  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-Nikaido-type theorem, with a considerably
weakened Walras law in its hypothesis. Further, we consider a non-cooperative
n-person game and prove the existence of a Nash equilibrium, under assumptions
that are less restrictive than the classical ones.
