Equilibrium problems when the equilibrium condition is missing

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.