International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 670410, 12 pages

doi:10.1155/2012/670410

Research Article

Coupled Fixed Point Results in Complete Partial Metric Spaces

H. Alaeidizaji¹ and V. Parvaneh²

¹Department of Mathematics, Payame Noor University, Tehran 19395-4697, Iran

²Department of Mathematics, Islamic Azad University, Gilan-E-Gharb Branch, Gilan-E-Gharb, Iran

Received 23 March 2012; Revised 18 September 2012; Accepted 25 September 2012

Academic Editor: A. Zayed

Copyright © 2012 H. Alaeidizaji and V. Parvaneh. 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 coupled fixed point theorems for a mapping satisfying some contraction conditions in complete partial metric spaces. Our consequences extend the results of H. Aydi (2011).

1. Introduction and Mathematical Preliminaries

The notion of a partial metric space (PMS) was introduced in 1992 by Matthews [1, 2]. Matthews proved a fixed point theorem on this spaces, analogous to the Banach's fixed point theorem. Recently, many authors have focused on partial metric spaces and their topological properties (see, e.g., [3–9]).

The definition of a partial metric space is given by Matthews (see [1, 2]) as follows:

Definition 1.1. Let be a nonempty set and let satisfies

(P1), for all ,(P2), for all ,(P3) , for all ,(P4), for all .

Then the pair is called a partial metric space and is called a partial metric on .

The function defined by

satisfies the conditions of a metric on ; therefore it is a (usual) metric on .

Remark 1.2. if , may not be 0.

(1)A famous example of partial metric spaces is the pair , where for all . In this case, is the Euclidian metric .(2)Each partial metric on generates a topology on which has a base of open p-balls , where and ().

The following concepts has been defined as follows on a partial metric space.

Definition 1.3 (see e.g., [1, 2]). (i) A sequence in a PMS converges to if and only if .

(ii) A sequence in a PMS is called Cauchy if and only if exists (and is finite).

(iii) A PMS is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

The concept of coupled fixed point have been introduced in [10] by Bhaskar and Lakshmikantham as follows.

Definition 1.4 (see [10]). An element is called a coupled fixed point of mapping if and .

Aydi in [11] has obtained some coupled fixed point results for mappings satisfying different contractive conditions on complete partial metric spaces. Some of these results are the following cases.

Theorem 1.5 (see [11, 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.

Theorem 1.6 (see [11, Theorem 2.4]). Let (X, p) 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 1.7 (see [11, Theorem 2.5]). Let (X, p) 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.

For a survey of fixed point theory, its applications, and related results in partial metric spaces we refer the reader to [4, 5, 12–20] and the references mentioned therein. Also, many researchers have obtained coupled fixed point results for mappings under various contractive conditions in the framework of partial metric spaces (see, e.g., [21, 22]).

In this paper we establish some coupled fixed point results of contractive mappings in the framework of complete partial metric spaces. Our results extend and generalize the results of Aydi [11].

2. Main Results

We recall three easy lemmas which have an essential role in the proof of the main result. These results can be derived easily (see, e.g., [1, 2, 6]).

Lemma 2.1. A sequence is a Cauchy sequence in the PMS if and only if it is a Cauchy sequence in the metric space .

A PMS is complete if and only if the metric space is complete. Moreover,

Lemma 2.2 (see [3]). Assume that as in a PMS such that . Then, , for every .

Lemma 2.3 (see, e.g., [3, 4]). Let (X, p) be a complete PMS. Then,

(a)if then, ,(b)if , then .

Throughout this paper, we assume that all of the constants are nonnegative. Our main result is the following. The method of the proof can be found in [11].

Theorem 2.4. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Proof. Let be arbitrary. Define such that and and in this way, we construct the sequences and as and , for all .

We will complete the proof in three steps.

Step I. Let . We will show that .

Using (2.2) we obtain that

Analogously, starting from , we have

In a similar way, we have

Analogously, starting from , we have

Adding (2.3), (2.4), (2.5), and (2.6) we obtain that

or, equivalently,

where, .

Repeating the above mentioned process, we have

where, from our assumption about coefficients , ; hence,

Step II. and are Cauchy.

If then, . Hence, we get and ; that is, is a coupled fixed point of . Now, let . For each , we have

So, we have . This proves that and are Cauchy sequences in and hence and are Cauchy sequences in the metric space . From Lemma 2.1, is complete, so and converge to some , respectively; that is, and . Therefore, from Lemma 2.1 and (2.10), we have

Step III. We will show that has a unique coupled fixed point.

From the above step,

Next, we will prove that and .

We have

Taking the limit as in the above inequality, as and using triangle inequality and (2.12), we have

But, for all , from (2.2),

In the above inequality, if , using (2.12) and Lemma 2.2 we have

Analogously,

Taking the limit as in the above inequality, since and using triangle inequality and (2.13), we have

Similar to (2.17), we have

Adding (2.18) and (2.21) and using (2.15) and (2.19), we obtain that

Therefore, ; that is, and .

Remark 2.5. (1) If in the above theorem, we assume that , for all , then we obtain the result of Aydi in [11] which is noted here in Theorem 1.5.

(2) If in the above theorem, , for all , unless , then we obtain the result of Aydi in [11] which is mentioned here in Theorem 1.6.

(3) If in the above theorem, we assume that , for all , except that , then we obtain the result of Aydi in [11] (Theorem 1.7).

Many results can be deduced from the above theorem as follows.

Corollary 2.6. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Corollary 2.7. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Corollary 2.8. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Corollary 2.9. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Also, we have the following results, when the constants in the above corollaries are equal.

Corollary 2.10. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Corollary 2.11. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Corollary 2.12. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Corollary 2.13. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Corollary 2.14. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then, has a unique coupled fixed point in .

Corollary 2.15. Let be a complete partial metric space and be a mapping such that

for every pairs , where . Then has a unique coupled fixed point in .

Example 2.16. Let and on be given as . Obviously, the partial metric space is complete (see, e.g., Example  2.3 of [11]).

Define as , for all .

Now, we have

Thus, (2.2) is satisfied with . Obviously, all the conditions of Theorem 2.4 are satisfied. Moreover, is the unique coupled fixed point of .

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