An extension to a theorem of Himmelberg
type: Item
instance of: scholarly article
title: An extension to a theorem of Himmelberg
zbMATH Open document ID: 1109.54025
author: N. E. Zubov
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-01-09T00:00:00Z
review text: Let \(E=(E,\tau)\) be a topological vector space with topology \(\tau\), \(\widehat E=(E, \tau)^\wedge\) the completion of \(E\), \(E^*\) the dual of \(E\) and \((E^*, \tau_w)\) the weak topology of \(E\). We denote by \(co\) and \(cl\) the convex hull and the closure. The following theorem from this paper is an extension of a fixed-point theorem of \textit{C. J. Himmelberg} [J. Math. Anal. Appl. 38, 205--207 (1972; Zbl 0204.23104)]. Theorem: Let \(E\) be a topological vector space, \(C\) a convex subset of \(E\) and \(K\) a compact subset of \(C\) such that \(cl(co(K))\) is \(\widehat E\)-compact. Let \(f:C\to K\) be an usc multifunction such that for each \(x\in X\), \(f(x)\) is a nonempty, convex and closed subset of \(K\). If \((\widehat E,\tau_w)\) separates the points of \(\widehat E\), then \(f\) has a fixed-point in \(K\).
reviewed by: Valeriu Popa
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5083277
zbMATH Keywords: topological vector space
zbMATH Keywords: fixed points
zbMATH Keywords: upper semicontinuous multifunctions
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.04.048
OpenAlex ID: W1985303960
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DOI: 10.1016/J.JMAA.2006.04.048
description: scientific article; zbMATH DE number 5083277