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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 763952, 13 pages
http://dx.doi.org/10.1155/2012/763952
Research Article

Coupled Fixed Point Theorems for Nonlinear Contractions in Partial Metric Spaces

1Department of Mathematics, Yarmouk University, Irbed 21163, Jordan
2Department of Mathematics, Al-Hashmia University, Zarqa 13115, Jordan

Received 28 March 2012; Revised 22 August 2012; Accepted 26 August 2012

Academic Editor: Hari Mohan Srivástava

Copyright © 2012 Sharifa Al-Sharif 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 establish some results on the existence and uniqueness of coupled fixed point involving nonlinear contractive conditions in complete-ordered partial metric spaces.

1. Introduction

The concept of partial metric which is a generalized metric space was introduced by Matthews [1] in 1994, in which the distance between two identical elements needs not be zero. The existence of fixed point for contraction-type mappings on such spaces was considered by many authors [112]. A modified version of a Banach contraction mapping principle, more suitable to solve certain problems arising in computer science using the concept of partial metric space is given in [1].

Gnana Bhaskar and Lakshmikantham [13] introduced the concept of coupled fixed point of a mapping and proved some interesting coupled fixed point theorems for mapping satisfying the mixed monotone property. Later in [14], Lakshmikantham and Ćirić investigated some more coupled fixed point theorems in partially ordered sets. For more on coupled fixed point theory, we refer the reader to [2, 1420].

First, we start by recalling some definitions and properties of partial metric spaces.

Definition 1.1 (see [9]). A partial metric on a nonempty set is a function such that for all :, , , .

A partial metric space is a pair such that is a non empty set and is a partial metric on . Each partial metric on generates a topology on which has as a base the family of open -balls , , where for all and . Matthews observed in [1, page 187] that a sequence in a partial metric space converges to some with respect to if and only if . It is clear that if , then from , and , . But if may not be .

If is a partial metric on , then the function given by is a metric on .

Example 1.2 (see, e.g., [1, 7]). Consider with . Then, is a partial metric space.

It is clear that is not a (usual) metric. Note that in this case .

Definition 1.3 (see [1, Definition 5.2]). Let be a partial metric space and let be a sequence in . Then, is called a Cauchy sequence if exists (and is finite).

Definition 1.4 (see [1, Definition 5.3]). A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point , such that .

Example 1.5 (see [12]). Let and define by Then, is a complete partial metric space.

It is well known (see, e.g., [1, page 194]) that a sequence in a partial metric space is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space , and that a partial metric space is complete if and only the metric space is complete. Furthermore, if and only if

Let be a partial metric. We endow with the partial metric defined for by

A mapping is said to be continuous at , if for every , there exists such that .

In this paper, we establish some results on the existence and uniqueness of a coupled fixed point involving nonlinear contractive conditions in complete-ordered partial metric spaces analogous to some other results in [17, 18].

Before presenting our main results, we recall some basic concepts.

Definition 1.6 (see [8]). An element is said to be a coupled fixed point of the mapping if and .

Definition 1.7 (see, Gnana Bhashkar and Lakshmikantham [13]). Let be a partially ordered set and . The mapping is said to has the mixed monotone property if

Now, let us denote by the set of all nondecreasing continuous functions that satisfy(i) if and only if ,(ii), for all .

Again, let denote all functions which satisfy for all and . It is an easy matter to see that and for any .

2. Main Results

The aim of this work is to prove the following theorem.

Theorem 2.1. Let be a partially ordered set and suppose that there is a partial metric on such that is a complete partial metric space. Let be a mapping having the mixed monotone property on and assume that there exist and such that for all with , and . Suppose either is continuous or has the following properties:(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all .
If there exist such that , and , then there exist such that and , that is, has a coupled fixed point. Furthermore, .

Proof. Choose and set and . Since and , letting and , we denote and due to the mixed monotone property of , we have Further, for , we can easily verify that Since and , from (2.1), we have Similarly, since and from (2.1), we also have Consequently, since is nondecreasing, using (2.5) and (2.6), we get By adding (2.7) to (2.8), we have Now, we will show that both and are Cauchy sequences. Note that Consequently, if , then and , that means is a coupled fixed point of . If , for each , combining (2.7) and (2.8) using property , we have Thus, By definition of , we have , so Consequently, which implies that both and are Cauchy sequences in the metric space . Since the metric space is complete, it follows that there exist such that Therefore, using property and the fact that is complete if and only if is complete, using (1.3), we have From and (2.10), we have Since , we get, . Similarly, one can show that . Therefore, Finally, we will show that and .
(a) Assume that is continuous on . In particular, is continuous at , hence for any , there exists such that if verifying , meaning that because , then we have Since , for , there exist such that, for and . Therefore, for , we have so we get Now, for any , On the other hand, inserting in (2.1), we get which implies , so for any . This implies that . Similarly, we can show that .
(b) Assume that satisfies the two conditions given by (i) and (ii). Since are a nondecreasing sequences and , we have and for all . By the condition , we have Therefore, Taking the limit as in the above inequality, using (2.18), and the properties of and , we get . Thus, . Hence, . Similarly, one can show that . Thus, we proved that has a coupled fixed point.

Corollary 2.2. Let be a partially ordered set and suppose that there is a partial metric on such that is a complete partial metric space. Let be a mapping having the mixed monotone property on . Supposed that for all with and . Suppose either is continuous or has the following properties:(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all . If there exist such that and then, there exist such that and , that is, has a coupled fixed point. Also, .

Proof. For , taking and , we get the result.

The following main theorem for Gnana Bhaskar and Lakshmikantham in [13] proved the next theorem.

Theorem 2.3 (see Gnana Bhaskar and Lakshmikantham [13]). Let be a partially ordered set and suppose there is a metric d on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Assume that there exists a with for all with and . suppose either is continuous or has the following properties:(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all .
If there exist such that and then there exist such that and , that is, has a coupled fixed point.

Note that for , and in Corollary 2.2, we get analogous to Theorem 2.3 in complete-ordered partial metric space.

Theorem 2.4. Let be a partially ordered set having the property that for every , there exists in such that and and suppose that there is a partial metric on such that is a complete partial metric space. Let be a mapping having the mixed monotone property on and assume that there exist and such that for all with , and . Suppose either is continuous or has the following properties:(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all .
If there exist such that  and then has a unique coupled fixed point.

Proof. From Theorem 2.1, the set of coupled fixed points of is non-empty. Suppose and are coupled fixed points of , that is,   and  . We shall show that and .
By assumption, there exists such that and . We define sequences as follows: Since we may assume that . By using the mathematical induction, it is easy to prove that and for any . From (2.1), we have Since is nondecreasing, from the above inequalities, we have Adding (2.32) to (2.33), we get Therefore, that is, since , the sequences and are convergent and Similarly, one can show that . Since letting , we obtain , so and .

Theorem 2.5. Let be a partially ordered set such that for every , there exists in such that and and suppose there is a partial metric on such that is a complete partial metric space. Let be a mapping having the mixed monotone property on and assume that there exist and such that for all with   and . Suppose either is continuous or has the following properties:(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all .
If there exist such that  and , then has a unique coupled fixed point. In addition, if or , then where is a coupled fixed point of .

Proof. Following the proof of Theorem 2.4, has a unique coupled fixed point . We only have to show that . Assume . Using the mathematical induction, one can show that for any . Note that, by , Therefore, using the condition , (2.1), and a property of , From , we have . Assume that . Letting in (2.40) we get Since , and is nondecreasing it follows that that is, , which is a contradiction. Thus, , so .

Example 2.6 (see [17]). Let with usual order. Define by and by . Then,(i) is a complete partially ordered partial metric space,(ii) has the mixed monotone property,(iii)for with and , we have (iv) is continuous.

Proof. The proofs of (i), (ii), and (iii) are clear. To prove (iv), letting and , we claim . To prove our claim, let , then So, Since and , we have . Therefore, and hence . So, . We deduce that all the hypotheses of Theorem 2.1 are satisfied with , and . Therefore, has a coupled fixed point. Here, is the coupled fixed point of .

3. Application

In this part, from previous obtained results, we will deduce some coupled fixed point results for mappings satisfying a contraction of integral type in a complete partial metric space.

Let be the set of all functions satisfying the following conditions:(i) is a Lebesgue integrable mapping on each compact subset of ,(ii)for all , we have,(iii) is subadditive on each , that is,

Let be fixed. Let be a family of functions that belong to . For all , we denote as follows: We have the following result.

Theorem 3.1. Let be a partially ordered set and suppose there is a partial metric on such that is a complete partial metric space. Let be a mapping having the mixed monotone property on . Assume that there exist and such that for all with , , and . Suppose either is continuous, or has the following properties:(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all .
If there exist such that and , then there exist such that and , that is, has a coupled fixed point.

Proof. Take and . Note that the are taken to be subadditive on each in order to get . Moreover, it is easy to show that is continuous, nondecreasing and verifies . We get that . Also, we can find that . From (3.3), we have Now, applying Theorem 2.1, we obtain the desired result.

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