A Generalized KKM Theorem and its Applications to Saddle Point and Nash Equilibrium Problem

The classical Knaster-Kuratowski-Mazurkiewicz (in short, KKM) theorem is a basic result for combinatorial mathematics; it is equivalent to many basic theorems such as Brouwer´s fixed point theorem, Sperner´s lemma, and Fan´s minimax inequality. In 1961, Ky Fan generalized the classical KKM theorem from finite dimensional spaces to infinite dimensional spaces, and since then, this theorem has become a very versatile tool in nonlinear analysis. The main purpose of this paper is to generalize the KKM theorem under the non-convexity setting of topological space. Furthermore, as its applications, existence theorems for a saddle point problem and the Nash equilibrium problem for non-cooperative games are obtained in general topological spaces without any convexity structure and linear structure. Our results improve and unify the corresponding results in the recently existing literatures.