New fixed-point theorems for two maps and applications to problems on sets with convex sections and minimax inequalities
Let \(X\) and \(Y\) be nonempty convex subsets of two Hausdorff topological vector spaces, let \(S: X\to 2^Y\) and let \(T: Y\to 2^X\). Using \textit{A. Tychonoffs} fixed point theorem [Math. Ann. 111, 767--776 (1935; Zbl 0012.30803/JFM 61.1195.01)], the author derives the following. Theorem: Assume that (1) For each \((x, y)\in X\times Y\), \(S(x)\) and \(T(y)\) are nonempty and convex and \(S^{-1}(y)\) and \(T^{-1}(x)\) are open; (2) There is a nonempty compact convex \(X_0\subseteq X\) and a nonempty compact \(D\subseteq Y\) such that for all \(y\in Y-D\), \(X_0\cap T(u)\neq\varnothing\). Then there is \((x_0, y_0)\in X\times Y\) such that \((x_0, y_0)\in T(y_0)\times S(x_0)\). The author generalizes this result and makes applications to: maps which have the local intersection property; an intersection theorem for transfer-closed maps; sets with convex sections; and a generalization of \textit{M. Sion's} minimax inequality [Pac. J. Math. 8, 171--176 (1958; Zbl 0081.11502)].
scientific article; zbMATH DE number 2075451
scientific article