A semilocal convergence of a secant-type method for solving generalized equations
type: Item
instance of: scholarly article
title: A semilocal convergence of a secant-type method for solving generalized equations
zbMATH Open document ID: 1118.47052
author: Alain Pietrus
author: Saïd Hilout
published in: Positivity
publication date: 2007-02-13T00:00:00Z
review text: Let \(X,Y\) be two Banach spaces and let \(f:X\rightarrow Y\) be continuous and \(G:X\rightarrow \mathcal{P}(Y)\) be a set-valued map with closed graph. In order to solve the inclusion \[ 0\in f(x)+G(x), \] the authors consider the iterative method defined by \(x_0,x_1\in X\) and \[ y_k=\alpha x_k+(1-\alpha)x_{k-1},\;0\in f(x_k)+[y_k,x_k;f](x_{k+1}-x_k)+G(x_{k+1}),\;k\geq 1, \] where \(\alpha\) is fixed in \([0,1)\) and \([y_k,x_k;f]\) denotes a first order divided difference of \(f\) on the points \(y_k\) and \(x_k\). The main result of the paper (Theorem~3.1) is a semilocal convergence theorem for the iterative process considered.
reviewed by: Vasile Berinde
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 49J53
zbMATH DE Number: 5125397
zbMATH Keywords: Banach space
zbMATH Keywords: set-valued map
zbMATH Keywords: divided difference
zbMATH Keywords: iterative method
zbMATH Keywords: semilocal convergence theorem
zbMATH Keywords: metric regularity
zbMATH Keywords: calmness properties
zbMATH Keywords: superlinear convergence
zbMATH Keywords: Aubin continuity
MaRDI profile type: Publication
full work available at URL: s11117-006-0044-3
OpenAlex ID: W2007533038
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DOI: 10.1007/S11117-006-0044-3
description: scientific article; zbMATH DE number 5125397