Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 457069, 10 pages
doi:10.1155/2008/457069
Research Article

Best Proximity Sets and Equilibrium Pairs for a Finite Family of Multimaps

M. A. Al-Thagafi and Naseer Shahzad

Department of Mathematics, King AbdulAziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 12 May 2008; Accepted 16 October 2008

Academic Editor: Jerzy Jezierski

Copyright © 2008 M. A. Al-Thagafi and Naseer Shahzad. 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 establish the existence of a best proximity pair for which the best proximity set is nonempty for a finite family of multimaps whose product is either an 𝔄𝐜κ-multimap or a multimap T:A→2B such that both T and S∘T are closed and have the KKM property for each Kakutani multimap S:B→2A. As applications, we obtain existence theorems of equilibrium pairs for free n-person games as well as for free 1-person games. Our results extend and improve several well-known and recent results.