Berge's maximum theorem to vector-valued functions with some applications

Abstract

In this paper, we introduce pseudocontinuity for Berge's maximum theorem for vector-valued functions which is weaker than semicontinuity. We prove the Berge's maximum theorem for vector-valued functions with pseudocontinuity and obtain the set-valued mapping of the solutions is upper semicontinuous with nonempty and compact values. As applications, we derive some existence results for weakly Pareto-Nash equilibrium for multiobjective games and generalized multiobjective games both with pseudocontinuous vector-valued payoffs. Moreover, we obtain the existence of essential components of the set of weakly Pareto-Nash equilibrium for these discontinuous games in the uniform topological space of best-reply correspondences. Some examples are given to investigate our results.