Local convergence of generalized Mann iteration
type: Item
instance of: scholarly article
title: Local convergence of generalized Mann iteration
zbMATH Open document ID: 1386.65151
published in: Numerical Algorithms
publication date: 2018-01-12T00:00:00Z
full work available at URL: Local L.pdf
review text: The article deals with generalized Mann iterations \[ x_{n+1} = (I - D_n)x_n + D_nT(x_n), \qquad n = 0,1,2,\ldots,\eqno(1) \] for the approximative construction of fixed points of a nonlinear operator \(T:\;C \to H\), where \(H\) is a real Hilbert space, \(C\) an open subset of \(H\), \(\{D_n\} \subset L(H)\) is a generalized control sequence. Let \(\text{Fix}\, T \neq \emptyset\). The main result is the following: the sequence (1), with starting point in the ball \(B_r \subset C\), remains in \(B_r\) and converges weakly to a fixed point of \(T\), provided that: {\parindent=0.7cm \begin{itemize}\item[(i)] \(T\) is demiclosed at zero on \(B_r \subset C\); \item[(ii)] \(D_x\) is invertible and \(\|D_x^{-1}\| \leq M\) (\(x \in B_r\)); \item[(iii)] \(T\) is generalized demicontractive: \[ \langle D_x(x - T(x)),x - p \rangle \geq \lambda \|D_x(x - T(x))\|^2, \qquad x \in B_r,\quad p \in \operatorname{Fix}T; \] \end{itemize}} with \(\lambda > 0.5\) (\(D_x\) is the control linear mapping, \(D:\;C \to L(C)\)); \noindent moreover, the sequence (1) converges strongly to the unique fixed point \(p\) of \(T\) if, in addition, \(T\) is quasi-expansive on \(B_r\): \[ \|Tx - p\| \geq \frac{1 - \rho}{\rho} k\|x - p\|, \qquad x \in B_r,\quad p \in \operatorname{Fix}T. \] At the end of the article, some numerical experiments are presented. There are some vague places in the article. In particular, the relations between \(D_n\) and \(D_x\) is not explained; the definition of the constant \(\rho\) is incomprehensible.
reviewed by: Peter P. Zabreĭko
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 6825425
zbMATH Keywords: fixed point
zbMATH Keywords: iterative methods
zbMATH Keywords: Mann iteration
zbMATH Keywords: local convergence
author: Şt. Măruşter
author: L. Maruster
MaRDI profile type: Publication
OpenAlex ID: W2592179347
cites work: Local convergence radius for the Mann-type iteration
cites work: An optimization of Chebyshev's method
cites work: Q4157882
cites work: On the local convergence of a third order family of iterative processes
cites work: Q3237489
cites work: A Note on the Convergence of Newton’s Method
cites work: Convergence and Complexity of Newton Iteration for Operator Equations
cites work: Q4356579
cites work: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space
cites work: Concerning the ``terra incognita'' between convergence regions of two Newton methods
cites work: Q3397208
cites work: Local convergence of Newton's method in Banach space from the viewpoint of the majorant principle
cites work: On the local convergence of a deformed Newton's method under Argyros-type condition
cites work: Convergence of Newton's method and uniqueness of the solution of equations in Banach space
cites work: Q4602630
cites work: Approximating Fixed Points of Nonexpansive Mappings
cites work: Mean value iteration of nonexpansive mappings in a Banach space
cites work: The Solution by Iteration of Nonlinear Equations in Hilbert Spaces
cites work: Estimating the radius of an attraction ball
DOI: 10.1007/S11075-017-0289-X
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description: scientific article; zbMATH DE number 6825425