Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 184534, 9 pages

http://dx.doi.org/10.1155/2012/184534

## Coupled Fixed Point Theorems under Weak Contractions

^{1}Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea^{2}Faculty of Mathematics, University of Belgrade, 11000 Beograd, Serbia^{3}Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran^{4}Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

Received 6 October 2011; Accepted 28 December 2011

Academic Editor: Binggen Zhang

Copyright © 2012 Y. J. Cho et al. 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

Cho et al. [Comput. Math. Appl. 61(2011), 1254–1260] studied common fixed point
theorems on cone metric spaces by using the concept of *c*-distance. In this paper, we prove
some coupled fixed point theorems in ordered cone metric spaces by using the concept of
*c*-distance in cone metric spaces.

#### 1. Introduction

Many fixed point theorems have been proved for mappings on cone metric spaces in the sense of Huang and Zhang [1]. For some more results on fixed point theory and applications in cone metric spaces, we refer the readers to [2–15]. Recently, Bhaskar and Lakshmikantham [16] introduced the concept of a coupled coincidence point of a mapping from into and a mapping from into and studied fixed point theorems in partially ordered metric spaces. For some more results on couple fixed point theorems, refer to [17–23].

Recently, Cho et al. [7] introduced a new concept of -distance in cone metric spaces, which is a cone version of -distance of Kada et al. [24] (see also [25]) and proved some fixed point theorems for some contractive type mappings in partially ordered cone metric spaces using the -distance.

In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept of -distance.

#### 2. Preliminaries

In this paper, assume that is a real Banach space. Let be a subset of with . Then is called a *cone* if the following conditions are satisfied:(1) is closed and ;(2), implies ;(3) implies .

For a cone , define the *partial ordering * with respect to by if and only if . We write to indicate that but , while stand for .

It can be easily shown that for all positive scalars .

*Definition 2.1 (see [1]). *Let be a nonempty set. Suppose that the mapping satisfies the following conditions:(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. Let be a sequence in and .(1)If, for any with , there exists such that for all , then is said to be *convergent* to a point and is the *limit* of . We denote this by or as .(2)If, for any with , there exists such that for all , then is called a *Cauchy sequence* in .(3)The space is called a complete cone metric space if every Cauchy sequence is convergent.

*Definition 2.3 (see [7]). *Let be a partially ordered set, and let be a function. Then the mapping is said to have the *mixed monotone property* if is monotone nondecreasing in and is monotone nonincreasing in ; that is,
for all and
for all .

*Definition 2.4 (see [7]). *An element is called a *coupled fixed point* of a mapping if and .

Recently, Cho et al. [7] introduced the concept of -distance on cone metric space which is a generalization of -distance of Kada et al. [24].

*Definition 2.5 (see [7]). *Let be a cone metric space. Then a function is called a *-distance* on if the following are satisfied:(q1) for all ;(q2) for all ;(q3) for any , if there exists such that for each , then whenever is a sequence in converging to a point ;(q4) for any with , there exists with such that and imply .

Cho et al. [7] noticed the following important remark in the concept of -distance on cone metric spaces.

*Remark 2.6 (see [7]). *Let be a -distance on a cone metric space . Then(1) does not necessarily hold for all ,(2) is not necessarily equivalent to for all .

The following lemma is crucial in proving our results.

Lemma 2.7 (see [7]). *Let be a cone metric space, and let be a -distance on . Let and be sequences in and . Suppose that is a sequence in converging to . Then the following hold:*(1)*if and , then ;*(2)*if and , then converges to a point ;*(3)*if for each , then is a Cauchy sequence in ;*(4)*If , then is a Cauchy sequence in .*

#### 3. Main Results

In this section, we prove some coupled fixed point theorems by using -distance in partially ordered cone metric spaces.

Theorem 3.1. *Let be a partially ordered set, and suppose that is a complete cone metric space. Let be a -distance on , and let be a continuous function having the mixed monotone property such that
**
for some and all with or . If there exist such that and , then has a coupled fixed point . Moreover, one has and .*

*Proof. *Let be such that and . Let and . Since has the mixed monotone property, we have and . Continuing this process, we can construct two sequences and in such that
Let . Now, by (3.1), we have
From (3.3), it follows that

Similarly, we have
Thus it follows from (3.4) and (3.5) that
Repeating (3.6) -times, we get
Thus we have
Let with . Since
and , we have
From Lemma 2.7 (3), it follows that and are Cauchy sequences in . Since is complete, there exist such that and . Since is continuous, we have
By the uniqueness of the limits, we get and . Thus is a coupled fixed point of .

Moreover, by (3.1), we have
Therefore, we get
Since , we conclude that , and hence and . This completes the proof.

Theorem 3.2. *In addition to the hypotheses of Theorem 3.1, suppose that any two elements and in are comparable. Then the coupled fixed point has the form , where .*

*Proof. *As in the proof of Theorem 3.1, there exists a coupled fixed point . Here and . By the additional assumption and (3.1), we have
Thus we have
Since , we get . Hence and . Let and . Then
From Lemma 2.7 (1), we have . Hence the coupled fixed point of has the form . This completes the proof.

Theorem 3.3. *Let be a partially ordered set, and suppose that is a complete cone metric space. Let be a -distance on , and let be a function having the mixed monotone property such that
**
for some and all with or . Also, suppose that has the following properties:*(a)*if is a nondecreasing sequence in with , then for all ;*(b)*if is a nonincreasing sequence in with , then for all .**Assume there exist such that and . If , then has a coupled fixed point.*

*Proof. *As in the proof of Theorem 3.1, we can construct two Cauchy sequences and in such that

Moreover, we have that converges to a point and converges to ,
for each . By (q3), we have
and so
By the properties (a) and (b), we have
By (3.17), we have
Thus we have
By (3.21), we get
Therefore, we have
By using (3.20) and (3.26), Lemma 2.7 (1) shows that and . Therefore, is a coupled fixed point of . This completes the proof.

*Example 3.4. *Let with and . Let (with usual order), and let be defined by . Then is an ordered cone metric space (see [7, Example 2.9]). Further, let be defined by . It is easy to check that is a -distance. Consider now the function defined by
Then it is easy to see that
for all with or . Note that and . Thus, by Theorem 3.1, it follows that has a coupled fixed point in . Here is a coupled fixed point of .

#### Acknowledgments

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No.: 2011–0021821).

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