Mann iterates of directionally nonexpansive mappings in hyperbolic spaces
type: Item
instance of: scholarly article
title: Mann iterates of directionally nonexpansive mappings in hyperbolic spaces
zbMATH Open document ID: 1038.47037
DOI: 10.1155/S1085337503212021
author: Ulrich Kohlenbach
author: Leuştean Laurenţiu
published in: Abstract and Applied Analysis
publication date: 2003-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/50781
review text: Let \((X, \rho)\) be a metric space. It is assumed that \((X,\rho)\) contains a nonempty family \(M\) of metric lines such that for each pair of distinct points \(x\) and \(y\) in \(X\) and for each \(t \in [0, 1]\) there is a unique point \(z\) on the line segment joining \(x\) and \(y, [x, y]\), satisfying \[ \rho(x, z) = t\rho(x, y), \qquad \rho(y, z) = (1 - t)\rho(x, y).\tag{1} \] The unique point satisfying (1) is denoted by \((1-t)x\oplus ty\). The authors define a hyperbolic space in terms of \(\oplus\) and establish the convergence of a Mann-type iteration scheme of the form \[ x_0 := x \in X, \qquad x_{n+1} := (1 - \lambda_n)x_n\oplus \lambda_nf(x_n),\tag{2} \] where \(f\) is a directional nonexpansive selfmap of a nonempty convex subset of \(X\). They show that the iteration scheme converges to a fixed point of \(T\) and obtain error bounds. While of theoretical interest, using (2) in a computational situation would be very challenging.
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 03F10
Mathematics Subject Classification ID: 03F60
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 1929992
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: directionally nonexpansive maps
zbMATH Keywords: hyperbolic spaces
zbMATH Keywords: logical analysis
zbMATH Keywords: Mann iteration
zbMATH Keywords: uniform asymptotic regularity
reviewed by: Billy E. Rhoades
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1929992