Existence of maximal elements and applications
type: Item
instance of: scholarly article
zbMATH Open document ID: 0996.47058
publication date: 2002-10-27T00:00:00Z
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H04
zbMATH DE Number: 1657012
zbMATH Keywords: best approximation
zbMATH Keywords: fixed point
zbMATH Keywords: maximal element
zbMATH Keywords: inward set
zbMATH Keywords: the KKM-map principle
zbMATH Keywords: variational inequalities
zbMATH Keywords: monotone map
zbMATH Keywords: Ky Fan's lemma
zbMATH Keywords: Ky Fan's KKM-principle
MaRDI profile type: Publication
author: A. Carbone
title: Existence of maximal elements and applications
published in: Nonlinear Analysis Forum
review text: If \(C\) is a nonempty subset of a vector space \(X\) and \(F:C \to 2^X\) is a set-valued map from \(C\) to \(X\), then \(F\) is said to be a KKM-map if for each finite subset \(S \subset C\), \(conv(S) \subset \bigcup_{x \in S} Fx\). Ky Fan's Lemma says that if \(C\) is a nonempty convex compact subset of \(\mathbb{R}^n\) and \(F:C \to 2^C\) is a set-valued function with convex values and an open graph such that \(x \notin Fx\) for each \(x \in C\), then there exists an \(x \in C\) such that \(Fx= \emptyset\). The author proves a number of results related to Ky Fan's KKM-principle, such as the following one. NEWLINENEWLINENEWLINETheorem. \(C\) is a nonempty closed convex subset of a topological Hausdorff vector space \(X\) and \(F:C \to 2^C\) be such that NEWLINENEWLINENEWLINE(i) \(x \notin Fx\) for each \(x \in C\); NEWLINENEWLINENEWLINE(ii) \(Fx\) is closed for each \(x \in C\); NEWLINENEWLINENEWLINE(iii) \(F^{-1}(y)=\{x \in C \mid y \in Fx\}\) is convex for each \(y \in C\); NEWLINENEWLINENEWLINE(iv) \(C\) can be covered by a finite number of values of \(F\). NEWLINENEWLINENEWLINEThen there exists an \(x \in C\) such that \(Fx= \emptyset\). NEWLINENEWLINENEWLINEThe author uses his results to prove some known theorems on the existence of fixed points and of zeroes for single valued maps.
reviewed by: Jan Jaworowski
description: scientific article; zbMATH DE number 1657012