Minimax problems of uniformly same-order set-valued mappings (Q2873002)

Let \(X,Y\) and \(V\) be real topological vector spaces. Assume that \(S\) is a closed convex cone in \(V\) with \(\text{int S}\neq \emptyset\). Let \(X_0\subset X\) and \(Y_0\subset Y\) be nonempty sets, and NEWLINE\[NEWLINE F:X_0\times Y_0\rightarrow 2^{V} \tag{1} NEWLINE\]NEWLINE be a set-valued mapping with nonempty values.NEWLINENEWLINEThe mapping (1) is said to be \(S(\text{int S})\)-uniformly same-order on \(X_0\) with respect to \(y_0\in Y_0\) if there exists \(x_0\in X_0\) such that NEWLINE\[NEWLINE F(x_0,y_0)\subset F(x_0,Y_0)+S \setminus \{0_V\}(\text{int S}). NEWLINE\]NEWLINE Then for all \(x\in X_0\) NEWLINE\[NEWLINE F(x,y_0)\subset F(x,Y_0)+S \setminus \{0_V\}(\text{int S}). NEWLINE\]NEWLINENEWLINENEWLINE\(F\) is said be \(S(\text{int S})\)-uniformly same-order on \(X_0\) if \(F\) is \(S(\text{int S})\)-uniformly same-order on \(X_0\) with respect to any \(y_0\in Y_0.\)NEWLINENEWLINEIn this paper, for the introduced mappings some minimax problems are investigated without the assumption of convexity, and the minimization and maximization of mappings are taken in the sense of vector optimization.