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

The classical Knaster-Kuratowsid-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.