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Abstract and Applied Analysis
Volume 2012, Article ID 324874, 15 pages
http://dx.doi.org/10.1155/2012/324874
Research Article

Coupled Fixed Point Theorems for Weak Contraction Mappings under -Invariant Set

1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand
2Department of Mathematics Education and RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea

Received 30 January 2012; Accepted 27 February 2012

Academic Editor: Jean Michel Combes

Copyright © 2012 Wutiphol Sintunavarat 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

We extend the recent results of the coupled fixed point theorems of Cho et al. (2012) by weakening the concept of the mixed monotone property. We also give some examples of a nonlinear contraction mapping, which is not applied to the existence of the coupled fixed point by the results of Cho et al. but can be applied to our results. The main results extend and unify the results of Cho et al. and many results of the coupled fixed point theorems.

1. Introduction

Since Banach's fixed point theorem in 1922, because of its simplicity and usefullnes, it has become a very popular tool in solving the existence problems in many branches of nonlinear analysis. For some more results of the generalization of this principle, refer to [19] and references mentioned therein. Ran and Reurings [10] extended the Banach contraction principle to metric spaces endowed with a partial ordering, and they gave applications of their results to matrix equations. Afterward, Nieto and Rodriguez-López [11] extended Ran and Reurings’s theorems in [10] for nondecreasing mappings and obtained a unique solution for a first order-ordinary differential equation with periodic boundary conditions.

In 2006, Gnana Bhaskar and Lakshmikantham [12] introduced the concept of the mixed monotone property and a coupled fixed point. They also established some coupled fixed point theorems for mappings that satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. Because of their important role in the study of nonlinear differential equations, nonlinear integral equations and differential inclusions, a wide discussion on coupled fixed point theorems has been studied by many authors. A number of articles in this topic have been dedicated to the improvement and generalization; see [1322] and references therein.

on the other hand, Huang and Zhang [23] introduced the notion of cone metric spaces and established the fixed point theorems for mappings on this space. In 2011, Cho et al. [24] introduced a new concept of the -distance in cone metric spaces, which is a cone version of the -distance of Kada et al. [25] and proved some fixed point theorems for some contractive type mappings in partially ordered cone metric spaces using the -distance. Afterward, Sintunavarat et al. [26] studied and established the fixed point theorems for the generalized contraction mappings by using this concept. For some more results on the fixed point theory and applications in cone metric spaces by using the cone metric, -cone-distance, and -distance, we refer the readers to [2735].

Recently, Cho et al. [36] established new coupled fixed point theorems under weak contraction mappings by using the concept of mixed monotone property and -distance in partially cone metric spaces as follows.

Theorem 1.1 (see [36]). Let be a partially ordered set, and suppose that is a complete cone metric space. Let be a -distance on and a continuous function having the mixed monotone property such that for some and all with If there exist such that then has a coupled fixed point . Moreover, one has and .

Theorem 1.2 (see [36]). In addition to the hypotheses of Theorem 1.1, suppose that any two elements and in are comparable. Then, the coupled fixed point has the form , where .

Theorem 1.3 (see [36]). Let be a partially ordered set, and suppose that is a complete cone metric space. Let be a -distance on and a function having the mixed monotone property such that for some and all with 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 If , then has a coupled fixed point.

In this paper, we weaken the condition of the mixed monotone property in results of Cho et al. [36] by using the concept of the -invariant set due to Samet and Vetro [37] in the cone version. We also give the example of a nonlinear contraction mapping, which is not applied by the main results of Cho et al. but can be applied to our results. The presented results extend and complement some recent results of Cho et al. [36] and some classical coupled fixed point theorems and several results in the literature.

2. Preliminaries

Throughout this paper denotes a partially ordered set with the partial order . By , we mean but . A mapping is said to be nondecreasing (non-increasing) if, for all , implies , resp.).

Definition 2.1 (see [12]). Let be a partial ordered set. A mapping is said to have a mixed monotone property if is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any

Definition 2.2 (see [12]). Let be a nonempty set. An element is called a coupled fixed point of mapping if
Next, we give some terminology of cone metric spaces and the concept of -distance in cone metric spaces due to Cho et al. [24], which is a generalization of the -distance of Kada et al. [25].
Let be a real Banach space, a zero element in , and 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 stands for .
It can be easily shown that for all positive scalars .
The cone is called normal if there is a number such that, for all , The least positive number satisfying the above is called the normal constant of .

Definition 2.3 (see [23]). 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 2.4 (see [23]). 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 . One denotes 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 2.5 (see [24]). 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 .

Remark 2.6. The -distance is a -distance on if we take is a metric space, , , and (q3) is replaced by the following condition:
for any , is lower semicontinuous.
Moreover, (q3) holds whenever is lower semi-continuous. Thus, if is a metric space, , and , then every -distance is a -distance. But the converse is not true in the general case. Therefore, the -distance is a generalization of the -distance.

Example 2.7. Let be a cone metric space and a normal cone. Define a mapping by for all . Then, is -distance.

Example 2.8. Let with and (this cone is not normal). Let , and define a mapping by for all , where such that . Then, is a cone metric space. Define a mapping by for all . Then, is a -distance.

Example 2.9. Let be a cone metric space and a normal cone. Define a mapping by for all , where is a fixed point in . Then, is a -distance.

Example 2.10. 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 -distance.

Remark 2.11. From Examples 2.9 and 2.10, we have two important results. For the -distance, does not necessarily hold and is not necessarily equivalent to for all .

The following lemma is crucial in proving our results.

Lemma 2.12 (see [24]). Let be a cone metric space and 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. Coupled Fixed Point under -Invariant Set

In this section, we prove some coupled fixed point theorems by using the -distance under the concept of the -invariant in cone metric spaces and apply our results in partially ordered cone metric spaces. First of all, we give the concept of the -invariant set in the cone version.

Definition 3.1. Let be a cone metric space and a given mapping. Let be a nonempty subset of . One says that is the -invariant subset of if and only if, for all , one has(a);(b).
We obtain that the set is trivially -invariant.
The next example plays a key role in the proof of our main results in partially ordered set.

Example 3.2. Let be a cone metric space endowed with a partial order . Let be a mapping satisfying the mixed monotone property, that is, for all , we have Define the subset by Then, is the -invariant of .
Next, we prove the main results of this work.

Theorem 3.3. Let be a complete cone metric space. Let be a -distance on , a nonempty subset of , and a continuous function such that for some and all with If is an -invariant and there exist such that then has a coupled fixed point . Moreover, if , then and .

Proof. As , we can construct sequences and in such that Since and is an -invariant set, we get Again, using the fact that is an -invariant set, we have By repeating the argument to the above, we get for all . From (3.3), we have Combining (3.10) and (3.11), we get Since for all and is an -invariant set, we get for all . From (3.3), we have From (3.14), we have Adding (3.12) and (3.15), we get We repeat the above process for -times, and we get From (3.17), we can conclude that Let with . Since and , we have Using Lemma 2.12(3), we have and are Cauchy sequences in . By the completeness of , we get and for some .
Since is continuous, taking in (3.6), we get By the uniqueness of the limits, we get and . Therefore, is a coupled fixed point of .
Finally, we assume that and so . By (3.3), we have Therefore, we get Since , we conclude that and hence and . This completes the proof.

Theorem 3.4. In addition to the hypotheses of Theorem 3.3, suppose that any two elements and in , we have Then, the coupled fixed point has the form , where .

Proof. As in the proof of Theorem 3.3, there exists a coupled fixed point . Hence, From the additional hypothesis and (3.3), we get Therefore, we have Since , we get . Therefore, and .
Let and . Then, From Lemma 2.12(1), we have . Therefore, the coupled fixed point of has the form . This completes the proof.

Next, we apply Theorems 3.3 and 3.4 in partially ordered cone metric spaces. If we set as Example 3.2, then we get to Theorems 3.1 and 3.2 of Cho et al. [36].

Corollary 3.5 (see [36, Theorem 3.1]). Let be a partially ordered set, and suppose that is a complete cone metric space. Let a -distance on and be a continuous function having the mixed monotone property such that for some and all with If there exist such that then has a coupled fixed point . Moreover, one has and .

Corollary 3.6 (see [36, Theorem 3.2]). In addition to the hypotheses of Corollary 3.5, suppose that any two elements and in are comparable. Then the coupled fixed point has the form , where .

Theorem 3.7. Let be a complete cone metric space. Let be a -distance on , a subset of , and a function such that for some and all with Also, suppose that(i)there exist such that ,(ii)if two sequences , are such that for all and , then for all .If is an -invariant set, then has a coupled fixed point.

Proof. As in the proof of Theorem 3.3, we can construct two Cauchy sequences and in such that for all . Moreover, we have that converges to a point and converges to : for each . By (q3), we have and so By assumption (ii), we have and . From (3.32), we have Thus, we have By (3.37), we get Therefore, we have By using (3.36) and (3.41), Lemma 2.12(1) shows that and . Therefore, is a coupled fixed point of . This completes the proof.

Corollary 3.8 (see [36, Theorem 3.3]). Let be a partially ordered set, and suppose that is a complete cone metric space. Let be a -distance on and a function having the mixed monotone property such that for some and all with Also, suppose that has the following properties:(a)if is a non-decreasing sequence in with , then for all ;(b)if is a non-increasing sequence in with , then for all . Assume there exist such that If , then has a coupled fixed point.

Proof. Let . We obtain that is an -invariant set. By (3.42), we have for some and all with or . From assumptions (a) and (b), we know that, for two sequences , such that is a non-decreasing sequence in with and is a non-increasing sequence in with , for all . Therefore, we have for all . Since , we have for all . Therefore, we can conclude that for all . Now, all the hypotheses of Theorem 3.7 hold. Thus, has a coupled fixed point.

Example 3.9. Let with and Let (with usual order), and let be defined by Then, is an ordered cone metric space (see [24, Example 2.9]). Further, let be defined by It is easy to check that is a -distance. Consider now the function defined by For and , we have but for all . So the mapping does not satisfy the mixed monotone property. Hence, the main results of Cho et al. [36] cannot be applied to this example. But it is easy to see that for all . Note that there exists such that Now, we can apply Theorem 3.3 with . Therefore, has a unique coupled fixed point, that is, a point .

Remark 3.10. Although the main results of Cho et al. [36] are an essential tool in the partially ordered cone metric spaces to claim the existence of the coupled fixed points the mappings do not have the mixed monotone property in general case such as the mapping in the above example. Therefore, it is very interesting to consider our theorems as another auxiliary tool to claim the existence of a coupled fixed point.

Acknowledgments

The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST), the second 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), and the third author would like to thank the Commission on Higher Education, the Thailand Research Fund, and King Mongkut’s University of Technology Thonburi (KMUTT) (Grant no. MRG5380044) for financial support during the preparation of this paper.

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