Nonlinear ergodic theorems for almost nonexpansive curves over commutative semigroups
type: Item
instance of: scholarly article
title: Nonlinear ergodic theorems for almost nonexpansive curves over commutative semigroups
zbMATH Open document ID: 0876.47047
DOI: 10.12775/TMNA.1995.021
author: Osamu Kada
author: Wataru Takahashi
published in: Topological Methods in Nonlinear Analysis
publication date: 1997-11-25T00:00:00Z
review text: The author proves the following theorem: Let \(S\) be a commutative semigroup of nonlinear operators on a real Hilbert space \(H\) with identity. Let \(u\) be an almost nonexpansive curve (that is, \(u\) is a function from \(S\) into \(H\) such that there exists a real valued function \(\varepsilon(\cdot,\cdot)\) on \(S\times S\) such that \(|u(s+h)- u(t+h)|^2\leq|u(s)- u(t)|^2+ \varepsilon(s,t)\), where \(\lim_{s,t\to\infty} \varepsilon(s,t)=0\)), let \(\mu\) be an invariant mean on a subspace \(X\) of \(\ell^\infty(S)\), (that is, \(\mu\in X^*\) and \(|\mu|=\mu(1)=1\) where 1 is the constant function in \(\ell^\infty(S)\)), let \(P\) be the metric projection of \(H\) onto \(F_\mu(u)\), where \(F_\mu(u)\) is the set of generalized fixed points (that is, \(q\in F_\mu(u)\) if and only if \(|u(t)-q|^2\leq|u(s)-q|^2+ \varepsilon(s)\), where \(\varepsilon(s)= \mu\varepsilon(s,t)\)). Then \(Pu(t)\) converges strongly to \(u(\mu)\), which is the \(\mu\)-asymptotic center of \(u\) in \(H\).
reviewed by: Glenn Webb
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 856955
zbMATH Keywords: nonlinear ergodic theorems
zbMATH Keywords: asymptotic center
zbMATH Keywords: commutative semigroup of nonlinear operators
zbMATH Keywords: almost nonexpansive curve
zbMATH Keywords: invariant mean
zbMATH Keywords: metric projection
zbMATH Keywords: generalized fixed points
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 856955