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

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 -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.