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 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.