Intersection theorems for weak KKM set-valued mappings in the finite dimensional setting

Abstract Let X be a convex set in a vector space, Y be a nonempty set and S , T : X ⇉ Y two set-valued mappings. S is said to be a weak KKM mapping w.r.t. T if for each nonempty finite subset A of X and any x ∈ conv  A , T ( x ) ∩ S ( A ) ≠ ∅ . Recently, the authors obtained two intersection theorems for a pair of such mappings, when X is a compact convex subset of a topological vector space. In this paper, we obtain open versions of the above mentioned theorems when X is a compact convex set in R n . As applications, we establish several minimax inequalities and existence criteria for the solutions of three types of set-valued equilibrium problems.