New fixed-point theorems for two maps and applications to problems on sets with convex sections and minimax inequalities
type: Item
instance of: scholarly article
title: New fixed-point theorems for two maps and applications to problems on sets with convex sections and minimax inequalities
zbMATH Open document ID: 1057.54030
DOI: 10.1016/S0898-1221(04)90016-8
publication date: 2004-06-15T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 47H05
zbMATH DE Number: 2075451
zbMATH Keywords: fixed point
zbMATH Keywords: intersection theorem
zbMATH Keywords: minimax inequalities
zbMATH Keywords: local intersection property
zbMATH Keywords: transfer closed maps
zbMATH Keywords: sets with convex sections
author: K. Q. Lan
reviewed by: Thomas B. Muenzenberger
MaRDI profile type: Publication
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published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 2075451