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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 Ã°ÂÂ”Â„Ã°ÂÂÂœÃŽÂº-multimap or a multimap T:AÃ¢Â†Â’2B such that both T and SÃ¢ÂˆÂ˜T are closed and have the KKM property for each Kakutani multimap S:BÃ¢Â†Â’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

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