Research Article | Open Access

# Coupled Fixed Point Theorems of Single-Valued Mapping for *c*-Distance in Cone Metric Spaces

**Academic Editor:**Francis T.K. Au

#### Abstract

A new concept of the *c*-distance in cone metric space has been
introduced recently in 2011. The aim of this paper is to extend and generalize
some coupled fixed-point theorems on *c*-distance in cone metric space. Some
examples are given.

#### 1. Introduction

In 2007, Huang and Zhang [1] introduced the concept of cone metric space where each pair of points is assigned to a member of a real Banach space with a cone. Then, several authors have studied the existence and uniqueness of the fixed point and common fixed point for self-map by considered different types of contractive conditions. Some of these works are noted in [2â€“12].

In [13], Bhaskar and Lakshmikantham introduced the concept of coupled fixed point for a given partially ordered set . Lakshmikantham and Ä†iriÄ‡ [14] proved some more coupled fixed-point theorems in partially ordered set.

In [15], Sabetghadam et al. considered the corresponding definition of coupled fixed point for the mapping in complete cone metric space and proved some coupled fixed point theorems. Then, several authors have studied the existence and uniqueness of the coupled fixed point and coupled common fixed point by considered different types of contractive conditions. Some of these works are noted in [16â€“23].

Recently, Cho et al. [23] introduced a new concept of the *c*-distance in cone metric spaces (also see [24]) and proved some fixed-point theorems in ordered cone metric spaces. This is more general than the classical Banach contraction mapping principle. Sintunavarat et al. [25] extended and developed the Banach contraction theorem on *c*-distance of Cho et al. [23]. Wang and Guo [24] proved some common fixed point theorems for this new distance. Several authors have studied on the generalized distance in cone metric space. Some of this works are noted in [26â€“29].

In [28], Cho et al. proved some coupled fixed point theorems in ordered cone metric spaces by using the concept of *c*-distance.

Recall the following definition.

*Definition 1.1 (see [15]). *Let be a cone metric space. An element is said to be a coupled fixed point of the mapping if and .

The following theorems are the main results given in [15].

Theorem 1.2 (see [15]). *Let be a complete cone metric space. Suppose that the mapping satisfies the following contractive condition for all :
**
where are nonnegative constants with . Then has a unique coupled fixed point.*

Theorem 1.3 (see [15]). *Let be a complete cone metric space. Suppose the mapping satisfies the following contractive condition for all :
**
where are nonnegative constants with . Then has a unique coupled fixed point.*

In this paper we proved some coupled fixed point results for *c*-distance in cone metric space. Our theorems extend and develop some theorems of Sabetghadam et al. [15] on *c*-distance of Cho et al. [23] in cone metric space.

#### 2. Preliminaries

Let be a real Banach space and denote to the zero element in . A cone is a subset of such that (1) is nonempty set closed and , (2)if are nonnegative real numbers and then , and(3) and implies .For any cone , the partial ordering with respect to is defined by if and only if . The notation of stands for but . Also, we used to indicate that , where denotes the interior of . A cone is called normal if there exists a number such that for all . The least positive number satisfying the above condition is called the normal constant of .

*Definition 2.1 (see [1]). *Let be a nonempty set and be a real Banach space equipped with the partial ordering with respect to the cone . Suppose that the mapping satisfies the following condition: (1) for all and if and only if ,(2) for all , and (3) for all . Then is called a cone metric on and is called a cone metric space.

*Definition 2.2 (see [1]). *Let be a cone metric space and be a sequence in and .

One has the following:(1)for all with , if there exists a positive integer such that for all , then is said to be convergent and is the limit of . We denote this by , (2)for all with , if there exists a positive integer such that for all then is called a Cauchy sequence in , and (3)a cone metric space is called complete if every Cauchy sequence in is convergent.

Lemma 2.3 (see [8]). *(1) If be a real Banach space with a cone and where and , then .**(2) If , and , then there exists a positive integer such that for all .*

Next we give the notation of *c*-distance on a cone metric space which is a generalization of -distance of Kada et al. [30] with some properties.

*Definition 2.4 (see [23]). *Let be a cone metric space. A function is called a *c*-distance on if the following conditions hold: (q1) for all , (q2) for all , (q3) for each and , if for some , then whenever is a sequence in converging to a point , and (q4) for all with , there exists with such that and imply .

*Example 2.5 (see [23]). *Let and . Let and define a mapping by for all . Then is a cone metric space. Define a mapping by for all . Then is a *c*-distance on .

Lemma 2.6 (see [23]). *Let be a cone metric space and is a -distance on . Let and be sequences in and . Suppose that is a sequences in converging to . Then the following hold.*(1)*If and , then . *(2)*If and , then converges to . *(3)*If for , then is a Cauchy sequence in . *(4)*If , then is a Cauchy sequence in . *

*Remark 2.7 (see [23]). *(1) does not necessarily for all .

(2) is not necessarily equivalent to for all .

#### 3. Main Results

In this section we prove some coupled fixed point theorems using *c-*distance in cone metric space. In whole paper cone metric space is over nonnormal cone with nonempty interior.

Theorem 3.1. *Let be a complete cone metric space, and is a c -distance on . Let be a mapping and suppose that there exists mappings such that the following hold:*(a)* and for all , *(b)* and for all , *(c)* for all , *(d)* for all .**Then has a coupled fixed point . Further, if and , then and . Moreover, the coupled fixed point is unique and is of the form for some .*

*Proof. *Choose . Set . Then we have the following:

And similarly
Put, . Then we have
where .

Let . It follows that
Then we have

From (3.5) we have
and also
Thus, Lemma 2.6(3) shows that and are Cauchy sequences in . Since is complete, there exists and such that and as . By (q3) we have the following:
and also

On the other hand,
By Lemma 2.6 (1), (3.8), and (3.10), we have . By similar way we have . Therefore is a coupled fixed point of .

Suppose that and , then we have
and also
which implies that
Since , Lemma 2.3 (1) shows that . But and , hence and .

Finally, suppose that there is another coupled fixed point then we have
and also
which implies that
Since , Lemma 2.3 (1) shows that . But and . Hence and . Also we have and . Hence Lemma 2.6 part 1 shows that and , which implies that . Similarly, we prove that and . Hence, . Therefore, the coupled fixed point is unique and is of the form for some .

From above Theorem we have the following corollaries.

Corollary 3.2. *Let be a complete cone metric space, and is a c -distance on . Suppose that the mapping satisfies the following contractive condition:
**
for all , where are nonnegative constants with . Then has a coupled fixed point . Further, if and then and . Moreover, the coupled fixed point is unique and is of the form for some .*

Corollary 3.3. *Let be a complete cone metric space and is a c -distance on . Suppose that the mapping satisfies the following contractive condition:
**
for all , where . Then has a coupled fixed point . Further, if and , then and . Moreover, the coupled fixed point is unique and is of the form for some .*

Theorem 3.4. *Let be a complete cone metric space, and is a c -distance on . Suppose that the mapping is continuous, and suppose that there exists mappings such that the following hold: *(a)*, for all ,*(b)* for all , and *(c)* for all .**Then has a coupled fixed point . Further, if and , then and . Moreover, the coupled fixed point is unique and is of the form for some .*

*Proof. *Choose . Set . Then we have the following:
Then, we have
where .

Similarly we have
where .

Let . Then it follows that
and also
Thus, Lemma 2.6 (3) shows that and are Cauchy sequences in . Since is complete, there exists and such that and as . Since is continuous, then . Similarly, . Therefore, is a coupled fixed point of .

Suppose that and , then we have
Since , Lemma 2.3 (1) shows that . By similar way, .

Finally, suppose that there is another coupled fixed point , then we have
Since , Lemma 2.3 (1) shows that . Also we have . Hence Lemma 2.6 (1) show that . By similar way we have which implies that . Similarly, we prove that and . Hence, . Therefore, the coupled fixed point is unique and is of the form for some .

From the above theorem, we have the following corollaries.

Corollary 3.5. *Let be a complete cone metric space, and is a c -distance on . Suppose that the mapping is continuous, and suppose that there exist mappings such that the following hold: *(a)* and for all ,*(b)* for all , and*(c)* for all .**Then has a coupled fixed point . Further, if and , then and . Moreover, the coupled fixed point is unique and is of the form for some .*

Corollary 3.6. *Let be a complete cone metric space, and is a c -distance on . Suppose that the mapping is continuous and satisfies the following contractive condition:
**
for all , where are nonnegative constants with . Then has a coupled fixed point . Further, if and , then and . Moreover, the coupled fixed point is unique and is of the form for some .*

Corollary 3.7. *Let be a complete cone metric space, and is a c -distance on . Suppose that the mapping is continuous and satisfies the following contractive condition for all:
**
for all , where are nonnegative constants with . Then has a coupled fixed point . Further, if and then and . Moreover, the coupled fixed point is unique and is of the form for some .*

Corollary 3.8. *Let be a complete cone metric space, and is a c -distance on . Suppose that the mapping is continuous and satisfies the following contractive condition for all:
**
for all , where is a constant. Then has a coupled fixed point . Further, if and , then and . Moreover, the coupled fixed point is unique and is of the form for some .*

Finally, we provide another result without condition (b) in Theorem 3.1, and we do not require that is continuous.

Theorem 3.9. *Let be a complete cone metric space, and is a c -distance on . Let be a mapping, and suppose that there exists mappings such that the following hold: *(a)*, and for all , *(b)* for all , and*(c)* for all .*

*Proof. *Choose . Set . Observe that
equivalently
Then we have the following:
Then, we have
where .

Similarly we have
where .

Let . Then it follows that
and also
Thus, Lemma 2.6 (3) shows that and are Cauchy sequences in . Since is complete, there exists and such that and as . By (q3) we have

On the other hand, we have
Then, we have
By Lemma 2.6 (1), (3.36), and (3.39), we have .

By similar way we have
By Lemma 2.6 (1), (3.37), and (3.40), we have . Therefore, is a coupled fixed point of .

Suppose that and , then we have