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New fixed-point theorems for two maps and applications to problems on sets with convex sections and minimax inequalities. (English) Zbl 1057.54030

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 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 M. Sion’s minimax inequality [Pac. J. Math. 8, 171–176 (1958; Zbl 0081.11502)].

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
47H05 Monotone operators and generalizations
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