On mean nonexpansive mappings and the Lifshitz constant
type: Item
instance of: scholarly article
title: On mean nonexpansive mappings and the Lifshitz constant
zbMATH Open document ID: 1266.47073
author: Łukasz Piasecki
author: Víctor Pérez García
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-10-19T00:00:00Z
review text: A mapping \(T\) of a metric space \(X\) into itself is called Lipschitzian if there exists a positive number \(k\) such that \(d(Tx,Ty) \leq k d(x,y)\) for \(x, y \in X.\) The smallest such \(k\) is called the Lipschitz constant \(k(T)\) of \(T\). If \(\alpha = (\alpha_1, \dots,\alpha_n\)), where the entries are positive real numbers adding up to 1, the mapping \(T\) is called an \(\alpha\)-nonexpansive mapping if, for each \(x,y \in M\), \(\{\Sigma \alpha_j d(T^j(x), T^j(y)) \leq d(x,y): j = 1, \dots,n\}\). The map \(T\) is called mean expansive when the multi-index \(\alpha\) is not specified. With these definitions, it is shown how the theory of polynomials can be used to describe the asymptotic behavior of the sequence of Lipschitz constants of the iterates of mean nonexpansive mappings. As consequences, using the Lifshitz theorem, some fixed point theorems are proved. Further, it is shown that the fixed point set of every mean nonexpansive mapping on a strictly convex Banach space is closed and convex.
reviewed by: Srinivasa Swaminathan
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 6096039
zbMATH Keywords: mean nonexpansive mapping
zbMATH Keywords: fixed point
zbMATH Keywords: Lifshitz constant
zbMATH Keywords: Lipschitz constant
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.06.045
OpenAlex ID: W1976980189
cites work: Q3646731
cites work: Q3581626
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DOI: 10.1016/J.JMAA.2012.06.045
description: scientific article; zbMATH DE number 6096039