AN EXISTENCE THEOREM FOR A VARIATIONAL RELATION PROBLEM WITH APPLICATIONS TO MINIMAX INEQUALITIES

Variational relation problems, a recent concept introduced by Luc, are general models for a large class of problems in optimization and nonlinear analysis. In this paper we establish an existence theorem for the solution of the following variational relation problem: Find x* is an element of X such that (x*, y) is an element of R for every y is an element of Y, where X is a nonempty convex subset of a vector topological space, Y is a compact convex subset of a Hausdorff topological vector space and R is a relation between the elements of the two sets. As applications, we obtain an intersection theorem, a fixed point theorem and several minimax inequalities.