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.