Some minimax theorems of set-valued maps and their applications

In the paper, we consider two types of scalarization functions of sets and investigate their properties. Moreover, based on two set-relations, we propose two kinds of minimax and maximin values of set-valued maps, respectively, and show some minimax theorems of set-valued maps with respect to those minimax and maximin values by using several properties of the above two functions. As an application of these results, we give common saddle point theorems of vector-valued functions.