Asymptotic behavior of almost-orbits of semigroups of Lipschitzian mappings in Banach spaces
type: Item
instance of: scholarly article
title: Asymptotic behavior of almost-orbits of semigroups of Lipschitzian mappings in Banach spaces
zbMATH Open document ID: 0673.47046
author: Peijun Zhang
author: Wataru Takahashi
published in: Kodai Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let G be a right-reversible semitopological semigroup, and let C be a nonempty closed convex subset of a uniformly convex Banach space E; let \(S=\{S(t):\) \(t\in G\}\) be a family of Lipschitzian mappings \(C\to C\) with Lipschitzian constants \(k_ t\) such that \(\lim_{t} \sup k_ t\leq 1\) and with semigroup properties (i) \(S(st)x=S(s)S(t)x\) and (ii) \(u(t)=S(t)\) x is continuous function on G. The authors show that the set F(S) of common fixed points of S(t) is a closed and convex subset of C. Moreover, having \(\{\) u(t): \(t\in G\}\) an almost-orbit of S they prove that, if E has a Fréchet-differentiable norm than the set \(M=\cap_{s\in G}\overline{co}\{u(t):\) \(t\geq s\}\) contains at most one point of F(S). (An almost-orbit is defined as a function u: \(G\to C\) such that \(\lim_{t}\sup_{s}| u(st)- S(s)u(t)| =0.)\) Using this fact and other properties of M and F(S), they establishes some theorems concerning weak convergence of an almost-orbit and prove strong convergence of its metric projection on F(S).
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 4101950
zbMATH Keywords: right-reversible semitopological semigroup
zbMATH Keywords: family of Lipschitzian mappings
zbMATH Keywords: Fréchet-differentiable norm
zbMATH Keywords: weak convergence of an almost- orbit
zbMATH Keywords: strong convergence of its metric projection
reviewed by: Pavel Doktor
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138038824
OpenAlex ID: W2079631693
cites work: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces
cites work: Applications of the Hahn-Banach Theorem in Approximation Theory
cites work: Geometry of Banach spaces. Selected topics
cites work: Uniformly Lipschitzian Families of Transformations in Banach Spaces
cites work: A note on segmenting Mann iterates
cites work: Nonlinear ergodic theorems for an amenable semigroup of nonexpansive mappings in a Banach space
cites work: Fixed point theorems for uniformly Lipschitzian semigroups in Hilbert spaces
cites work: A Nonlinear Ergodic Theorem for a Reversible Semigroup of Lipschitzian Mappings in a Hilbert Space
cites work: Weak convergence and nonlinear ergodic theorems for reversible semigroup of nonexpansive mappings
cites work: On the asymptotic behaviour of almost-orbits of nonlinear contraction semigroups in Banach spaces
cites work: Asymptotic behavior of solutions of differential equations associated to monotone operators
cites work: Q3788597
cites work: Asymptotic behavior of almost-orbits of reversible semigroups of Lipschitzian mappings
DOI: 10.2996/KMJ/1138038824
description: scientific article; zbMATH DE number 4101950