Abstract and Applied Analysis

Volume 2012 (2012), Article ID 495892, 17 pages

doi:10.1155/2012/495892

Research Article

On Weakly -Contractive Mappings in Ordered Partial Metric Spaces

Erdal Karapınar¹ and Wasfi Shatanawi²

¹Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

²Department of Mathematics, Hashemite University, Zarqa, Jordan

Received 12 May 2012; Revised 17 July 2012; Accepted 26 July 2012

Academic Editor: Paul Eloe

Copyright © 2012 Erdal Karapınar and Wasfi Shatanawi. 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 introduce the notion of weakly -contractive mappings in ordered partial metric spaces and prove some common fixed point theorems for such contractive mappings in the context of partially ordered partial metric spaces under certain conditions. We give some common fixed point results of integral type as an application of our main theorem. Also, we give an example and an application of integral equation to support the useability of our results.

1. Introduction and Preliminaries

In 1994, Matthews [1] introduced the notion of a partial metric space as a generalization of the usual metric space. In partial metric space self distance, that is. is not necessarily equal a zero. In this interesting paper, Matthews [1] prove the Banach contraction mapping principle in the frame of partial metric spaces. After this initial paper, many authors have studied various type contractions and related fixed point results in partial metric spaces (see, [2–32]).

Definition 1.1 (see [1]). A partial metric on a nonempty set is a function such that for all :

 ,  ,  ,  .

A pair is called a partial metric space where 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 1.2 (see [1]). Let be a partial metric space. Then one has the following.

(1)A sequence in a partial metric space converges 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 .

The following lemma is crucial in proving our main results.

Lemma 1.3 (see [1]). 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

In 1972, Chatterjea [33] introduced the concept of -contraction as follows.

Definition 1.4 (see [33]). Let be a metric space and be a mapping. Then is called a -contraction if there exists such that

holds for all .

In this interesting paper, Chatterjea [33] proved the following theorem.

Theorem 1.5 (see [33]). Every -contraction in a complete metric space has a unique fixed point.

Choudhury [34] introduced the concept of weakly -contractive mapping as a generalization of -contractive mapping and prove that every weakly -contractive mapping in a complete metric space has a unique fixed point.

Definition 1.6 (see [34]). Let be a metric space and be a mapping. Then is called a weakly -contractive if there exists a continuous function such that for all , we have

Harjani et al. [35] announced some fixed point results for weakly C-contractive mappings in a complete metric space endowed with a partial order. Meanwhile, Shatanawi [36] proved some fixed point and coupled fixed point theorems for a nonlinear weakly -contraction type mapping in metric and ordered metric spaces.

In this paper, we introduce the concept of weakly -contractive mappings in ordered partial metric spaces, and we prove some existence theorems of common fixed point for such mapping in the context of complete partial metric spaces under certain conditions.

2. The Main Result

We start this section with the following definitions.

Definition 2.1. Suppose that is a partial metric space. A mapping is said to be continuous at if for every , there exists such that . We say that is continuous if is continuous at all .

It is easy to see that if is a partial metric space, is continuous, is a sequence in , and

Altun and Simsek [37] introduce the notion of weakly increasing of two mappings in the following way.

Definition 2.2 (see [37]). Let be a partially ordered set. Two mappings are said to be weakly increasing if and for all .

For more details on weakly increasing mappings, we refer the reader to [24, 38–41]. Let denote all functions such that

  is continuous,   if and only if .

Similarly, we denote by all functions such that

  is continuous and nondecreasing,   if and only if .

Inspired the definitions above, we introduce the following definition.

Definition 2.3. Let be an partially ordered metric space. Then the mappings are said to be weakly -contractive mappings if and are weakly increasing with respect to and for any comparable and , we have

where and .

Now, we introduce and prove our first results.

Theorem 2.4. Let be a partially ordered set and suppose that there exists a partial metric on such that is complete. Suppose that are weakly -contractive mappings. If and are continuous, then and have a common fixed point; that is, there exists such that .

Proof. Given . Set and . Continuing this process, we construct sequences in such that and . Using the fact that that and are weakly increasing with respect to , we obtain that

Now, we will prove that

Since and are comparable, by (2.2), we have

By , we have

Thus (2.5) becomes

Using the fact that is nondecreasing, we get that

Hence, we have

Similarly, we may show that

From (2.9) and (2.10), we have

By (2.11), we get that is a non increasing sequence. Hence there is such that

Letting in (2.7), we get

Thus

and hence

Using the continuity of , we conclude that

Again, on taking limit sup in (2.5), we have and hence . From the definition of , we have

Next, we will show that is a Cauchy sequence in the metric space . It is sufficient to show that is a Cauchy sequence in . Suppose to the contrary, that is, is not a Cauchy sequence in . Then there exists for which we can find two subsequences of positive integers and such that is the smallest index for which

This means that

From (2.18), (2.19), and the triangular inequality, we get that

Letting in above inequalities and using (2.17), we have

Again, from (2.18) and the triangular inequality, we get that

Letting in above inequalities and using (2.4) and (2.21), we get that

By the fact that

for all , and the expression above, we conclude that

By (2.2), we have

Letting and using the continuity of and , we get that

Therefore, we get that . Hence which is a contradiction. Thus, is a Cauchy sequence in . From Lemma 1.3, the sequence converges in the metric space , say . Again from Lemma 1.3, we have

Moreover, since is a Cauchy sequence in the metric space , we have

From the definition of , we have

Letting in the above equality and using (2.4) and (2.29), we get

Thus by (2.28), we have

Now,

Letting in above inequalities and using (2.4) and (2.29), we get that

Similarly, we may show that

Since is continuous and

then by (2.34), we have

Similarly, we may show that . By () and () we derive that

The above inequality yields that

Since and are comparable and is nondecreasing, then by (2.2), we have

Thus, we have and hence . By using () and () of Definition 1.1, we find that . That is, is a common fixed point of and .

The continuity of and in Theorem 2.4 can be dropped. For this instance, suppose that satisfies the following property:

: if is a nondecreasing sequence in such that , then for all .

Theorem 2.5. Let be a partially ordered set and suppose that there exists a partial metric on such that is complete. Suppose that be weakly -contractive mappings. If satisfies property , then and have a common fixed point, that is, there exists such that .

Proof. Following the proof of Theorem 2.4 step by step to construct a nondecreasing sequence in such that ,

By property, we have for all . By (2.2), we have

Letting in above inequalities, and using (2.41) we get . Since is nondecreasing we get that . Hence . By and , we conclude that . By similar arguments, we can show that . Thus is a common fixed point of and .

By taking (the identity function on ) and defining via where in Theorems 2.4 and 2.5, we have the following results.

Corollary 2.6. Let be a partially ordered set and suppose that there exists a partial metric on such that is complete. Suppose that be weakly increasing mappings with respect to such that for any comparable and , one has

If and are continuous and , then and have a common fixed point, that is, there exists such that .

Corollary 2.7. Let be a partially ordered set and suppose that there exists a partial metric on such that is complete. Suppose that be weakly increasing mappings with respect to such that for any comparable and , one has

If satisfies property and , then and have a common fixed point, that is, there exists such that .

Corollary 2.8. Let be a partially ordered set and suppose that there exists a partial metric on such that is complete. Suppose that are mapping such that for any comparable and in , one has

where is a continuous function such that if and only if and is a continuous nondecreasing function such that if and only if . Also, suppose that for all . If is continuous, then has a fixed point, that is, there exists such that .

Proof. It follows from Theorem 2.4 by taking and noting that and are weakly -contractive mappings.

Corollary 2.9. Let be a partially ordered set and suppose that there exists a partial metric on such that is complete. Suppose that be mapping such that for any comparable and in , one has

where is a continuous function such that if and only if and is a continuous nondecreasing function such that if and only if . Also, suppose that for all . If satisfies property , then has a fixed point, that is, there exists such that .

Proof. It follows from Theorem 2.5 by taking and noting that and are weakly -contractive mappings.

We present the following common fixed points of integral type as an application of our results.

Denote by the set of functions satisfying the following hypotheses:

(1) is a Lebesgue integrable function on each compact subset of , (2)for every , we have .

It is easy to see that the mapping

We have the following result.

Corollary 2.10. Let be an partially ordered metric space. Suppose that are weakly increasing mappings with respect to and for any comparable and , one has

If and are continuous, then and have a common fixed point.

Proof. It follows from Theorem 2.4 by defining via and via

Corollary 2.11. Let be an partially ordered metric space. Suppose that are weakly increasing mappings with respect to and for any comparable and , one has

If satisfies property , then and have a common fixed point.

Proof. It follows from Theorem 2.5 by defining via and via

Example 2.12. Let . Define the partial metric space on by and the relation on by if and only if . Also, define the mappings by , and the functions by and . Then one has the following.

(1) is a complete ordered partial metric space. (2) and are continuous. (3) and are weakly increasing. (4)For any two comparable elements and in , we have

Proof. The proof of (1) and (2) is clear. To prove (3), given . Since

we have . Similarly, we can show that . Thus and are weakly increasing mappings. To prove (4), given two comparable elements and in . Without loss of generality, we assume that , that is, . So,

From (3) and (4), we conclude that and are weakly -contractive mappings. Note that Example 2.12 satisfies all the hypotheses of Theorem 2.4. Thus and have a common fixed point. Here is a common fixed point of and .

3. Application

In this section, we apply our results to prove an existence solution of the following integral equation:

Let be the space of all continuous functions defined on . Define a partial metric space:

by

Also, define a relation on by

Then is an ordered complete partial metric space.

Now, we will give an existence theorem for the solution of the integral equation (3.1).

Theorem 3.1. Suppose the following hypotheses hold.

(i) for all . (ii)There exists a continuous function such that

(iii)There exists such that

(iv) for all .

Then the integral equation (3.1) has a solution .

Proof. Define the operators:

by

Given . Then from (i), we have

Thus and are weakly increasing mappings with respect to . Again, for with , we have

By (iv), we have

By using (iii), we get

Hence

Take . Since , we have . Also, we have

Moreover if is a nondecreasing sequence in such that as , then for all (see [25]). Thus, all the required hypotheses of Corollary 2.7 are satisfied. Therefore, has a fixed point and hence the integral equation (3.1) has a solution.

Acknowledgments

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

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