Semi-continuity of the solution maps of set-valued equilibrium problems with equilibrium constraints

Abstract

In this paper, we study the parametric set-valued equilibrium problems with equilibrium constraints based on the comparison of objective values of set-valued maps by set order relations. We introduce new notions of generalized concavity for set-valued maps and study their properties as well as their relationship with other existing well-known notions. By using the generalized concavity and semi-continuity of set-valued maps, we study the existence of solutions for set-valued equilibrium problems when the equilibrium condition is missing. We further establish sufficient conditions for lower/upper semi-continuity of the solution maps of the set-valued equilibrium problems involving set order relations. Several examples are provided to illustrate the derived results.