Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 648985, 5 pages doi:10.1155/2008/648985 Research Article Best Proximity Pairs for Upper Semicontinuous Set-Valued Maps in Hyperconvex Metric Spaces A. Amini-Harandi, 1 A. P. Farajzadeh, 2 D. O’Regan, 3 and R. P. Agarwal 4 1 Department of Mathematics, Faculty of Basic Sciences, University of Shahrekord, Shahrekord 88186-34141, Iran 2 Department of Mathematics, School of Science, Razi University, Kermanshah 67149, Iran 3 Department of Mathematics, College of Arts, Social Sciences and Celtic Studies, National University of Ireland, Galway, Ireland 4 Department of Mathematical Sciences, College of Science, Florida Institute of Technology, Melbourne, FL 32901, USA Correspondence should be addressed to A. Amini-Harandi, aminih a@yahoo.com Received 14 July 2008; Accepted 27 October 2008 Recommended by Nan-jing Huang A best proximity pair for a set-valued map F : A ⊸ B with respect to a map g : A → A is defined, and new existence theorems of best proximity pairs for upper semicontinuous set-valued maps with respect to a homeomorphism are proved in hyperconvex metric spaces. Copyright q 2008 A. Amini-Harandi 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. 1. Introduction and preliminaries Let M, d be a metric space and let Aand B be nonempty subsets of M. Let g : A → A and let F : A ⊸ B be a set-valued map. Now, g a,Fa is called a best proximity pair for F with respect to g if dg a,Fa  dA, B, where dA, B inf{da, b : a ∈ A, b ∈ B}. Best proximity pair theorems establish conditions under which the problem of minimizing the real-valued function x → dg x,Fx has a solution. In the setting of normed linear spaces, the best proximity pair problem has been studied by many authors for g  I , see 1–5. Very recently, Al-Thagafi and Shahzad 1 proved some existence theorems for a finite family of Kakutani set-valued maps in a normed space setting. In the present paper, our aim is to prove new results in hyperconvex metric spaces. In the rest of this section, we recall some definitions and theorems which are used in Section 2. Let X and Y be topological spaces with A ⊆ X and B ⊆ Y . Let F : X ⊸ Y be a set- valued map with nonempty values. The image of A under F is the set FA  x∈A Fx and the inverse image of B under F is F - B {x ∈ X : Fx ∩ B /  ∅}. Now, F is said to be upper semicontinuous, if for each closed set B ⊆ Y , F - B {x ∈ X : Fx ∩ B /  ∅} is closed in X.