Weak convergence theorems for asymptotically nonexpansive mappings and semigroups
type: Item
instance of: scholarly article
title: Weak convergence theorems for asymptotically nonexpansive mappings and semigroups
zbMATH Open document ID: 0983.47040
DOI: 10.1016/S0362-546X(99)00200-X
author: Wiesława Kaczor
author: Tadeusz Kuczumow
author: Simeon Reich
author: Jesus García-Falset
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-04-21T00:00:00Z
review text: The authors consider weak convergence in mean ergodic theorems for asymptotically nonexpansive semigroups and the weak convergence of the modified Mann-Ishikawa iteration processes for asymptotically nonexpansive mappings. Their results are new even for nonexpansive mappings and semigroups. The paper consists of nine sections. The first one is an introduction. Section 2 contains some preliminary results. In Section 3 the main Lemma is formulated: Let \(X\) be a uniformly convex Banach space whose dual \(X^*\) has the KK-property (Kadec-Klee). If \(\{x_n\}\) is a bounded sequence such that \(\lim_{n\to\infty} \|\alpha x_n+ (1-\alpha)f_1- f_2\|\) exists for all \(\alpha\in [0,1]\) and \(f_1,f_2\in \omega_w(\{x_n\})\), then \(\omega_w\{x_n\}\) is a singleton. Section 4 deals with asymptotically nonexpansive semigroups. The main result here is: Let \(X\) and \(X^*\) be as in the main Lemma and let \(C\subset X\) be bounded closed convex set. If \({\mathcal F}\) is an asymptotically nonexpansive semigroup acting on \(C\) and \(u\) is an almost orbit of \({\mathcal F}\), then the following conditions are equivalent: (1) \(\omega_w(u)\subset F({\mathcal F})\); (2) \(\text{w-}\lim_{t\to\infty} u(t)= x\in F({\mathcal F})\); (3) \(\text{w-}\lim_{t\to\infty}(u(t)- u(t+ s))= 0\) for each \(s\in [0,\infty)\). In Section 5 the authors consider the weak convergence of almost orbits of asymptotically nonexpansive mappings. The basic theorem here is: If \(T:C\to C\) is asymptotically nonexpansive in the intermediate sense and \(\{x_n\}\) is an almost orbit of \(T\), then \(\{x_n\}\) is weakly convergent to a fixed point of \(T\) iff \(w\)-\(\lim_{n\to\infty}(x_n- x_{n+1})= 0\). Section 6 is devoted to the investigations on modified Mann-Ishikawa iteration processes defined by \[ x_{i+1}= \alpha_i T^{n_i} y_i+ (1-\alpha_i)x_i\quad\text{and}\quad y_i= \beta_i T^{n_i}x_i+ (1- \beta_i) x_i, \] where \(\{\alpha_i\}\) and \(\{\beta_i\}\) are chosen so that \(\alpha_i\in [a,b]\) and \(\bet…
reviewed by: Vasil G. Angelov
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 1556814
zbMATH Keywords: Kadec-Klee
zbMATH Keywords: weak convergence
zbMATH Keywords: mean ergodic theorems
zbMATH Keywords: asymptotically nonexpansive semigroups
zbMATH Keywords: modified Mann-Ishikawa iteration processes
zbMATH Keywords: almost orbit
zbMATH Keywords: theorem of Reich-Nevanlinna
MaRDI profile type: Publication
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Weak convergence theorems for asymptotically nonexpansive mappings and semigroups (full work available at URL)
full work available at URL: s0362-546x(99)00200-x

Weak convergence theorems for asymptotically nonexpansive mappings and semigroups (OpenAlex ID)
OpenAlex ID: W2028658523