Normal structure and orbital fixed point conditions
type: Item
instance of: scholarly article
title: Normal structure and orbital fixed point conditions
zbMATH Open document ID: 1392.54032
author: Nazeer Shahzad
author: W. A. Kirk
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-05-03T00:00:00Z
review text: The main aim of this survey paper is to stress that the ``old'' concepts of normal structure (for a Banach space \(X\)) and diminishing orbital diameter (for a selfmap \(T\)) are strongly related to a number of recent similar concepts. The following is the main result in this exposition. Theorem. Let \(X\) be a Banach space, \(D\) be a weakly compact subset of \(X\) and let the selfmap \(T:D\to D\) be such that i)\ \(T\) is nonexpansive with respect to orbits ii)\ \(\inf_{m\in N}\limsup_n ||T^m(x)-T^n(x)||< \operatorname{diam}(O(x))\), \(\forall x\in D\), \(x\neq Tx\). Then, \(T\) has a fixed point in \(D\). Further aspects occasioned by these developments are also discussed.
reviewed by: Mihai Turinici
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 6866175
zbMATH Keywords: Banach space
zbMATH Keywords: normal structure
zbMATH Keywords: fixed point
zbMATH Keywords: orbitally nonexpansive map
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.02.022
OpenAlex ID: W2792771147
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DOI: 10.1016/J.JMAA.2018.02.022
description: scientific article; zbMATH DE number 6866175