Nonlinear ergodic theorems of almost-orbits of non-Lipschitzian semigroups
type: Item
instance of: scholarly article
title: Nonlinear ergodic theorems of almost-orbits of non-Lipschitzian semigroups
zbMATH Open document ID: 0781.47043
author: Tae Hwa Kim
published in: Kodai Mathematical Journal
publication date: 1993-05-16T00:00:00Z
review text: Let \(G\) be a semitopological semigroup and let \(\{S(t):t\in G\}\) be a continuous representation of \(G\) in the family of continuous mappings of a nonempty closed convex subset \(C\) of a Banach space into itself. The representation is said to be a semigroup of weakly asymptotically nonexpansive type if for each \(x\in C\) and each bounded \(D\subset C\) \[ \limsup_ t\sup_{y\in D}(| S(t)x-S(t)y|-| x-y|)\leq 0. \] If \(G\) is right reversible then a function \(u:G\to C\) is an almost orbit of \(\{S(t):t\in G\}\) if \[ \lim_ t\bigl(\sup_{s\in G}| u(st)-S(s)u(t)|\bigr)=0. \] The author of the paper under review proves theorems on fixed points of almost-orbits in the case of uniformly convex Banach spaces. Also some results on the weak convergence of almost-orbits and ergodic theorems are proved. The theorems generalize previously known commutative versions.
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H20
zbMATH DE Number: 167946
zbMATH Keywords: semitopological semigroup
zbMATH Keywords: continuous representation
zbMATH Keywords: semigroup of weakly asymptotically nonexpansive type
zbMATH Keywords: fixed points of almost-orbits
zbMATH Keywords: uniformly convex Banach spaces
zbMATH Keywords: weak convergence
zbMATH Keywords: ergodic theorems
reviewed by: Maciej Sablik
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039604
OpenAlex ID: W2093131069
cites work: Q5655874
cites work: Applications of the Hahn-Banach Theorem in Approximation Theory
cites work: Asymptotic Behavior of Iterates of Nonexpansive Mappings in Banach Spaces with Opial's Condition
cites work: Q3322631
cites work: Some geometric properties related to the fixed point theory for nonexpansive mappings
cites work: A note on segmenting Mann iterates
cites work: Non-Expansive Actions of Topological Semigroups and Fixed Points
cites work: Fixed Point Theorems for Proximately Nonexpansive Semigroups
cites work: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type
cites work: Semigroup of nonexpansive mappings on a Hilbert space
cites work: Weak convergence and nonlinear ergodic theorems for reversible semigroup of nonexpansive mappings
cites work: Asymptotic behavior of iterates of nonexpansive mappings in Banach spaces
cites work: On the asymptotic behaviour of almost-orbits of nonlinear contraction semigroups in Banach spaces
cites work: Weak convergence of the sequence of successive approximations for nonexpansive mappings
cites work: Asymptotic behavior of almost-orbits of semigroups of Lipschitzian mappings in Banach spaces
DOI: 10.2996/KMJ/1138039604
description: scientific article; zbMATH DE number 167946