Compactness and the fixed point property in \(\ell_{1}\)
type: Item
instance of: scholarly article
title: Compactness and the fixed point property in \(\ell_{1}\)
zbMATH Open document ID: 1348.54049
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Journal of Mathematical Analysis and Applications
publication date: 2016-07-29T00:00:00Z
review text: The authors prove that the following conditions are equivalent for a convex closed subset of~\(\ell^1\): (1)~\(C\) is a compact set; (2)~\(C\) satisfies the fixed point property for Lipschitzian mappings \(T:C\to C\) which are nonexpansive on \(\overline{\mathrm{co}}\,T(C)\); (3)~\(C\) satisfies the fixed point property for cascading nonexpansive mappings; (4)~for every \(L>2\), \(C\) satisfies the fixed point property for uniformly \(L\)-Lipschitzian mappings. This result is extended to some classes of Banach spaces having a boundedly complete Schauder basis, and also to the space of trace class operators on a Hilbert space, with the trace norm.
reviewed by: Zoran Kadelburg
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 6607762
zbMATH Keywords: nonexpansive mappings
zbMATH Keywords: uniformly Lipschitzian mappings
zbMATH Keywords: cascading nonexpansive mappings
zbMATH Keywords: fixed point
zbMATH Keywords: compact domain
MaRDI profile type: Publication
full work available at URL: j.jmaa.2016.06.019
OpenAlex ID: W2433800128
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DOI: 10.1016/J.JMAA.2016.06.019
description: scientific article; zbMATH DE number 6607762