Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 457069, 10 pages
Best Proximity Sets and Equilibrium Pairs for a Finite Family of Multimaps
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.
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 -multimap or a multimap such that both and are closed and have the KKM property for each
Kakutani multimap . As applications, we obtain existence
theorems of equilibrium pairs for free -person games as well as for free
1-person games. Our results extend and improve several well-known
and recent results.