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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 313675, 11 pages
http://dx.doi.org/10.1155/2012/313675
Research Article

Fixed Point of -Hardy-Rogers Contractive Mappings in Partially Ordered Partial Metric Spaces

1Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan
2Institut Supérieur d'Informatique et de Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1, 4011 Hammam Sousse, Tunisia
3Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11 120 Beograd, Serbia

Received 20 March 2012; Accepted 29 August 2012

Academic Editor: Brigitte Forster-Heinlein

Copyright © 2012 Mujahid Abbas 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

We prove some fixed point theorems for a T-Hardy-Rogers contraction in the setting of partially ordered partial metric spaces. We apply our results to study periodic point problems for such mappings. We also provide examples to illustrate the results presented herein.

1. Introduction and Preliminaries

The notion of a partial metric space was introduced by Matthews in [1]. In partial metric spaces, the distance of a point in the self may not be zero. After the definition of a partial metric space, Matthews proved the partial metric version of Banach fixed point theorem. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle, more suitable in this context [1]. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions (e.g., [221], [22]). Existence of fixed points in partially ordered metric spaces has been initiated in 2004 by Ran and Reurings [23]. Subsequently, several interesting and valuable results have appeared in this direction [14]. The aim of this paper is to study the necessary conditions for existence of fixed point of mapping satisfying -Hardy-Rogers conditions in the framework of partially ordered partial metric spaces. Our results extend and strengthen various known results [8, 24]. In the sequel, the letters , , and will denote the set of real numbers, the set of nonnegative real numbers, and the set of nonnegative integer numbers, respectively. The usual order on (resp., on will be indistinctly denoted by or by .

Consistent with [1, 8] (see [2529]) the following definitions and results will be needed in the sequel.

Definition 1.1 (see [1]). A partial metric on a nonempty set is a mapping such that for all ,,, , .
A partial metric space is a pair such that is a nonempty set and is a partial metric on . If , then and imply that . But converse does not hold always. A trivial example of a partial metric space is the pair , where is defined as . Each partial metric on generates a topology on which has as a base the family open -balls , where .

On a partial metric space the concepts of convergence, Cauchy sequence, completeness, and continuity are defined as follows.

Definition 1.2 (see [1]). Let be a partial metric space and let be a sequence in . Then (i) converges to a point if and only if (we may still write this as or ); (ii) is called a Cauchy sequence if there exists (and is finite) .

Definition 1.3 (see [1]). A partial metric space is said to be complete if every Cauchy sequence in converges to a point , such that . If is a partial metric on , then the function given by is a metric on .

Lemma 1.4 (see [1, 20]). Let be a partial metric space. Then(a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space ;(b) is complete if and only if the metric space is complete. Furthermore, if and only if

Remark 1.5. (1) (see [19]) Clearly, a limit of a sequence in a partial metric space does not need to be unique. Moreover, the function does not need to be continuous in the sense that and implies . For example, if and for , then for for each and so, for example, and when .

(2) (see [7]) However, if then for all .

Definition 1.6 (see [30]). Suppose that is a partial metric space. Denote its topology. We say is continuous if both and are continuous.

Remark 1.7. It is worth to notice that the notions -continuous and -continuous of any function in the context of partial metric spaces are incomparable, in general. Indeed, if ,  ,  ,  , and for all and , then is a -continuous and -discontinuous at point ; while is a -discontinuous and -continuous at .

According to [31], we state the following definition.

Definition 1.8. Let be a partial metric space. A mapping is said to be(i)sequentially convergent if for any sequence in such that is convergent in implies that is convergent in ,(ii)subsequentially convergent if for any sequence in such that is convergent in implies that has a convergent subsequence in .

Consistent with [24, 31] we define a -Hardy-Rogers contraction in the framework of partial metric spaces.

Definition 1.9. Let be a partial metric space and be two mappings. A mapping is said to be a -Hardy-Rogers contraction if there exist , with such that for all

Putting and , (resp., and ) in the previous definition, then the inequality (1.2) is said a -Kannan (resp., -Chatterjea) type contraction. Also, if and , (1.2) is said the -Reich type contraction.

Definition 1.10. Let be a nonempty set. Then is called a partially ordered partial metric space if and only if (i) is a partial metric on and (ii) is a partial order on .
Let be a partial metric space endowed with a partial order and let be a given mapping. We define sets by

A point is called a fixed point of mapping if . The set of all fixed points of the mapping is denoted by .

2. Fixed Point Results

In this section, we obtain fixed point results for a mapping satisfying a -Hardy-Rogers contractive condition defined on a partially ordered partial metric space which is complete.

We start with the following result.

Theorem 2.1. Let be an partially ordered partial metric space which is complete. Let be a continuous, injective mapping and a nondecreasing -Hardy-Rogers contraction for all . If there exists with , and one of the following two conditions is satisfied(a) is a continuous self-map on ;(b)for any nondecreasing sequence in with it follows for all ;
then provided that is subsequentially or sequentially convergent. Moreover, has a unique fixed point if .

Proof. As is nondecreasing, therefore by given assumption, we have Define a sequence in with and so for . Since therefore by replacing by and by in (1.2), we have that is, Similarly, replacing by and by in (1.2), we obtain Summing (2.3) and (2.4), we obtain , where /. Obviously . Therefore, for all , Now, for any with , we have which implies that as . Hence is a Cauchy sequence in and in . Since is complete, therefore from Lemma 1.4, is a complete metric space. Hence converges to some with respect to the metric , that is, or equivalently, Suppose that is subsequentially convergent, therefore convergence of in implies that has a convergent subsequence in . So for some . As is continuous, so (2.9) and Definition 1.6. imply that . From (2.7) and by the uniqueness of the limit in metric space , we obtain . Consequently, If is a continuous self-map on , then and as . Since as , we obtain that . As is injective, so we have . If is not continuous then by given assumption we have for all . Thus for a subsequence of we have and . Now, On taking limit as and applying Remark 1.5. (2) we get which implies that , and so . Now injectivity of gives . Following similar arguments to those given above, the result holds when is sequentially convergent.
Suppose that . Let be a fixed point of . As , therefore . From (1.2), we have and hence , which further implies that as is injective.

Example 2.2. Let be endowed with usual order and let be the complete partial metric on defined by for all . Let be defined by and . Note that . For any , we have Therefore, is a -Hardly-Rogers contraction with ,   . Obviously, is continuous and sequentially convergent. Thus, all the conditions of Theorem 2.1 are satisfied. Moreover, is the unique fixed point of .

Example 2.3. Let be endowed with usual order and let be a partial metric on defined by for all . Define by and. Note that . For any , we have Therefore, is a -Hardly-Rogers contraction with . Also, is continuous and sequentially convergent. Thus all the conditions of Theorem 2.1 are satisfied. Moreover, is the unique fixed point of .

Taking in (1.2) and Theorem 2.1, we get the Hardy-Rogers type [32] (and so the Kannan, Chatterjea, and Reich) fixed point theorem on partially ordered partial metric spaces.

Corollary 2.4. Let be a partially ordered partial metric space which is complete. Let be a nondecreasing mapping such that for all , we have where , with . If there exists with , and one of the following two conditions is satisfied.(a) is a continuous self map on ;(b)for any nondecreasing sequence in with it follows for all ; then . Moreover, has a unique fixed point if .

Remark 2.5. Corollary 2.4 corresponds to Theorem 2 of Altun et al. [8] in partially ordered partial metric spaces. For particular choices of the coefficients in Theorem 2.1, we obtain the -Kannan, -Chatterjea, and -Reich type fixed point theorems. Also, Theorem 2.1 is an extension of Theorem 2.1 of Filipović et al. [24] from the cone metric spaces to partial metric spaces.

3. Periodic Point Results

Let . If the map satisfies for each , then it is said to have the property , for more details see [33].

Definition 3.1. Let be a partially ordered set. A mapping is called (1) a dominating map on if for each in and (2) a dominated map on if for each in .

Example 3.2. Let be endowed with usual ordering. Let defined by , then for all . Thus is a dominating map.

Example 3.3. Let be endowed with usual ordering. Let defined by for and let for , for any , then for all ,    that is is the dominating map. Note that if is a dominating or a dominated mapping.
We have the following result.

Theorem 3.4. Let be a partially ordered partial metric space which is complete. Let be an injective mapping and a nondecreasing such that for all , we have for some and for all , . Then has the property provided that is nonempty and is a dominating map on .

Proof. Let for some . Now we show that . Since is dominating on , therefore which further implies that as is nondecreasing. Hence . Now by using (3.1), we have Repeating the above process, we get Taking limit as , we obtain and . As is injective, so , that is, .

Theorem 3.5. Let be a partially ordered partial metric space which is complete. let be mappings satisfy the condition of Theorem 2.1. If is dominating on , then has the property .

Proof. From Theorem 2.1, . We will prove that (3.1) is satisfied for all . Indeed, is a dominating map so that and also is nondecreasing so that and hence . Now from (1.2), that is, Again by using (1.2), we have which implies that Summing (3.5) and (3.7) implies ,/. Obviously, . By Theorem 3.4, has the property .

Acknowledgment

S. Radenović is thankful to The Ministry of Science and Technology Development of Serbia.

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