Densely Defined Equilibrium Problems

In the present work, we deal with set-valued equilibrium problems, for which we provide sufficient conditions for the existence of a solution. The conditions, that we consider, are imposed not on the whole domain, but rather on a self-segment-dense subset of it, a special type of dense subset. As an application, we obtain a generalized Debreu–Gale–Nikaïdo-type theorem, with a considerably weakened Walras law in its hypothesis. Furthermore, we consider a noncooperative $$n$$n-person game and prove the existence of a Nash equilibrium, under assumptions that are less restrictive than the classical ones.