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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 181598, 6 pages

http://dx.doi.org/10.1155/2014/181598
Research Article

Best Proximity Point for - -Proximal Contractive Multimaps

1Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology H-12, Islamabad 44000, Pakistan

2Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan

3Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 27 April 2014; Accepted 21 June 2014; Published 24 July 2014

Academic Editor: Poom Kumam

Copyright © 2014 Muhammad Usman Ali 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 extend the notions of - -proximal contraction and -proximal admissibility to multivalued maps and then using these notions we obtain some best proximity point theorems for multivalued mappings. Our results extend some recent results by Jleli and those contained therein. Some examples are constructed to show the generality of our results.

1. Introduction and Preliminaries

Samet et al. [1] introduced the notion of - -contractive type mappings and proved some fixed point theorems for such mappings in the frame work of complete metric spaces. Karapınar and Samet [2] generalized - -contractive type mappings and obtained some fixed point theorems for generalized - -contractive type mapping. Some interesting multivalued generalizations of - -contractive type mappings are available in [312]. Recently, Jleli and Samet [13] introduced the notion of - -proximal contractive type mappings and proved some best proximity point theorems. Many authors obtained best proximity point theorems in different setting; see, for example, [1335]. Abkar and Gbeleh [16] and Al-Thagafi and Shahzad [18, 20] investigated best proximity points for multivalued mappings. The purpose of this paper is to extend the results of Jleli and Samet [13] for nonself multivalued mappings. To demonstrate generality of our main result we have constructed some examples.

Let be a metric space. For , we use the following notations: , , , , is the set of all nonempty subsets of , is the set of all nonempty closed subsets of , and is the set of all nonempty compact subsets of . For every , let Such a map is called the generalized Hausdorff metric induced by . A point is said to be the best proximity point of a mapping if . When , the best proximity point reduces to fixed point of the mapping .

Definition 1 (see [28]). Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the weak -property if and only if, for any and ,

Example 2. Let , endowed with the usual metric . Let and . Then for we have Also, . Thus, the pair satisfies weak -property.

Definition 3 (see [13]). Let and . We say that is an -proximal admissible if where .

Example 4. Let , endowed with the usual metric . Let be any fixed positive real number, and . Define by Define by Let , , , and be arbitrary points from satisfying It follows from (8) that . Further, from (9), and , which implies that . Hence, . Therefore, is an -proximal admissible map.

Let denote the set of all functions : satisfying the following properties:(a) is monotone nondecreasing;(b) for each .

Definition 5 (see [13]). A nonself mapping is said to be an - -proximal contraction, if where and .

Example 6. Let us consider Example 4 again with for each . Then it is easy to see that, for each , we have Thus, is an - -proximal contraction.

The following are main results of [13].

Theorem 7 (see [13], Theorem 3.1). Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let and . Suppose that be a mappings satisfying the following conditions:  (i) and satisfies the -property; (ii) is an -proximal admissible; (iii)there exist elements such that  (iv) is a continuous - -proximal contraction.Then there exists an element such that .

(C) If is a sequence in such that for all and as , then there exists a subsequence of such that for all .

Theorem 8 (see [13], Theorem 3.2). Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let and . Suppose that is a mapping satisfying the following conditions:  (i) and satisfies the -property; (ii) is an -proximal admissible; (iii)there exist elements and such that  (iv)property (C) holds and is an - -proximal contraction.Then there exists an element such that .

Definition 9 (see [16]). An element is said to be the best proximity point of a multivalued nonself mapping , if .

2. Main Result

We start this section by introducing following definition.

Definition 10. Let and be two nonempty subsets of a metric space . A mapping is called -proximal admissible if there exists a mapping such that where , , and .

Definition 11. Let and be two nonempty subsets of a metric space . A mapping is said to be an - -proximal contraction, if there exist two functions and such that

Lemma 12 (see [5]). Let be a metric space and . Then for each with and , there exists an element such that

Now we are in position to state and prove our first result.

Theorem 13. Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let and be a strictly increasing map. Suppose that is a mapping satisfying the following conditions:  (i) for each and satisfies the weak -property; (ii) is an -proximal admissible; (iii)there exist elements and such that  (iv) is a continuous - -proximal contraction.Then there exists an element such that .

Proof. From condition (iii), there exist elements and such that Assume that ; for otherwise is the best proximity point. From condition (iv), we have For , it follows from Lemma 12 that there exists such that From (19) and (20), we have As , there exists such that for otherwise is the best proximity point. As satisfies the weak -property, from (18) and (22), we have From (21) and (23), we have Since is strictly increasing, we have . Put . Also, we have , , and . Since is an -proximal admissible, then . Thus we have Assume that ; for otherwise is the best proximity point. From condition (iv), we have For , it follows from Lemma 12 that there exists such that From (26) and (27), we have As , there exists such that for otherwise is the best proximity point. As satisfies the weak -property, from (25) and (29), we have From (28) and (30), we have Since is strictly increasing, we have . Put . Also, we have , , and . Since is an -proximal admissible then . Thus, we have Continuing in the same way, we get sequences in and in , where for each such that As , there exists such that Since satisfies the weak -property form (33) and (35), we have . Then from (34), we have For we have Hence, is a Cauchy sequence in . Similarly, we show that is a Cauchy sequence in . Since and are closed subsets of a complete metric space, there exist in and in such that and as . By (35), we conclude that as . Since is continuous and , we have . Hence, . Therefore, is the best proximity point of the mapping .

Theorem 14. Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let and be mappings satisfying the following conditions:  (i) for each and satisfies the weak -property; (ii) is an -proximal admissible; (iii)there exist elements and such that  (iv) is a continuous - -proximal contraction.Then there exists an element such that .

Theorem 15. Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let and be a strictly increasing map. Suppose that is a mapping satisfying the following conditions:  (i) for each and satisfies the weak -property; (ii) is an -proximal admissible; (iii)there exist elements and such that  (iv)property (C) holds and is an - -proximal contraction.Then there exists an element such that .

Proof. Following the proof of Theorem 13, there exist Cauchy sequences in and in such that (33) holds and and as . From the condition (C), there exists a subsequence of such that for all . Since is an - -proximal contraction, we have Letting in the above inequality, we get . By continuity of the metric , we have Since , , and , then . Hence, . Therefore, is the best proximity point of the mapping .

Theorem 16. Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let and be mappings satisfying the following conditions:  (i) for each and satisfies the weak -property; (ii) is an -proximal admissible; (iii)there exist elements and such that  (iv)property (C) holds and is an - -proximal contraction.Then there exists an element such that .

Example 17. Let be endowed with the usual metric . Suppose that and . Define by and by Let for all . Notice that , , and for each . Also, the pair satisfies the weak -property. Let ; then . Consider , , and such that and . Then we have . Hence, is an -proximal admissible map. For and in , we have such that and . If , then we have for otherwise Hence, is an - -proximal contraction. Moreover, if is a sequence in such that for all and as , then there exists a subsequence of such that for all . Therefore, all the conditions of Theorem 15 hold and has the best proximity point.

Example 18. Let be endowed with the usual metric . Let be any fixed real number, and . Define by and by Let for all . Notice that , , and for each . If with either or or both are nonzero, we have for otherwise For and in , we have such that and . Furthermore, it is easy to see that remaining conditions of Theorem 13 also hold. Thus, has the best proximity point.

3. Consequences

From results of previous section, we immediately obtain the following results.

Corollary 19. Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let and be mappings satisfying the following conditions:  (i) and satisfies the weak -property; (ii) is an -proximal admissible; (iii)there exist elements and in such that  (iv) is a continuous - -proximal contraction.Then there exists an element such that .

Corollary 20. Let and be two nonempty closed subsets of a complete metric space such that is nonempty. Let and be mappings satisfying the following conditions:  (i) and satisfies the weak -property; (ii) is an -proximal admissible; (iii)there exist elements and in such that  (iv)property (C) holds and is an - -proximal contraction.Then there exists an element such that .

Remark 21. Note that Corollaries 19 and 20 generalize Theorems 7 and 8 in Section 1, respectively.

Conflict of Interests

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

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The third author, therefore, acknowledges with thanks DSR for financial support. The authors would like to thank the anonymous reviewers for their valuable comments.

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