Implicit and explicit algorithms for minimum-norm fixed points of pseudocontractions in Hilbert spaces
type: Item
instance of: scholarly article
title: Implicit and explicit algorithms for minimum-norm fixed points of pseudocontractions in Hilbert spaces
zbMATH Open document ID: 1293.47068
author: Yonghong Yao
author: Vittorio Colao
author: Giuseppe Marino
author: Hong-Kun Xu
published in: Taiwanese Journal of Mathematics
publication date: 2013-02-21T00:00:00Z
full work available at URL: http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/2229
review text: Let \(C\) be a nonempty convex subset of a real Hilbert space \(H\) with inner product \(\left\langle\cdot,\cdot\right\rangle\) and norm \(\left\| \cdot \right\|\). A mapping \(T:C \rightarrow C\) is said to be pseudocontractive (or a pseudocontraction) if \[ \left\langle Tx-Ty,x-y\right\rangle \leq \left\| x-y \right\|^2 \text{ for all }x,y \in C. \] In order to approximate fixed points of pseudocontractions, there exist various iterative methods, e.g., Mann iteration, Ishikawa iteration, etc. In the paper under review, the authors consider two new iterative methods (one implicit and another one explicit) for approximating the fixed points of Lipschitzian pseudocontractions \(T\). The main results of the paper show that these algorithms converge strongly to the minimum-norm fixed point of \(T\). Examples are also given to illustrate the theoretical results and to show that the new results are effective improvements of those existing in the literature.
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 6137979
zbMATH Keywords: Hilbert space
zbMATH Keywords: pseudocontraction
zbMATH Keywords: Lipschitzian mapping
zbMATH Keywords: fixed point
zbMATH Keywords: implicit algorithm
zbMATH Keywords: explicit algorithm
zbMATH Keywords: convergence theorem
zbMATH Keywords: metric projection
zbMATH Keywords: minimum-norm
reviewed by: Vasile Berinde
MaRDI profile type: Publication
OpenAlex ID: W1929067491
DOI: 10.11650/TWJM/1500406745
description: scientific article; zbMATH DE number 6137979