Strong convergence for hybrid implicit \(S\)-iteration scheme of nonexpansive and strongly pseudocontractive mappings
type: Item
instance of: scholarly article
title: Strong convergence for hybrid implicit \(S\)-iteration scheme of nonexpansive and strongly pseudocontractive mappings
zbMATH Open document ID: 1472.47073
DOI: 10.1155/2014/735673
author: A. Rafiq
author: Faisal Ali
author: Shin Min Kang
author: Young Chel Kwun
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: Let \(K\) be a nonempty closed convex subset of a real Banach space \(E\), let \(S : K \rightarrow K\) be nonexpansive, and let \(T : K \rightarrow K\) be Lipschitz strongly pseudocontractive mappings such that \(p \in F \left(S\right) \cap F \left(T\right) = \left\{x \in K : S x = T x = x\right\}\) and \(\| x-Sy\| \leq \| x-Sy\|\) and \(\|x-Ty\| \leq \|Tx-Ty\|\) for all \(x, y \in K\). Let \(\left\{\beta_n\right\}\) be a sequence in \(\left[0, 1\right]\) satisfying (i) \(\sum_{n = 1}^\infty \beta_n = \infty\); (ii) \(\lim_{n \rightarrow \infty} \beta_n = 0 \). For arbitrary \(x_0 \in K\), let \(\left\{x_n\right\}\) be a sequence iteratively defined by \(x_n = S y_n, y_n = \left(1 - \beta_n\right) x_{n - 1} + \beta_n T x_n, n \geq 1 \). Then the sequence \(\left\{x_n\right\}\) converges strongly to a common fixed point \(p\) of \(S\) and \(T\).
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 7022976
zbMATH Keywords: strong convergence
zbMATH Keywords: real Banach space
zbMATH Keywords: Lipschitz strongly pseudocontractive mappings
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/735673
OpenAlex ID: W2155019053
Wikidata QID: Q59041050
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description: scientific article; zbMATH DE number 7022976