Iterative sequence for asymptotically demicontractive maps in Banach spaces
type: Item
instance of: scholarly article
title: Iterative sequence for asymptotically demicontractive maps in Banach spaces
zbMATH Open document ID: 1062.47070
author: Chika Moore
author: B. V. C. Nnoli
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-02-22T00:00:00Z
review text: Let \(E\) be a real Banach space. A mapping \(T: E\to E\) is called (i) uniformly \(L\)-Lipschitzian if there exists a constant \(L> 0\) such that for all \(x,y\in E\) and \(n\in\mathbb{N}\), \(\| T^nx- T^ny\|\leq L\| x-y\|\), (ii) asymptotically demicontractive if there exists a sequence \(\{a_n\}^\infty_{n=0}\) with \(\lim_{n\to\infty} a_n= 1\) such that \(\| T^nx- x^*\|^2\leq a^2_n\| x- x^*\|^2+ k\| x- T^n x\|^2\) for some \(k\in [0,1)\) and for all \(x\in E\), \(x^*\in F(T)\) and \(n\in\mathbb{N}\), where \(F(T):= \{x\in E: x= Tx\}\) denotes the set of fixed points of \(T\). In this paper, the authors prove necessary and sufficient conditions for the strong convergence of the Mann iteration sequence to a fixed point of an asymptotically demicontractive map in real Banach spaces. The main result goes as follows. Let \(E\) be a real Banach space and \(T: E\to E\) be a uniformly \(L\)-Lipschitzian asymptotically demicontractive map with sequence \(\{a_n\}\) and \(F(T)\neq \varnothing\). Let \(\{c_n\}_{n\geq 0}\subset [0,1]\) be a real sequence such that \(\sum_{n\geq 0} c^2_n<\infty\) and \(\sum_{n\geq 0} c_n(a^2_n- 1)< \infty\). Let \(\{x_n\}_{n\geq 0}\) be the sequence generated from an arbitrary \(x_0\in E\) by \(x_{n+1}= (1- c_n) x_n+ c_n T^n x_n\), \(n\geq 0\). Then \(\{x_n\}_{n\geq 0}\) converges strongly (i) to a fixed point of \(T\) if and only if \(\liminf_{n\to\infty}\, d(x_n, F(T))= 0\), (ii) to \(x^*\in F(T)\) if and only if there exists an infinite subsequence of \(\{x_n\}_{n\geq 0}\) which converges strongly to \(x^*\in F(T)\).
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 2136668
zbMATH Keywords: uniformly \(L\)-Lipschitzian
zbMATH Keywords: fixed point
zbMATH Keywords: asymptotically hemicontractive maps
zbMATH Keywords: strong convergence
zbMATH Keywords: Banach spaces
reviewed by: T. D. Narang
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.03.006
OpenAlex ID: W2081139462
cites work: Iterative sequences for asymptotically quasi-nonexpansive mappings.
cites work: Iterative sequences for asymptotically quasi-nonexpansive mappings with error member
cites work: On Naimpally and Singh's open questions
cites work: Iterative approximation of fixed points of asymptotically demicontractive mappings
cites work: Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings
cites work: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process
DOI: 10.1016/J.JMAA.2004.03.006
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description: scientific article; zbMATH DE number 2136668