Abstract and Applied Analysis

Volume 2012 (2012), Article ID 901792, 24 pages

doi:10.1155/2012/901792

Research Article

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

Zaid Mohammed Fadail and Abd Ghafur Bin Ahmad

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

Received 30 May 2012; Accepted 8 July 2012

Academic Editor: Ngai-Ching Wong

Copyright © 2012 Zaid Mohammed Fadail and Abd Ghafur Bin Ahmad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The existence and uniqueness of the common coupled fixed point in cone metric spaces have been studied by considering different types of contractive conditions. A new concept of the c-distance in cone metric space has been recently introduced in 2011. Then, coupled fixed point results for contraction-type mappings in ordered cone metric spaces and cone metric spaces have been considered. In this paper, some common coupled fixed point results on c-distance in cone metric spaces are obtained. Some supporting 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. Subsequently, several authors have studied the existence and uniqueness of the fixed point and common fixed point for self-map by considering different types of contractive conditions. Some of these works are noted in [2–12].

In [13], Bhaskar and Lakshmikantham introduced the concept of coupled fixed point for a given partially ordered set . Lakshmikantham and Ćirić [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. Subsequently, several authors have studied the existence and uniqueness of the coupled fixed point and common coupled fixed point by considering different types of contractive conditions. Some of these works are noted in [16–23].

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. Subsequently, several authors have studied on the generalized distance in cone metric space. Some of this works are noted in [26–31].

In [30], Fadail and Ahmad proved some coupled fixed point theorems in cone metric spaces by using the concept of c-distance.

Recall that an element is said to be a coupled fixed point of the mapping if and .

Definition 1.1. An element is called

(1)a coupled coincidence point of mappings and if and , and is called coupled point of coincidence. (2)a common coupled fixed point of mappings and if .

Abbas et al. [20] introduced the following definition.

Definition 1.2. The mappings and are called w-compatible if , whenever and .

The aim of this paper is to continue the study of common coupled fixed points of mappings but now for c-distance in cone metric space. Our theorems extend and develop some theorems in literature to c-distance in cone metric spaces. In this paper, we do not impose the normality condition for the cones, the only assumption is that cone has nonempty interior.

2. Preliminaries

Let be a real Banach space and denote 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 , (3) and implies .

For any cone , the partial ordering with respect to is defined by if and only if . The notation of stand 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 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 , (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 a sequence in and .

(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 . (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. [32] 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:

for all , for all , for each and , if for some , then whenever is a sequence in converging to a point , 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 hold for all . (2) is not necessarily equivalent to for all .

3. Main Results

In this section, we prove some common coupled fixed point results using c-distance in cone metric space. Also, we generalize the contractive conditions in literature by replacing the constants with functions.

Theorem 3.1. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Let and be two mappings 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 .

If and is a complete subspace of , then and have a unique coupled point of coincidence in . Further, if and , then and . Moreover, if and are w-compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

Proof. Choose . Set , this can be done because . Continuing this process, we obtain two sequences and such that , . Then we have

Similarly, we have

Put . Then we have

where .

Let . It follows that

Then we have

Consequently,

Thus, Lemma 2.6 (3) shows that and are Cauchy sequences in . Since is complete, there exists and in such that and as . Using q3, we have

On the other hand,

Thus, Lemma 2.6 (1), (3.5), and (3.8) show that . By similar way, we can prove that . Therefore, is a coupled coincidence point of and .

Suppose that and . Then we have

This implies that

Since , Lemma 2.3 (1) shows that . But and . Consequently, and .

Finally, suppose there is another coupled point of coincidence of and such that and for some in . Then we have

and also,

This implies that

Since , Lemma 2.3 (1) shows that . But and . Hence and . Also we have and . Thus, Lemma 2.6 (1) shows that and , which implies that . Similarly, we can prove that and . Thus, . Therefore, is the unique coupled point of coincidence. Now, let . Since and are w-compatible, then we have

Thus is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that . Therefore, . Hence is the unique common coupled fixed point of and .

The following corollaries can be obtained as consequences of this theorem.

Corollary 3.2. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Suppose the mappings and satisfy the following contractive condition:

for all , where are nonnegative constants with . If and is a complete subspace of , then and have a coupled coincidence point in . Further, if and , then and . Moreover, if and are w -compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

Corollary 3.3. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Suppose the mappings and satisfy the following contractive condition:

for all , where is a constants. If and is a complete subspace of , then and have a unique coupled point of coincidence in . Further, if and , then and . Moreover, if and are w -compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

Theorem 3.4. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Let and be two mappings and suppose that there exists mappings such that the following hold:

(a) and for all , (b) for all , (c) for all .

If and is a complete subspace of , then and have a unique coupled point of coincidence in . Further, if and , then and . Moreover, if and are w -compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

Proof. Choose . Set . This can be done because . Continuing this process, we obtain two sequences and such that ,  .

Then we have

Hence

where . It follows that

Similarly, we have

where It follows that

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 in such that and as . Using (q3), we have

On the other hand and by using (3.20), we have

also by using (3.22), we have

Thus, Lemma 2.6 (1), (3.25), and (3.27) show that . Again, Lemma 2.6 (1), (3.26), and (3.28) show that . Therefore, is a coupled coincidence point of and .

Suppose that and . Then we have

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

Finally, suppose there is another coupled point of coincidence of and such that and for some in . Then we have

and also,

Also, we have and . Thus, Lemma 2.6 (1) shows that and , which implies that . Similarly, we can prove that and . Thus, . Therefore, is the unique coupled point of coincidence. Now, let . Since and are w-compatible, then we have

Then, is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that . Therefore, . Hence, is the unique common coupled fixed point of and .

The following corollaries can be obtained as consequences of Theorem 3.4.

Corollary 3.5. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Suppose the mappings and satisfy the following contractive condition:

for all , where are nonnegative constants with . If and is a complete subspace of , then and have a unique coupled point of coincidence in . Further, if , and , then and . Moreover, if and are w -compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form () for some .

Corollary 3.6. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Suppose the mappings and satisfy the following contractive condition:

for all , where is constants. If and is a complete subspace of , then and have a unique coupled point of coincidence in . Further, if and , then and . Moreover, if and are w-compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

Finally, we provide another result with another contractive type.

Theorem 3.7. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Let and be two mappings and suppose that there exists mappings such that the following hold:

(a), and   for all , (b) for all , (c) for all .

If and is a complete subspace of , then and have a unique coupled point of coincidence in . Further, if and , then and . Moreover, if and are w-compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

Proof. Choose . Set this can be done because . Continuing this process, we obtain to sequences and such that . Observe that

equivalently

Then, we have

Hence

where .

Similarly, we have

where .

Let . Then, it follows that

Thus, Lemma 2.6 (3) shows that and are Cauchy sequences in . Since is complete, there exists such that and as . Using (q3), we have

On the other hand,

Then,

Thus, Lemma 2.6 (1), (3.41), and (3.44) show that . Similarly, we can prove that . Therefore, is a coupled coincidence point of and . Hence, is the coupled point of coincidence.

Suppose that and . Then, we have

Since , Lemma 2.3(1) shows that . Similarly, we have .

Finally, suppose there is another coupled point of coincidence of and such that and for some in . Then, we have

Since , Lemma 2.3 (1) shows that . By similar way, . Also, we have and . Thus, Lemma 2.6 (1) shows that and , which implies that . Similarly, we can prove that and . Thus, . Therefore, is the unique coupled point of coincidence. Now, let . Since and are w-compatible, then we have

Then, is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that . Therefore, . Hence, is the unique common coupled fixed point of and .

The following corollaries can be obtained as consequences of Theorem 3.7.

Corollary 3.8. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Let and be two mappings and suppose that there exists mappings such that the following hold:

(a) and , (b), (c).

for all . If and is a complete subspace of , then and have a unique coupled point of coincidence in . Further, if and , then and . Moreover, if and are w-compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

Corollary 3.9. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Suppose the mappings and satisfy the following contractive condition:

for all , where are nonnegative constants with . If and is a complete subspace of , then and have a unique coupled point of coincidence in . Further, if and then and . Moreover, if and are w-compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

Corollary 3.10. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Suppose the mappings and satisfy the following contractive condition:

for all , where are nonnegative constants with . If and is a complete subspace of , then and have a unique coupled point of coincidence in . Further, if and , then and . Moreover, if and are w-compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

Corollary 3.11. Let be a cone metric space with a cone having nonempty interior and is a c-distance on . Suppose the mappings and satisfy the following contractive condition:

for all , where is a constants. If and is a complete subset of , then and have a unique coupled point of coincidence in . Further, if and , then and . Moreover, if and are w-compatible, then and have a unique common coupled fixed point and the common coupled fixed point of and is of the form for some .

4. Some Examples

Example 4.1. Let and . Let and define a mapping by for all . Clearly, is a cone metric space. Define a mapping by for all . Then is a c-distance on . Define the mappings by and by for all . Obviously that and is a complete subspace of . Take , where . Observe the following:

(a)            for all .(b) and for all .(c) for all .(d)For all , we have

Clearly that and are w-compatible. Therefore, and satisfy all the conditions of Theorem 3.1. Hence, and have a unique common coupled fixed point and with .

Example 4.2. Consider Example 2.5. Define the mappings by and by for all . Obviously that and is a complete subspace of . Then, we have with . Clearly that and are w-compatible. Therefore, and satisfy all the conditions of Corollary 3.3. Hence, and have a unique common coupled fixed point and with .

Example 4.3. Let and . Let and define a mapping by for all . Then is a complete cone metric space (see [15]). Define a mapping by for all . Then is a c-distance on . In fact (q1)–(q3) are immediate. Let with and put . If and , then we have

This show, that (q4) holds. Therefore, is a c-distance on . Define the mappings by and by for all . Obviously that and is a complete subspace of . Then we have

with . Clearly that and are w-compatible. Therefore, and satisfy all the conditions of Corollary 3.3. Hence, and have a unique common coupled fixed point and with .

Acknowledgments

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant no. OUP-UKM-FST-2012. The authors thank the referee for his/her careful reading of the paper and useful suggestions.

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