Minimax problems for set-valued mappings with set optimization

Abstract

In this paper, we introduce a class ofset-valued mappings with some set order relations, which is calleduniformly same-order. For this sort of mappings, we obtain someexistence results of saddle points and depict the structures of thesets of saddle points. Moreover, we obtain a minimax theorem andestablish an equivalent relationship between the minimax theorem anda saddle point theorem for the scalarset-valued mappings, in which the minimization and the maximizationof set-valued mappings are taken in the sense of set optimization.