such that is convergent in , then the cone metric space is called sequentially compact. Clearly, every sequentially compact cone metric space is complete. Huang and Zhang in [1] investigated the existence and uniqueness of the fixed point for a selfmap on a cone metric space . They considered different types of contractive conditions on . They also assumed to be complete when is a normal cone, and to be sequentially compact when is a regular cone. Later, in [7], Rezapour and Hamlbarani improved some of the results in [1] by omitting the normality assumption of the cone , when is complete. See [4, 6, 7, 9] for more related results about (complete) cone metric spaces and fixed point theorems for different types of mappings on these spaces.

In the rest of this paper, we always suppose that is a real Banach space, is a cone with and is partial ordering with respect to . We also note that the relations and () always hold true.

2. Main Results

For a given partially ordered set , Bhaskar and Lakshmikantham in [10] introduced the concept of coupled fixed point of a mapping . Later in [11] Lakshmikantham and Ćirić investigated some more coupled fixed point theorems in partially ordered sets. The following is the corresponding definition of coupled fixed point in cone metric spaces.

Definition 2.1.

Let be a cone metric space. An element is said to be a coupled fixed point of the mapping if and .

In the next theorems of this section, we investigate some coupled fixed point theorems in cone metric spaces.

Theorem 2.2.

Let be a complete cone metric space. Suppose that the mapping satisfies the following contractive condition for all :

(2.1)

where are nonnegative constants with . Then has a unique coupled fixed point.

Proof.

Choose and set , , . Then by (2.1) we have

(2.2)

and similarly,

(2.3)

Therefore, by letting

(2.4)

we have

(2.5)

Consequently, if we set then for each we have

(2.6)

If then is a coupled fixed point of . Now, let . For each we have

(2.7)

Therefore,

(2.8)

which implies that and are Cauchy sequences in , and there exist such that and . Let with . For every there exists such that and for all . Thus

(2.9)

Consequently, for all . Thus, and hence . Similarly, we have meaning that is a coupled fixed point of .

Now, if is another coupled fixed point of then

(2.10)

and therefore,

(2.11)

Since , (2.11) implies that . Hence, we have and the proof of the theorem is complete.

It is worth noting that when the constants in Theorem 2.2 are equal we have the following corollary.

Corollary 2.3.

Let be a complete cone metric space. Suppose that the mapping satisfies the following contractive condition for all :

(2.12)

where is a constant. Then has a unique coupled fixed point.

Example 2.4.

Let , and . Define with . Then is a complete cone metric space. Consider the mapping with . Then satisfies the contractive condition (2.12) for , that is,

(2.13)

Therefore, by Corollary 2.3, has a unique coupled fixed point, which in this case is . Note that if the mapping is given by , then satisfies the contractive condition (2.12) for , that is,

(2.14)

In this case, and are both coupled fixed points of and hence the coupled fixed point of is not unique. This shows that the condition in corollary (2.12) and hence in Theorem 2.2 are optimal conditions for the uniqueness of the coupled fixed point.

Theorem 2.5.

Let be a complete cone metric space. Suppose that the mapping satisfies the following contractive condition for all :

(2.15)

where are nonnegative constants with . Then has a unique coupled fixed point.

Proof.

Choose and set , , . Then by applying (2.15) we get

(2.16)

where . This implies that and are Cauchy sequences in and therefore by the completeness of , there exist , such that and . Let and choose a natural number such that for all . Thus,

(2.17)

which implies that

(2.18)

Since was arbitrary, or equivalently . Similarly, one can get showing that is a coupled fixed point of .

Now, if is another coupled fixed point of then by applying (2.15) we have

(2.19)

and therefore . Similarly, we can get and hence .

Theorem 2.6.

Let be a complete cone metric space. Suppose that the mapping satisfies the following contractive condition for all ,

(2.20)

where are nonnegative constants with . Then has a unique coupled fixed point.

Proof.

First, note that the uniqueness of the coupled fixed point is an obvious result of in (2.20). To prove the existence of the fixed point, let and choose the sequence and like in the proof of Theorem 2.5, that is , , . Then by applying (2.20) we have

(2.21)

which implies

(2.22)

Similarly, one can get

(2.23)

Therefore, and are Cauchy sequences in and hence by the completeness of , there exist such that and . Let with and for each choose a natural number such that for all . Thus,

(2.24)

which implies

(2.25)

Since was arbitrary, or equivalently . Similarly, one can get and hence is a coupled fixed point of .

When the constants in Theorems 2.5 and 2.6 are equal, we get the following corollaries.

Corollary 2.7.

Let be a complete cone metric space. Suppose that the mapping satisfies the following contractive condition for all :

(2.26)

where is a constant. Then has a unique coupled fixed point.

Corollary 2.8.

Let be a complete cone metric space. Suppose that the mapping satisfies the following contractive condition for all :

(2.27)

where is a constant. Then has a unique coupled fixed point.

Remark 2.9.

Note that in Theorem 2.5, if the mapping satisfies the contractive condition (2.15) for all , then also satisfies the following contractive condition:

(2.28)

Consequently, by adding (2.15) and (2.28), also satisfies the following:

(2.29)

which is a contractive condition of the type (2.26) in Corollary 2.7 (with equal constants). Therefore, one can also reduce the proof of general case (2.15) in Theorem 2.5 to the special case of equal constants. A similar argument is valid for the contractive conditions (2.20) in Theorem 2.6 and (2.27) in Corollary 2.8.