A remark on intersection of convex sets.
type: Item
instance of: scholarly article
title: A remark on intersection of convex sets.
zbMATH Open document ID: 1040.46014
DOI: 10.1016/S0022-247X(03)00330-5
author: Giuseppe Marino
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-09-07T00:00:00Z
review text: In this short note, the author shows that if \(\{G_\gamma:\gamma\in\Gamma\}\) is a collection of convex subsets of a real normed linear space with nonempty intersection and if \((G)^\varepsilon\) denotes the \(\varepsilon\)-neighborhood of a set \(G\), then either the intersection \(\bigcap_{\gamma\in\Gamma} (G_\gamma)^\varepsilon\) is bounded for each \(\varepsilon>0\) or the intersection is unbounded for each \(\varepsilon>0\). The author uses this to prove a result about fixed points of continuous pseudo-contractive mappings from closed convex subsets of a Hilbert space into the Hilbert space. In particular, if such a map \(T:K\to H\) satisfies a certain weak inward condition and if there exists a bounded subset \(A\) of \(K\) such that \(K \cap (\bigcap_{a\in A} G(a,T(a))) \neq \emptyset\) (where \(G(a,T(a))\) denotes the closed half-space of points in \(H\) closer to \(T(a)\) than to \(a\)) and \(\bigcap_{a\in A} (G(a,T(a)))^\varepsilon\) is bounded for some \(\varepsilon > 0\), then \(T\) has a fixed point in \(K\). Reviewer's remark: There is a misstatement in the proof of the theorem: the last line on page 777 states that an unbounded set is contained in a bounded set. If the reader ignores this line, the next inclusion holds with \(\delta\) equal to the diameter of \(A-T(A)\).
reviewed by: B. Turett
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A05
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 1976026
zbMATH Keywords: convex sets
zbMATH Keywords: pseudo-contractive map
zbMATH Keywords: fixed point
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1976026