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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 150363, 20 pages
http://dx.doi.org/10.1155/2012/150363
Research Article

Coupled Fixed-Point Theorems for Contractions in Partially Ordered Metric Spaces and Applications

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea
3Department of Mathematics, Islamic Azad University, Semnan Branch, Semnan, Iran
4Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

Received 30 September 2011; Revised 20 December 2011; Accepted 24 December 2011

Academic Editor: Stefano Lenci

Copyright © 2012 M. Eshaghi Gordji 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

Bhaskar and Lakshmikantham (2006) showed the existence of coupled coincidence points of a mapping from into and a mapping from into with some applications. The aim of this paper is to extend the results of Bhaskar and Lakshmikantham and improve the recent fixed-point theorems due to Bessem Samet (2010). Indeed, we introduce the definition of generalized -Meir-Keeler type contractions and prove some coupled fixed point theorems under a generalized -Meir-Keeler-contractive condition. Also, some applications of the main results in this paper are given.

1. Introduction

The Banach contraction principle [1] is a classical and powerful tool in nonlinear analysis and has been generalized by many authors (see [215] and others).

Recently, Bhaskar and Lakshmikantham [16] introduced the notion of a coupled fixed-point of the given two variables mapping. More precisely, let be a nonempty set and be a given mapping. An element is called a coupled fixed-point of the mapping if

They also showed the uniqueness of a coupled fixed-point of the mapping and applied their theorems to the problems of the existence and uniqueness of a solution for a periodic boundary value problem.

Theorem 1.1 (see Zeidler [15]). Let be a partially ordered set and suppose that there is a metric d on such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on . Assume that there exists such that for all and . Moreover, if there exist such that then there exist such that and .

Later, in [17], Lakshmikantham and Ćirić investigated some more coupled fixed-point theorems in partially ordered sets, and some others obtained many results on coupled fixed-point theorems in cone metric spaces, intuitionistic fuzzy normed spaces, ordered cone metric spaces and topological spaces (see, e.g., [1825]).

In [9], Meir and Keeler generalized the well-known Banach fixed-point theorem [1] as follows.

Theorem 1.2 (Meir and Keeler [9]). Let be a complete metric space and be a given mapping. Suppose that, for any , there exists such that for all . Then T admits a unique fixed-point and, for all , the sequence converges to .

Proposition 1.3 (see [17]). Let be a partially ordered metric space and be a given mapping. If the contraction (1.2) is satisfied, then is a generalized Meir-Keeler type contraction.

Motivated by the results of Bhaskar and Lakshmikantham [16], Lakshmikantham and Ćirić [17], and Samet [26], in this paper, we introduce the definition of -Meir-Keeler-contractive mappings and prove some coupled fixed-point theorems under a generalized -Meir-Keeler contractive condition.

2. Main Results

Let be a nonempty set. We note that an element is called a coupled coincidence point of a mapping and if and for all . Also, we say that and are commutative (or commuting) if for all .

We introduce the following two definitions.

Definition 2.1. Let be a partially ordered set and and . We say that has the mixed strict -monotone property if, for any ,

Definition 2.2. Let be a partially ordered set and be a metric on . Let and be two given mappings. We say that is a generalized -Meir-Keeler type contraction if, for all , there exists such that, for all with and ,

Lemma 2.3. Let be a partially ordered set and be a metric on . Let and be two given mappings. If is a generalized -Meir-Keeler type contraction, then we have for all with , or , .

Proof. Let such that and or and . Then . Since is a generalized -Meir-Keeler type contraction, for , there exists such that, for all with and , Therefore, putting , , and , we have This completes the proof.

From now on, we suppose that is a partially ordered set, and there exists a metric on such that is a complete metric space.

Theorem 2.4. Let and be such that , is continuous and commutative with . Also, suppose that(a) has the mixed strict -monotone property;(b) is a generalized -Meir-keeler type contraction;(c)there exist such that and .
Then there exist such that and ; that is, and have a coupled coincidence in .

Proof. Let be such that and . Since , we can choose such that and . Again, from , we can choose such that and .
Continuing this process, we can construct the sequences and in such that for all .
Now, we show that for all . For , we have Since has the mixed strict -monotone property, then we have It follows that , that is, .
Similarly, we have Thus it follows that , that is, .
Again, we have Thus it follows that , that is, .
Similarly, we have Thus it follows that , that is, .
Continuing this process for each , we get the following: Denote that Since and , it follows from (2.6) and Lemma 2.3 that Since and , it follows from (2.6) and Lemma 2.3 that Thus it follows from (2.14)–(2.16) that . This means that the sequence is monotone decreasing. Therefore, there exists such that , that is,
Now, we show that . Suppose that hold. Let . Then there exists a positive integer such that Then, by using (2.7) and the condition (b), we have and so, by (2.6), we have
On the other hand, by (2.15), we have which is a contradiction with (2.18). Thus we have , that is, that is,
Now, we prove that and are Cauchy sequences in . Suppose that at least one of or is not a Cauchy sequence. Then there exist and two subsequences , of integers such that and for all . Thus we have for all . Let be the smallest number exceeding such that (2.25) holds. Then we have Thus, from (2.14), (2.25), (2.26) and the triangle inequality, it follows that and so Hence, by (2.23), we have It follows from (2.6), (2.14), and the triangle inequality that Form (2.13) we have and . Now, it follows from Lemma 2.3 and (2.30) that that is, This is a contradiction. Therefore, and are Cauchy sequences. Since is complete, there exist such that Since is monotone increasing and is monotone decreasing, we have for all . Thus it follows from (2.33) and the continuity of that Thus, for all , there exists a positive integer such that, for all , Hence, from (2.6), the commutativity of and and the triangle inequality, we have Thus, it follows from (2.34), (2.36), and Lemma 2.3 that as . Therefore, we have . Similarly, we can show that . This means that and have a coupled coincidence point in . This completes the proof.

Corollary 2.5. Let be a mapping satisfying the following conditions:(a) has the mixed strict monotone property;(b) is a generalized Meir-Keeler type contraction;(c)there exists such that and .
Then there exist such that and .

Proof. The conclusion follows from Theorem 2.4 by putting (: the identity mapping) on .

Now, we introduce the product space with the following partial order: for all ,

Theorem 2.6. Suppose that all the hypotheses of Theorem 2.4 hold and, further, for all , there exists such that is comparable with and . Then and have a unique coupled common fixed-point, that is, there exists a unique such that

Proof. By Theorem 2.4, the set of coupled coincidences of the mapping and is nonempty.
First, we show that, if and are coupled coincidence points of and , that is, if then we have Put , and choose such that and . Then, similarly as in the proof of Theorem 2.4, we can inductively define the sequences and such that for all . Also, if we set , , , and , then we can define the sequences , , , and as follows: for all . Since are comparable each other, then and . It is easy to show that , and are comparable each other, that is, and for all . Thus it follows from Lemma 2.3 that and so as . Therefore, we have
Similarly, we can prove that Thus, by the triangle inequality, (2.48) and (2.49), we have as , which imply that and .
Now, we prove that and . Denote that and . Since and , by the commutativity of and , we have Thus, is a coupled coincidence point of and .
Putting and in (2.52), it follows from (2.42) that and so, from (2.51) and (2.52), Therefore, is a coupled common fixed-point of and .
Finally, to prove the uniqueness of the coupled common fixed-point of and , assume that is another coupled common fixed-point of and . Then, by (2.42), we have and . This completes the proof.

Corollary 2.7. Suppose that all the hypotheses of Corollary 2.5 hold and, further, for all and , there exists that is comparable with and . Then there exists a unique such that .

3. Applications

Now, we give some applications of the main results in Section 2.

Theorem 3.1. Let and be two given mappings. Assume that there exists a function satisfying the following conditions:(a) and for any ;(b) is nondecreasing and right continuous;(c)for any , there exists such that, for all with and , Then is a generalized -Meir-Keeler type contraction.

Proof. For any , it follows from (a) that and so there exists such that, for all with and , From the right continuity of , there exists such that . For any such that , and since is nondecreasing function, we get the following: By (3.2), we have and so . Therefore, it follows that is a generalized -Meir-Keeler type contraction. This completes the proof.

Corollary 3.2 (see [26, Theorem  3.1]). Let be a given mapping. Assume that there exists a function satisfying the following conditions:(a) and for any ;(b) is nondecreasing and right continuous;(c)for any , there exists such that , and Then is a generalized Meir-Keeler type contraction.

The following result is an immediate consequence of Theorems 2.4 and 3.1.

Corollary 3.3. Let and be two given mappings such that , is continuous and commutative with . Also, suppose that(a) has the mixed strict -monotone property;(b)for any , there exists such that, for all with and , where is a locally integrable function from into itself satisfying the following condition: for all ;(c)there exist such that and .Then there exists such that and . Moreover, if and are comparable to each other, then and have a unique coupled common fixed-point in .

Corollary 3.4. Let be a mapping satisfying the following conditions:(a) has the mixed strict monotone property;(b) for any , there exists such that , and where is a locally integrable function from into itself satisfying for all ;(c) there exist such that and .Then there exists such that and . Moreover, if and are comparable to each other, then has a unique coupled common fixed-point in .

Corollary 3.5. Let and be two given mappings such that , is continuous and commutes with . Also, suppose that(a) has the mixed strict -monotone property;(b)for any with and , where and is a locally integrable function from into itself satisfying for all ;(c)there exist such that and .Then there exists such that and . Moreover, if and are comparable to each other, then and have a unique coupled common fixed-point in .

Proof. For any , if we take and apply Corollary 3.3, then we can get the conclusion.

Corollary 3.6. Let be a mapping satisfying the following conditions:(a) has the mixed strict monotone property,(b)for any with and , where and is a locally integrable function from into itself satisfying for all ;(c)there exist such that and .Then there exist such that and . Moreover, if and are comparable to each other, then has a unique coupled common fixed-point in .

Finally, by using the above results, we show the existence of solutions for the following integral equation: where (: the set of continuous functions from into ), , is a continuous function and

Definition 3.7. A lower solution for the integral equation (3.14) is an element such that where denotes the set of differentiable functions from into .

Now, we prove the existence of solutions for the integral equation (3.14) by using the existence of a lower solution for the integral equation (3.14).

Theorem 3.8. Let be the class of the functions satisfying the following conditions:(a) is increasing;(b)for any , there exists such that .In the integral equation (3.14), suppose that there exists such that, for all with , where . If a lower solution of the integral equation (3.14) exists, then a solution of the integral equation (3.14) exists.

Proof. Define a mapping by
Note that, if is a coupled fixed-point of , then is a solution of the integral equation (3.14).
Now, we check the hypotheses in Corollary 2.5 as follows:(1) is a partially ordered set if we define the order relation in as follows: for all and .(2) is a complete metric space if we define a metric as follows: (3)The mapping has the mixed strict monotone property. In fact, by hypothesis, if , then we have which implies that, for any , that is,
Similarly, if , then we have which implies that, for any , that is,
Now, we show that satisfies (1.2). In fact, let and . Then we have Since the function is increasing and , we have we obtain the following: Then, by Proposition 1.3, is a generalized Meir-Keeler type contraction.
Finally, let be a lower solution for the integral equation (3.14). Then we show that Indeed, we have for any and so for any . Multiplying by in (3.31), we get the following: for any , which implies that for any . This implies that and so Thus it follows from (3.35) and (3.33) that and so Hence we have for any .
Similarly, we have . Therefore, by Corollary 2.5, has a coupled fixed-point.

Example 3.9. In the integral equation (3.14), we put , for all and . Then is a continuous function, and we have