International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 686801, 10 pages

doi:10.1155/2012/686801

Research Article

Generalization of Some Coupled Fixed Point Results on Partial Metric Spaces

Wasfi Shatanawi,¹ Hemant Kumar Nashine,² and Nedal Tahat¹

¹Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

²Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Naradha, Mandir Hasaud, Chhattisgarh Raipur 492101, India

Received 21 March 2012; Accepted 3 May 2012

Academic Editor: Heinz Gumm

Copyright © 2012 Wasfi Shatanawi 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.

Using the setting of partial metric spaces, we prove some coupled fixed point results. Our results generalize several well-known comparable results of H. Aydi (2011). Also, we introduce an example to support our results.

1. Introduction and Preliminaries

The notion of coupled fixed point of a mapping was introduced by Gnana Bhaskar and Lakshmikantham in [1]. Later on, many authors investigated many coupled fixed point results in different spaces such as usual metric spaces, fuzzy metric spaces, generalized metric spaces, partial metric spaces, and partially ordered metric spaces (see [1–20]).

Definition 1.1 (see [1]). An element is called a coupled fixed point of a mapping if

Matthews [21] in 1994 introduced the notion of partial metric spaces in such a way that each object does not necessarily have to have a zero distance from itself. Consistent with Matthews [21], the following definitions and results will be needed in the sequel.

Definition 1.2 (see [21]). 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 nonempty set and is a partial metric on .

Each partial metric on generates a topology on . The set , where for all and , forms the base of .

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

is a metric on .

Definition 1.3 (see [21]). Let be a partial metric space. Then:

(1)a sequence in a partial metric space converges, with respect to , to a point if and only if ,(2)a sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) ,(3)A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

Lemma 1.4 (see [21]). Let be a partial metric space.

(1) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .(2)A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if

Abdeljawad et al. [22–24], Altun et al. [25], Karapinar and Erhan [26–28], Oltra and Valero [29] and Romaguera [30] studied fixed point theorems in partial metric spaces. For more works in partial metric spaces, we refer the reader to [31–40].

Aydi [2] proved the following coupled fixed point theorems in partial metric spaces.

Theorem 1.5. 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.

Theorem 1.6. 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.

In this paper, we prove some coupled fixed point results. Our results generalize Theorems 1.5 and 1.6. Also, we introduce an example to support our results.

2. The Main Result

Theorem 2.1. Let be a complete partial metric space. Suppose that the mapping satisfies

for all . If , then has a unique coupled fixed point.

Proof. Choose . Let and . Again let and . By continuing in the same way, we construct two sequences and in such that

Then by (2.1), we have

Thus from (2.3), we have

By repeating (2.4) -times, we get that

Letting in (2.5), we get that

Therefore, we have

For with , we have

By (2.5) and (2.8), we have

Letting in (2.9), we get that

Thus exists and is finite. Hence is a Cauchy sequence in . Similarly, we may show that

and hence is a Cauchy sequence in . By Lemma 1.4 there exist such that (resp., ) if and only if

Now, we prove that . By (2.1), we have

Letting in the above inequality and using (2.12), we get that

Since , we conclude that . By and , we have . Similarly, we may show that . Thus is a coupled fixed point of . To prove the uniqueness of the fixed point, we let be a coupled fixed point of . We will show that and . By (2.1), we have

Since and , we have

Also, from (2.1), we have

Since and , we have

From (2.16) and (2.18), we have

Since , we have . Hence and . By and , we have and .

Corollary 2.2. Let be a complete partial metric space. Suppose that there are with such that the mapping satisfies

for all . Then has a unique coupled fixed point.

Proof. The proof follows from Theorem 2.1 by noting that:

Remarks.

(1)Theorem 1.5 [2, Theorem 2.1] is a special case of Corollary 2.2.(2)[2, Corollary 2.2] is a special case of Corollary 2.2.(3)Theorem 1.6 [2, Theorem 2.4] is a special case of Corollary 2.2.(4)[2, Corollary 2.6] is a special case of Corollary 2.2.

Now, we introduce an example satisfying the hypotheses of Theorem 2.1 but not the hypotheses of Theorems 2.1 and 2.4 of [2].

Example 2.3. Define by . Then is a complete partial metric space. Let be the mapping defined by

Then,

(a) for all .(b) There are no with such that for all .(c) There are no with such that for all .

Proof. To prove part (a), given . Then:

To prove part (b), suppose that there are with such that for all .

Since

we have , which is a contradiction.

To prove part (c), suppose that there are with such that for all .

Since

we have , which is a contradiction.

Thus by Theorem 2.1, has a unique coupled fixed point. Here, is the unique fixed point of .

Acknowledgments

The authors thank the editor and the referees for their valuable comments and suggestions.

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