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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 230708, 9 pages
http://dx.doi.org/10.1155/2014/230708
Research Article

Fixed Points of Multivalued Contractive Mappings in Partial Metric Spaces

1Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2Department of Mathematics and Applied Mathematics, University Pretoria, Lynnwood Road, Pretoria 0002, South Africa
3Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan
4Faculty of Applied Sciences, University Politehnica of Bucharest, 313 Splaiul Independentei, 060042 Bucharest, Romania

Received 23 November 2013; Accepted 7 January 2014; Published 20 February 2014

Academic Editor: Hassen Aydi

Copyright © 2014 Abdul Rahim Khan 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

The aim of this paper is to present fixed point results of multivalued mappings in the framework of partial metric spaces. Some examples are presented to support the results proved herein. Our results generalize and extend various results in the existing literature. As an application of our main result, the existence and uniqueness of bounded solution of functional equations arising in dynamic programming are established.

1. Introduction

In 1922, Banach proved his celebrated contraction principle [1]. As it is well known, there have been published remarkable research articles about fixed points theory for different classes of contractive mappings, on some spaces such as quasi-metric spaces [2], cone metric spaces [3], convex metric spaces [4], partially ordered metric spaces [510], -metric spaces [1115], partial metric spaces [16, 17], quasi-partial metric spaces [18], fuzzy metric spaces [19], and Menger spaces [20]. Also, studies either on approximate fixed point or on qualitative aspects of numerical procedures for approximating fixed points are available in literature; please, see [4, 2123].

The concept of a partial metric space is introduced by Matthews [24], as a part of the study of denotational semantics of dataflow networks. He gave a modified version of the Banach contraction principle, more suitable in this context (see also [25, 26]). In fact, (complete) partial metric spaces constitute a suitable framework to model several distinguished examples of the theory of computation and also to model metric spaces via domain theory (see [24, 2731]).

It was shown that, in some cases, the results of fixed point in partial metric spaces can be obtained directly from their induced metric counterparts [3234]. However, some conclusions important for the application of partial metrics in information sciences cannot be obtained in this way. For example, if is a fixed point of map , then, by using the method from [32], we cannot conclude that . For further details, we refer the reader to [35, 36].

Recently, Aydi et al. [37] introduced the concept of a partial Hausdorff metric. They initiated study of fixed point theory for multivalued mappings on partial metric space using the partial Hausdorff metric and proved an analogue of the well-known Nadler fixed point theorem.

In this paper, we obtain several fixed point results of multivalued mappings in partial metric spaces. Our results extend, unify, and generalize the comparable results in [3841].

2. Preliminaries

In the sequel the letters ,  , and will denote the set of all real numbers, the set of all nonnegative real numbers, and the set of all positive integer numbers, respectively.

Consistent with [24, 42], the following definitions and results will be needed in the sequel.

Definition 1. Let be a nonempty set. A function is said to be a partial metric on if, for any , the following conditions hold:(p1) if and only if ;(p2); (p3); (p4).
The pair is called a partial metric space.

If , then (p1) and (p2) imply that . But the converse does not hold in general.

A trivial example of a partial metric space is the pair , where

Example 2 (see [24]). If , then defines a partial metric on .

For more examples of partial metric spaces, we refer to [17, 29, 31, 4345].

Each partial metric on generates a topology on , whose base is the family of open -balls , where , for all and .

Observe (see [24, p. 187]) that a sequence in a partial metric space converges to a point , with respect to , if and only if .

If is a partial metric on , then the function defines a metric on .

Furthermore, a sequence converges in to a point if and only if

Definition 3 (see [24]). Let be a partial metric space. (a)A sequence in is said to be a Cauchy sequence if exists and is finite.(b) is said to be complete if every Cauchy sequence in converges with respect to to a point such that . In this case, we say that the partial metric is complete.

Lemma 4 (see [24, 42]). Let be a partial metric space. Then, (i)a sequence in is a Cauchy sequence in if and only if it is a Cauchy sequence in metric space ;(ii)a partial metric space is complete if and only if the metric space is complete.

Let be a partial metric space. Let  and  () be the family of all nonempty and nonempty and closed (nonempty, closed, and bounded) subsets of the partial metric space . Note that here closedness is considered in ( is the topology induced by ) while boundedness is given as follows: is a bounded subset in if there exist and such that, for all , we have ; that is, .

For and , [37] defines It is easy to check that , where .

Remark 5 (see [42]). Let be a partial metric space and let be any nonempty set in . Then where denotes the closure of with respect to the partial metric . Note that is closed in if and only if .

Let be any nonempty set and let be a given mapping. For any fixed , a sequence in such that is called a -orbital sequence about . Collection of all such sequences will be denoted by . Further a point is called a fixed point of if and only if [46]. The set of all fixed points of multivalued mapping is denoted by .

We have the following partial metric space version of Definition 1.8 in [47].

Definition 6. Let be any nonempty set, , and let . A mapping is said to be -orbitally lower semicontinuous at with respect to if and converges to implying that .

3. Fixed Points of Multivalued Mapping

In this section, we obtain several fixed point results of multivalued mappings satisfying more general contractive conditions than those of Cho et al. [47], [48], and Feng and Liu [38] in the frame work of partial metric spaces.

Theorem 7. Let be a partial metric space and let be a multivalued mapping. Suppose that there exist functions and such that where . If, for any , there exists satisfying then, for each , there exists in such that is Cauchy sequence. Further, if converges to and the function is -orbitally lower semicontinuous at with respect to , then is a fixed point of . If , then .

Proof. Let be a given point in . Since , we can choose such that Then, we have We define by for all . Then by definition of and , it follows that for all and for all . From (11), we have Continuing this way, we can obtain a sequence in such that which satisfies
Now by (13) and (14), we have As , we have for all . Thus, is (strictly) decreasing sequence of positive real numbers. Consequently, there exists such that converges to . Since for all , it follows from (13) that , and hence we have On taking upper limit as on both sides of (17), we have which implies that ; that is, .
Now we show that is a Cauchy sequence. Put . We can choose a real number such that there exists a positive integer such that for all . Thus, from (15), we have for all . So for all with , Also, from (17) and (19), we have for all . Now Thus for all . Using , we get that is a Cauchy sequence in the metric space . By Lemma 4, is a Cauchy sequence in .
Next, we assume that there exists an element in such that and the function is -orbitally lower semicontinuous at with respect to . Then it follows that Thus . Since is closed, .
Now, if , then from (9) we have where . Hence .

Example 8. Let and let be the partial metric defined by Define the mapping by Note that is closed and bounded for all in the partial metric space . Define and , where as Clearly, for all and for all . We will show that for all , (8) and (9) are satisfied. For this, we consider the following cases.(i)If , then there exists such that (ii)When , then there exists such that (iii)For there exists such that
Thus, all the conditions of Theorem 7 are satisfied. Moreover, and .

The next example shows that one cannot derive the conclusion of Theorem 7 using metric induced by a partial metric.

Example 9. Let be the partial metric space with . Define the mapping by Note that is closed and bounded for all in the partial metric space . Define and by where . Clearly, for all and for all . We will show that, for all , (8) and (9) are satisfied. For this, we consider the following cases.
When , then, for , (8) and (9) are satisfied.
For , take such that In case , taking we have Hence for all , there exists such that (8) and (9) are satisfied. Thus, all the conditions of Theorem 7 are satisfied. Moreover, and .
On the other hand, we have . If we take , then there does not exist any such that (8) and (9) are satisfied.

Hence we are justified in formulating the following result.

Theorem 10. Let be a partial metric space and let be a mapping. Suppose that there exist functions , such that where . If for any there exists satisfying then, for each , there exists in such that is a Cauchy sequence. Further, if converges to and the function is -orbitally lower semicontinuous at with respect to , then .

Proof. Let be a given point in . As in the proof of Theorem 7, we can obtain a sequence in such that , which satisfies From for all , we have As so we have for all , and it follows that is (strictly) decreasing sequence of positive real numbers. Consequently, there exists such that converges to . On taking upper limit as on both sides of (40), we have which implies that ; that is, .
Now we show that is a Cauchy sequence. Since , for all , it follows from (38) that , and hence we have Thus the sequence is bounded.
Following arguments similar to those in the proof of Theorem 7, we obtain that is a Cauchy sequence in and .

Now, in the next two results, we consider further generalization of the conditions (8), (9), and (37).

Theorem 11. Let be a partial metric space and let be a multivalued mapping. Suppose that there exist functions ,   such that is nondecreasing and subadditive and they satisfy If for any there exists satisfying then, for each , there exists in such that is a Cauchy sequence. Further, if converges to and the function is -orbitally lower semicontinuous at with respect to , then .

Proof. Let be a given point in . Since , we can choose such that As before by continuing this way, we can obtain a sequence in such that which satisfies By (47), we have for all . We define by for all . Then by the definitions of and , it follows that for all , and for all . From (48), we have As so we have for all . Also, By nondecreasing , it follows that for all . Thus, and are (strictly) decreasing sequences of positive real numbers. Consequently, there exist such that converges to and converges to . Now, by taking upper limit as in (49), we have which implies ; that is, .
Now we show that is a Cauchy sequence. There exists a real number such that, for a positive integer with , we have Clearly, for all . Now Thus for all . This implies that is a Cauchy sequence in the metric space . By Lemma 4, is a Cauchy sequence in . Using arguments as in the proof of Theorem 7, we can show that the limit point of is a fixed point of .

In same way, we can prove the following result.

Theorem 12. Let be a partial metric space and be a multivalued mapping. Suppose that there exist ,   such that is nondecreasing and subadditive and they satisfy If for any there exists satisfying then, for each , there exists in such that is a Cauchy sequence. Further, if converges to and the function is -orbitally lower semicontinuous at with respect to , then .

The following result generalizes and extends Theorem 3.1 in [38] to partial metric spaces.

Corollary 13. Let be a partial metric space, and let be a multivalued mapping. If there exist constants with such that, for any , there exists satisfying then, for each , there exists in such that is a Cauchy sequence. Further, if converges to and the function is -orbitally lower semicontinuous at with respect to , then .

The following corollary is an extension of [49] and in view of Corollary 3.2 in [38] is a special case of Theorem 10.

Corollary 14. Let be a partial metric space, and let be a multivalued mapping. If there exist constants such that, for any , there exists satisfying then, for each , there exists in such that is a Cauchy sequence. Further, if converges to and the function is -orbitally lower semicontinuous at with respect to , then .

Corollary 15. Let be a partial metric space and let be a self-mapping. Suppose that, there exist ,   such that is nondecreasing and subadditive and they satisfy If for any there exists satisfying then, for each , there exists in such that is a Cauchy sequence. Further, if converges to and the function is -orbitally lower semicontinuous at with respect to , then .

We remark that(1)if is a complete partial metric space in Theorems 7 and 10, is a multivalued mapping satisfying all the conditions of Theorems 7 and 10, and the function is lower semicontinuous on , then there exists in such that .(2)Theorems 7, 10, and 11 extend and generalize Theorems 2.1 and 2.4 in [47], Theorem 3.1 in [38], and Theorems 2.3, 2.4, 2.7 and 2.8 in [41] to partial metric spaces.

4. Application

Let and be the Banach spaces with and . Suppose that If we consider and as the state and decision spaces, respectively, then the problem of dynamic programming reduces to the problem of solving the functional equation Equation (65) can be reformulated as where .

For more on problems of dynamic programming involving such functional equations, we refer the reader to [25, 5052].

We study the existence and uniqueness of the bounded solution of the functional equation (66) arising in dynamic programming in the setup of partial metric spaces.

Let denote the set of all bounded real valued functions on . For an arbitrary , define . Then is a Banach space endowed with the metric defined as .

Now consider where . Then is a partial metric on (see also [53]).

We need the following two conditions.(A1) and are bounded.(A2)For , , and , define

Moreover, assume that there exist mappings and such that is nondecreasing and subadditive and they satisfy

Also for any , there exists such that hold for all .

Theorem 16. Assume that conditions (A1) and (A2) are satisfied. If is a closed convex subspace of , then the functional equation (66) has a unique and bounded solution.

Proof. Note that is a complete partial metric space. By (A1), is a self-map of . By (68) in (A2) it follows that for any , there exists such that hold for all . Now, we have Also Note that the above inequalities are true for all , and, for any , there exists such that Therefore by Corollary 15, the map has a fixed point ; that is, is a unique and bounded solution of functional equation (66).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors A. R. Khan and M. Abbas are grateful to King Fahd University of Petroleum and Minerals for supporting Research Project IN 121023.

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