A critical point theorem in bounded convex sets and localization of Nash-type equilibria of nonvariational systems

The localization of a critical point of minimum type of a smooth functional is obtained in a bounded convex conical set defined by a norm and a concave upper semicontinuous functional. A vector version is also given in order to localize componentwise solutions of variational systems. The technique is then used for the localization and multiplicity of Nash-type positive equilibria of nonvariational systems. Applications are given to periodic problems.