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Abstract and Applied Analysis

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

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

## Best Proximity Point for --Proximal Contractive Multimaps

^{1}Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology H-12, Islamabad 44000, Pakistan^{2}Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan^{3}Department 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 [3–12]. 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, [13–35]. 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. * 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.

#### References

- B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for
*α*-$\psi $-contractive type mappings,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 4, pp. 2154–2165, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. Karapınar and B. Samet, “Generalized
*α*-*ψ*contractive type mappings and related fixed point theorems with applications,”*Abstract and Applied Analysis*, vol. 2012, Article ID 793486, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - J. H. Asl, S. Rezapour, and N. Shahzad, “On fixed points of $\alpha $-$\psi $-contractive multifunctions,”
*Fixed Point Theory and Applications*, vol. 2012, article 212, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - B. Mohammadi, S. Rezapour, and N. Shahzad, “Some results on fixed points of (
*α*-*ψ*)-ciric generalized multifunctions,”*Fixed Point Theory and Applications*, vol. 2013, aricle 24, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. U. Ali and T. Kamran, “On $({\alpha}^{*},\psi )$-contractive multi-valued mappings,”
*Fixed Point Theory and Applications*, vol. 2013, article 137, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - P. Amiri, S. Rezapour, and N. Shahzad, “Fixed points of generalized (
*α*-*ψ*)-contractions,”*Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas*, 2013. View at Publisher · View at Google Scholar - G. Minak and I. Altun, “Some new generalizations of Mizoguchi-Takahashi type fixed point theorem,”
*Journal of Inequalities and Applications*, vol. 2013, article 493, 2013. View at Publisher · View at Google Scholar - M. U. Ali, T. Kamran, W. Sintunavarat, and P. Katchang, “Mizoguchi-Takahashi's fixed point theorem with $\alpha ,\eta $ functions,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 418798, 4 pages, 2013. View at MathSciNet - C. M. Chen and E. Karapinar, “Fixed point results for the $\alpha $-Meir-Keeler contraction on partial Hausdorff metric spaces,”
*Journal of Inequalities and Applications*, vol. 2013, no. 410, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. U. Ali, T. Kamran, and E. Karapınar, “(
*α*,$\psi $,$\xi $)-contractive multivalued mappings,”*Fixed Point Theory and Applications*, vol. 2014, article 7, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - M. U. Ali, T. Kamran, and E. Karapinar, “A new approach to ($\alpha $, $\psi $)-contractive nonself multivalued mappings,”
*Journal of Inequalities and Applications*, vol. 2014, article 71, 2014. - M. U. Ali, Q. Kiran, and N. Shahzad, “Fixed point theorems for multi-valued mappings involving $\alpha $-function,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 409467, 6 pages, 2014. View at Publisher · View at Google Scholar - M. Jleli and B. Samet, “Best proximity points for (
*α*-*ψ*)-proximal contractive type mappings and applications,”*Bulletin des Sciences Mathématiques*, vol. 137, no. 8, pp. 977–995, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Abkar and M. Gabeleh, “Best proximity points for asymptotic cyclic contraction mappings,”
*Nonlinear Analysis*, vol. 74, no. 18, pp. 7261–7268, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Abkar and M. Gabeleh, “Best proximity points for cyclic mappings in ordered metric spaces,”
*Journal of Optimization Theory and Applications*, vol. 151, no. 2, pp. 418–424, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Abkar and M. Gbeleh, “The existence of best proximity points for multivalued non-self mappings,”
*Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas*, vol. 107, no. 2, pp. 319–325, 2012. View at Publisher · View at Google Scholar - M. A. Alghamdi and N. Shahzad, “Best proximity point results in geodesic metric spaces,”
*Fixed Point Theory and Applications*, vol. 2012, article 234, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - M. A. Al-Thagafi and N. Shahzad, “Best proximity pairs and equilibrium pairs for Kakutani multimaps,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 3, pp. 1209–1216, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. A. Al-Thagafi and N. Shahzad, “Convergence and existence results for best proximity points,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 10, pp. 3665–3671, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. A. Al-Thagafi and N. Shahzad, “Best proximity sets and equilibrium pairs for a finite family of multimaps,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 457069, 10 pages, 2008. View at MathSciNet - M. Derafshpour, S. Rezapour, and N. Shahzad, “Best proximity points of cyclic $\phi $-contractions in ordered metric spaces,”
*Topological Methods in Nonlinear Analysis*, vol. 37, no. 1, pp. 193–202, 2011. View at MathSciNet · View at Scopus - C. di Bari, T. Suzuki, and C. Vetro, “Best proximity points for cyclic Meir-Keeler contractions,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 11, pp. 3790–3794, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. A. Eldred and P. Veeramani, “Existence and convergence of best proximity points,”
*Journal of Mathematical Analysis and Applications*, vol. 323, no. 2, pp. 1001–1006, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Markin and N. Shahzad, “Best proximity points for relatively $u$-continuous mappings in Banach and hyperconvex spaces,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 680186, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. Rezapour, M. Derafshpour, and N. Shahzad, “Best proximity points of cyclic $\varphi $-contractions on reflexive Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 946178, 7 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - S. S. Basha, N. Shahzad, and R. Jeyaraj, “Best proximity point theorems for reckoning optimal approximate solutions,”
*Fixed Point Theory and Applications*, vol. 2012, article 202, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - C. Vetro, “Best proximity points: convergence and existence theorems for $p$-cyclic mappings,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 73, no. 7, pp. 2283–2291, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Zhang, Y. Su, and Q. Cheng, “A note on “a best proximity point theorem for Geraghty-contractions”,”
*Fixed Point Theory and Applications*, vol. 2013, article 83, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - C. Mongkolkeha and P. Kumam, “Best proximity point theorems for generalized cyclic contractions in ordered metric spaces,”
*Journal of Optimization Theory and Applications*, vol. 155, no. 1, pp. 215–226, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. Sintunavarat and P. Kumam, “Coupled best proximity point theorem in metric spaces,”
*Fixed Point Theory and Applications*, vol. 2012, article 93, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. K. Nashine, C. Vetro, and P. Kumam, “Best proximity point theorems for rational proximal contractions,”
*Fixed Point Theory and Applications*, vol. 2013, article 95, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Y. J. Cho, A. Gupta, E. Karapinar, P. Kumam, and W. Sintunavarat, “Tripled best proximity point theorem in metric spaces,”
*Mathematical Inequalities & Applications*, vol. 16, no. 4, pp. 1197–1216, 2013. View at MathSciNet - C. Mongkolkeha, C. Kongban, and P. Kumam, “Existence and uniqueness of best proximity points for generalized almost contractions,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 813614, 11 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - P. Kumam, P. Salimi, and C. Vetro, “Best proximity point results for modified
*α*-proximal*C*-contraction mappings,”*Fixed Point Theory and Applications*, vol. 2014, article 99, 2014. - V. Pragadeeswarar, M. Marudai, P. Kumam, and K. Sitthithakerngkiet, “The existence and uniqueness of coupled best proximity point for proximally coupled contraction in a complete ordered metric space,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 274062, 7 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet