Equilibrium problems when the equilibrium condition is missing

Abstract

Abstract Given a nonempty convex subset X of a topological vector space and a real bifunction f defined on $$X \times X$$ X × X , the associated equilibrium problem consists in finding a point $$x_0 \in X$$ x 0 ∈ X such that $$f(x_0, y) \ge 0$$ f ( x 0 , y ) ≥ 0 , for all $$y \in X$$ y ∈ X . A standard condition in equilibrium problems is that the values of f to be nonnegative on the diagonal of $$X \times X$$ X × 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 $$g: X \times X \rightarrow \mathbb {R}$$ g : X × X → R , the two bifunctions being linked by a certain compatibility condition. Applications to variational inequality problems, quasiequilibrium problems and vector equilibrium problems are given.

Keywords

• 47J20
• 54C60