Some computational aspects of metric fixed-point theory
type: Item
instance of: scholarly article
title: Some computational aspects of metric fixed-point theory
zbMATH Open document ID: 1086.47021
author: Ulrich Kohlenbach
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-05-04T00:00:00Z
review text: Let \((X, \| \cdot \| )\) be a normed space, \(C \subset X\) a nonempty convex subset and \((\lambda_n)_{n \in \mathbb N}\) a sequence in \([0,1]\). An operator \(f:C \to C\) is said to be asymptotically nonexpansive with sequence \((k_n)_{n \in \mathbb N}\) if \( \lim_{n \to \infty} k_n=0\) and \[ \| f^n(x)-f^n(y)\| \leq (1+k_n) \| x-y \| ,\;\forall \;n \in \mathbb N, \;\forall \;x,y \in C. \] For such an operator, the Krasnoselskij--Mann iteration is defined as follows \(x_0:=x \in C, \;x_{n+1}:= (1-\lambda_n)x_n+ \lambda_n f^n (x_n) \). In this paper, the author shows how bounds on asymptotic regularity (\(\sum_{n=0}^{\infty}k_n \leq K)\) can effectively be converted into certain bounds on the convergence towards a fixed point.
reviewed by: Ioan A. Rus
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 03F10
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 2165059
zbMATH Keywords: fixed point
zbMATH Keywords: nonexpansive operator
zbMATH Keywords: Krasnoselskij-Mann iteration
zbMATH Keywords: computable analysis
MaRDI profile type: Publication
full work available at URL: j.na.2005.01.075
OpenAlex ID: W2086045987
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DOI: 10.1016/J.NA.2005.01.075
description: scientific article; zbMATH DE number 2165059