Existence of generalized equilibria

The problem of the existence of generalized equilibria and fixed-points of a correspondence F, defined on a compact subset M of Rn, with values in Rn, when the set M is neither assumed to be convex, nor smooth was considered. Let F be a correspondence from M to Rn, that is, a map from M to the set of all the subsets of Rn; the correspondence F is said to be upper semicontinuous (usc), if the set {x∈M|F(x)⊂V} is open in M for every open set V⊂Rn. Let N be a `normal cone' correspondence. Sufficient conditions were provided for the assertion to hold: for every usc correspondence F from M to Rn, with nonempty, convex, compact values, there exists a generalized equilibrium x*∈M of F.