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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 647091, 11 pages
http://dx.doi.org/10.1155/2011/647091
Research Article

Some Coupled Fixed Point Results on Partial Metric Spaces

Institut Supérieur d'Informatique de Mahdia, Université de Monastir, route de Réjiche, Km 4, BP 35, Mahdia 5121, Tunisia

Received 20 December 2010; Revised 19 February 2011; Accepted 19 March 2011

Academic Editor: Enrique Llorens-Fuster

Copyright © 2011 Hassen Aydi. 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 give some coupled fixed point results for mappings satisfying different contractive conditions on complete partial metric spaces.

1. Introduction and Preliminaries

For a given partially ordered set , Bhaskar and Lakshmikantham in [1] introduced the concept of coupled fixed point of a mapping . Later in [2], Círíc and Lakshmikantham investigated some more coupled fixed point theorems in partially ordered sets. The following is the corresponding definition of a coupled fixed point.

Definition 1.1 (see [3]). An element is said to be a coupled fixed point of the mapping if and .

Sabetghadam et al. [4] obtained the following.

Theorem 1.2. Let be a complete cone metric space. Suppose that the mapping satisfies the following contractive condition for all where are nonnegative constants with . Then, has a unique coupled fixed point.

In this paper, we give the analogous of this result (and some others in [4]) on partial metric spaces, and we establish some coupled fixed point results.

The concept of partial metric space was introduced by Matthews in 1994. In such spaces, the distance of a point in the self may not be zero. First, we start with some preliminaries definitions on the partial metric spaces [3, 513].

Definition 1.3 (see ([68])). A partial metric on a nonempty set is a function such that for all :(p1), (p2), (p3), (p4).

A partial metric space is a pair such that is a nonempty set and is a partial metric on .

Remark 1.4. It is clear that if , then from (p1), (p2), and (p3), . But if , may not be 0.

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

Definition 1.5 (see ([68])). Let be a partial metric space. Then, (i)a sequence in a partial metric space converges to a point if and only if ;(ii)a sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) ;(iii)a partial metric space is said to be complete if every Cauchy sequence in converges to a point , that is, .

Lemma 1.6 (see ([6, 7, 9])). Let be a partial metric space;(a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space , (b)a partial metric space is complete if and only if the metric space is complete; furthermore, if and only if

2. Main Results

Our first main result is the following.

Theorem 2.1. Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition for all where , are nonnegative constants with . Then, has a unique coupled fixed point.

Proof. Choose and set and . Repeating this process, set and . Then, by (2.1), we have and similarly Therefore, by letting we have Consequently, if we set , then, for each , we have If then . Hence, from Remark 1.4, we get and , meaning that is a coupled fixed point of . Now, let . For each , we have, in view of the condition (p4) Similarly, we have Thus, By definition of , we have , so, for any which implies that and are Cauchy sequences in because of . Since the partial metric space is complete, hence thanks to Lemma 1.6, the metric space is complete, so there exist such that Again, from Lemma 1.6, we get But, from condition (p2) and (2.6), so since , hence letting , we get . It follows that Similarly, we get Therefore, we have, using (2.1), and letting , then from (2.14) and (2.15), we obtain , so . Similarly, we have , meaning that is a coupled fixed point of .
Now, if is another coupled fixed point of , then It follows that In view of , this implies that , so and . The proof of Theorem 2.1 is completed.

It is worth noting that when the constants in Theorem 2.1 are equal, we have the following corollary

Corollary 2.2. Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition for all , where . Then, has a unique coupled fixed point.

Example 2.3. Let endowed with the usual partial metric defined by with . The partial metric space is complete because is complete. Indeed, for any , Thus, is the Euclidean metric space which is complete. Consider the mapping defined by . For any , we have which is the contractive condition (2.19) for . Therefore, by Corollary 2.2, has a unique coupled fixed point, which is . Note that if the mapping is given by , then satisfies the contractive condition (2.19) for , that is, In this case, and are both coupled fixed points of , and, hence, the coupled fixed point of is not unique. This shows that the condition in Corollary 2.2, and hence in Theorem 2.1 cannot be omitted in the statement of the aforesaid results.

Theorem 2.4. Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition for all where , are nonnegative constants with . Then, has a unique coupled fixed point.

Proof. We take the same sequences and given in the proof of Theorem 2.1 by Applying (2.23), we get where . By definition of , we have Since , hence , so the sequences and are Cauchy sequences in the metric space . The partial metric space is complete, hence from Lemma 1.6, is complete, so there exist such that From Lemma 1.6, we get By the condition (p2) and (2.25), we have so . It follows that Similarly, we find Therefore, by (2.23), and letting , then from (2.27)–(2.32), we obtain From the preceding inequality, we can deduce a contradiction if we assume that , because in that case we conclude that and now this inequality is, in fact, a contradiction, so , that is, . Similarly, we have , meaning that is a coupled fixed point of . Now, if is another coupled fixed point of , then, in view of (2.23), that is, since . It follows that . Similarly, we can have , and the proof of Theorem 2.4 is completed.

Theorem 2.5. Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition for all , where are nonnegative constants with . Then, has a unique coupled fixed point.

Proof. Since, , hence , and as a consequence the proof of the uniqueness in this theorem is as trivial as in the other results. To prove the existence of the fixed point, choose the sequences and like in the proof of Theorem 2.1, that is Applying again (2.37), we have It follows that for any Let us take . Hence, we deduce Under the condition , we get . From this fact, we immediately obtain that is Cauchy in the complete metric space . Of course, similar arguments apply to the case of the sequence in order to prove that and, thus, that the sequence is Cauchy in . Therefore, there exist such that Thanks to Lemma 1.6, we have The condition (p2) together with (2.41) yield that hence letting , we get . It follows that Similarly, we have Therefore, we have, using (2.37), Letting yields, using (2.46), and since , we have , that is, . Similarly, thanks to (2.47), we get , and hence is a coupled fixed point of .

When the constants in Theorems 2.4 and 2.5 are equal, we get the following corollaries.

Corollary 2.6. Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition for all where . Then, has a unique coupled fixed point.

Corollary 2.7. Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition for all where . Then, has a unique coupled fixed point.

Proof. The condition follows from the hypothesis on and given in Theorem 2.5.

Remark 2.8. (i) Theorem 2.1 extends the Theorem  2.2 of [4] on the class of partial metric spaces.
(ii) Theorem 2.4 extends the Theorem  2.5 of [4] on the class of partial metric spaces.

Remark 2.9. Note that in Theorem 2.4, if the mapping satisfies the contractive condition (2.23) for all , then also satisfies the following contractive condition: Consequently, by adding (2.23) and (2.52), also satisfies the following: which is a contractive condition of the type (2.50) in Corollary 2.6 with equal constants. Therefore, one can also reduce the proof of general case (2.23) in Theorem 2.4 to the special case of equal constants. A similar argument is valid for the contractive conditions (2.37) in Theorem 2.5 and (2.51) in Corollary 2.7.

Acknowledgment

The author thanks the editor and the referees for their kind comments and suggestions to improve this paper.

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