Strong and weak convergence theorems for asymptotically nonexpansive mappings.
type: Item
instance of: scholarly article
title: Strong and weak convergence theorems for asymptotically nonexpansive mappings.
zbMATH Open document ID: 1057.47071
DOI: 10.1016/S0022-247X(03)00061-1
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-27T00:00:00Z
review text: This article deals with iterations \(x_n\) defined by the formula \[ x_{n+1} = P(((1 - \alpha_n)x_n + \alpha_nT(PT)^{n-1}x_n), \quad n = 1,2,\ldots, \;x_1 \in K \] where \(K\) is a nonempty closed convex nonexpansive retract in a real uniformly convex Banach space \(E\), \(P\) a nonexpansive retraction, \(T: \;K \to E\) an asymptotically nonexpansive nonself-map such that \[ \| T^nx - T^ny\| \leq k_n\| x - y\| , \quad n = 1,2, \dots, \;x, y \in K, \] (\(k_n \to 1\) and \(\sum_{n=1}^\infty (k_n^2 - 1) < \infty\)) and such that \(F(T) = \{x \in K: \;Tx = x\} \neq \emptyset\), and \((\alpha_n)\) a sequence such that \(\varepsilon < 1 - \alpha_n < 1 - \varepsilon\) for some \(\varepsilon > 0\). It is proved the strong convergence of \((x_n)\) to some \(x_* \in F(T)\) provided that \(T\) is completely continuous and the weak convergence of \((x_n)\) to some \(x_* \in F(T)\) provided that the norm in \(E\) is Fréchet differentiable.
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 1917779
zbMATH Keywords: nonexpansive retracts
zbMATH Keywords: asymptotically nonexpansive retracts
zbMATH Keywords: demiclosed maps
zbMATH Keywords: modulus of convexity
author: C. E. Chidume
author: E. U. Ofoedu
author: H. Zegeye
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00061-1
OpenAlex ID: W2074474280
description: scientific article; zbMATH DE number 1917779