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

Let X be a convex set in a vector space, Y be a nonempty set and S,T : X paired right arrows 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 is an element of conv A, T(x) boolean AND S(A) not equal empty set. 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. (C) 2019 Elsevier B.V. All rights reserved.