Minimax inequalities for functions with noncompact domain and diagonally $GC$-quasiconcavity and their applications

Abstract

In this paper, we introduce the diagonally $GC$-quasiconcavity condition which generalizes and unifies the conditions of quasiconcavity, $CF$-quasiconcavity, diagonal transfer quasiconcavity, $C$-quasiconcavity, diagonally $C$-concavity, and diagonally $C$-quasiconcavity. By using the diagonally $GC$-quasiconcavity condition, we establish new minimax inequalities for functions with noncompact domain in topological spaces without linear structure. As applications of these minimax inequalities, we obtain existence theorems of saddle points in a zero-sum game with two players, existence theorems of solutions to the complementarity problem, and nonempty intersection theorems with equivalent forms.