Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 126205, 11 pages

http://dx.doi.org/10.1155/2011/126205

## Some Fixed Point Theorems in Ordered -Metric Spaces and Applications

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

Received 9 January 2011; Revised 22 March 2011; Accepted 19 April 2011

Academic Editor: D.Β Anderson

Copyright Β© 2011 W. 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 study a number of fixed point results for the two weakly increasing mappings and with respect to partial ordering relation in generalized metric spaces. An application to integral equation is given.

#### 1. Introduction

The existence of fixed points in partially ordered sets has been at the center of active research. In fact, the existence of fixed point in partially ordered sets has been investigated in [1]. Moreover, Ran and Reurings [1] applied their results to matrix equations. Some generalizations of the results of [1] are given in [2β6]. In [6], OβRegan and PetruΕel gave some existence results for the Fredholm and Volterra type.

The notion of -metric space was introduced by Mustafa and Sims [7] as a generalization of the notion of metric spaces. Mustafa et al. studied many fixed point results in -metric space [8β10] (also see [11β15]). In fact the study of common fixed points of mappings satisfying certain contractive conditions has been at the center of strong research activity. The following definition is introduced by Mustafa and Sims [7].

*Definition 1.1 (see [7]). *Let be a nonempty set and let be a function satisfying the following properties: if , , for all with , , for all with , , symmetry in all three variables, , for all . Then the function is called a generalized metric, or, more specifically, a -metric on , and the pair is called a -metric space.

*Definition 1.2 (see [7]). *Let be a -metric space, and let be a sequence of points of , a point is said to be the limit of the sequence , if , and we say that the sequence is -convergent to or -converges to .

Thus, in a -metric space if for any , there exists such that for all .

Proposition 1.3 (see [7]). *Let be a -metric space. Then the following are equivalent: *(1)* is -convergent to ; *(2)* as ; *(3)* as ; *(4)* as .*

*Definition 1.4 (see [7]). *Let be a -metric space, a sequence is called -Cauchy if for every , there is such that , for all ; that is as .

Proposition 1.5 (see [7]). *Let be a -metric space. Then the following are equivalent: *(1)*the sequence is -Cauchy;*(2)*for every , there is such that , for all .*

*Definition 1.6 (see [7]). *Let and be -metric spaces, and let be a function. Then is said to be -continuous at a point if and only if for every , there is such that and implies . A function is -continuous at if and only if it is -continuous at all .

Proposition 1.7 (see [7]). *Let be a -metric space. Then the function is jointly continuous in all three of its variables.*

Every -metric on will define a metric on by For a symmetric -metric space, However, if is not symmetric, then the following inequality holds:

The following are examples of -metric spaces.

*Example 1.8 (see [7]). *Let be the usual metric space. Define by
for all . Then it is clear that is a -metric space.

*Example 1.9 (see [7]). *Let . Define on by
and extend to by using the symmetry in the variables. Then it is clear that is a -metric space.

*Definition 1.10 (see [7]). *A -metric space is called -complete if every -Cauchy sequence in is -convergent in .

The notion of weakly increasing mappings was introduced in by Altun and Simsek [16].

*Definition 1.11 (see [16]). *Let be a partially ordered set. Two mappings are said to be weakly increasing if and , for all .

Two weakly increasing mappings need not be nondecreasing.

*Example 1.12 (see [16]). *Let , endowed with the usual ordering. Let defined by
Then and are weakly increasing mappings. Note that and are not nondecreasing.

The aim of this paper is to study a number of fixed point results for two weakly increasing mappings and with respect to partial ordering relation () in a generalized metric space.

#### 2. Main Results

Theorem 2.1. *Let be a partially ordered set and suppose that there exists -metric in such that is -complete. Let be two weakly increasing mappings with respect to . Suppose there exist nonnegative real numbers , , and with such that
**
for all comparative . If or is continuous, then and have a common fixed point .*

* Proof. * By inequality (2.2), we have
If is a symmetric -metric space, then by adding inequalities (2.1) and (2.3), we obtain
which further implies that
for all with and the fixed point of and follows from [2].

Now if is not a symmetric -metric space. Then by the definition of metric and inequalities (2.1) and (2.3), we obtain
for all . Here, the contractivity factor may not be less than 1.

Therefore metric gives no information. In this case, for given , choose such that . Again choose such that . Also, we choose such that . Continuing as above process, we can construct a sequence in such that , and , . Since and are weakly increasing with respect to , we have
Thus from (2.1), we have
By , we have
Also, we have
By , we get
Let
Then by (2.9) and (2.11), we have
Thus, if , we get for each . Hence for each . Therefore is -Cauchy. So we may assume that . Let with . By axiom of the definition of -metric space, we have
By (2.13), we get
On taking limit , we have
So we conclude that is a Cauchy sequence in . Since is -complete, then it yields that and hence any subsequence of converges to some . So that, the subsequences and converge to . First suppose that is -continuous. Since converges to , we get converges . By the uniqueness of limit we get . Claim: .

Since , by inequality (2.1), we have
Since , we get . Hence . If is -continuous, by similar argument as above we show that and have a common fixed point.

Theorem 2.2. *Let be a partially ordered set and suppose that there exists -metric in such that is -complete. Let be two weakly increasing mappings with respect to . Suppose there exist nonnegative real numbers , , and with such that
**
for all comparative . Assume that has the following property: **If is an increasing sequence converges to in , then for all .**Then and have a common fixed point .*

* Proof. * As in the proof of Theorem 3.1, we construct an increasing sequence in such that and . Also, we can show is -Cauchy. Since is -complete, there is such that is converges to . Thus and converge to . Since satisfies property , we get that , for all . Thus and are comparative. Hence by inequality (2.1), we have
On letting , we get
Since , we get . Hence . By similar argument, we may show that .

Corollary 2.3. *Let be a partially ordered set, and suppose that is a -complete metric space. Let be a continuous mapping such that , for all . Suppose there exist nonnegative real numbers , and with such that
**
for all comparative . Then has a fixed point .*

* Proof. *It follows from Theorem 2.1 by taking .

Corollary 2.4. *Let be a partially ordered set and suppose that there exists -metric in such that is -complete. Let be a mapping such that for all . Suppose there exist nonnegative real numbers , and with such that
**
for all comparative . Assume that has the following property: **If is an increasing sequence converges to in , then for all .**Then has fixed point .*

* Proof. * It follows from Theorem 2.2 by taking .

#### 3. Application

Consider the integral equation: where . The aim of this section is to give an existence theorem for a solution of the above integral equation using Corollary 2.4 This section is related to those [16β19].

Let be the set of all continuous functions defined on . Define by Then is a -complete metric space. Define an ordered relation on by Then is a partially ordered set. The purpose of this section is to give an existence theorem for solution of integral equation on (3.1). This section is inspired in [17β19].

Theorem 3.1. *Suppose the following hypotheses hold: *(1)* and are continuous, *(2)*for each , one has
*(3)*there exists a continuous function such that
for each comparable and each , *(4)* for some . **Then the integral equation (3.1) has a solution .*

* Proof. * Define by
Now, we have
Thus, we have , for all .

For with , we have
By using hypotheses (4), there is such that
Thus, we have . Thus all the required hypotheses of Corollary 2.4 are satisfied. Thus there exist a solution of the integral equation (3.1).

#### Acknowledgments

The author would like to thank the editor of the paper and the referees for their precise remarks to correct and improve the paper. Also, the author would like to thank the hashemite university for the financial assistant of this paper.

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