Equilibrium problems when the equilibrium condition is missing

Abstract

Given a nonempty convex subset X of a topological vector space and a real bifunction f defined on X times X, the associated equilibrium problem consists in finding a point x_0 in X such that f(x_0, y) ge 0, for all y in X. A standard condition in equilibrium problems is that the values of f to be nonnegative on the diagonal of X times X. In this paper, we deal with equilibrium problems in which this condition is missing. For this purpose, we will need to consider, besides the function f, another one g: X times X rightarrow mathbb {R}, the two bifunctions being linked by a certain compatibility condition. Applications to variational inequality problems, quasiequilibrium problems and vector equilibrium problems are given.