Common fixed point theorems and minimax inequalities in locally convex Hausdorff topological vector spaces

In this article, we introduce the concept of a family of set-valued mappings generalized Knaster-Kuratowski-Mazurkiewicz (KKM) w.r.t. other family of set-valued mappings. We then prove that if X is a nonempty compact convex subset of a locally convex Hausdorff topological vector space and T and S are two families of self set-valued mappings of X such that S is generalized KKM w.r.t. T, under some natural conditions, the set-valued mappings S∈S have a fixed point. Other common fixed point theorems and minimax inequalities of Ky Fan type are obtained as applications.