The existence of nonexpansive retractions in Banach spaces
type: Item
instance of: scholarly article
title: The existence of nonexpansive retractions in Banach spaces
zbMATH Open document ID: 0594.47052
author: Norimichi Hirano
author: Kazuo Kido
author: Wataru Takahashi
published in: Journal of the Mathematical Society of Japan
publication date: 1986-01-01T00:00:00Z
review text: Let C be a closed convex subset of a uniformly convex Banach space E, and T a nonexpansive mapping of C into itself. Assume that T has fixed points. Then we show the existence of a nonexpansive retraction P of C into the set of fixed points of T such that \(PT=TP=P\) and Px is implied in the convex closure of \(\{T^ nx:\) \(n=0,1,2,...\}\) for each \(x\in C\). If E has a Fréchet differentiable norm, then such retraction P is unique and the Cesàro means \(\frac{1}{n}\sum^{n-1}_{i=0}T^ ix\) converge, as \(n\to \infty\), weakly to Px for each \(x\in C\). This is a simple proof of a mean ergodic theorem for a nonexpansive mapping in a Banach space.
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 3955812
zbMATH Keywords: nonexpansive mapping
zbMATH Keywords: existence of a nonexpansive retraction
zbMATH Keywords: Fréchet differentiable norm
zbMATH Keywords: mean ergodic theorem for a nonexpansive mapping in a Banach space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03810001
OpenAlex ID: W2049963826
DOI: 10.2969/JMSJ/03810001
description: scientific article; zbMATH DE number 3955812