On the convergence of a certain class of iterative procedures under relaxed conditions with applications
type: Item
instance of: scholarly article
title: On the convergence of a certain class of iterative procedures under relaxed conditions with applications
zbMATH Open document ID: 0939.65087
DOI: 10.1016/S0377-0427(98)00060-0
author: Ioannis Konstantinos Argyros
published in: Journal of Computational and Applied Mathematics
publication date: 1998-01-01T00:00:00Z
review text: The paper deals with a locally unique solution \(x^*\) of the equation \(F(x)+ G(x)= 0\) in Banach spaces, where \(F\) is Fréchet differentiable and \(G\) is continuous. Inexact Newton-like methods of the form \[ x_{n+1}= x_n- A^{-1}(F(x_n)+ G(x_n))- z_n,\quad z_n= z(x_n)\tag{1} \] are studied. At each step the inverse of the same linear operator is used. The vectors \(z_n\) are chosen so that iteration \(\{x_n\}\) converges to \(x^*\). For \(A= F'(x_0)\), \(G(x)= 0\) and \(z_n= 0\) for all \(n\) one obtains the modified Newton's method. The iterative procedure is studied under natural assumptions. The assumed Ptak-like conditions are weaker than earlier ones. A semilocal convergence result is presented based on the contraction mapping principle. The result is applied to solve special nonlinear integral equation of Uryson type with the integral operator \[ [F(x)](t)= x(t)- \int^1_0 p_1(t) p_2(s)p_3(x(s)) ds \] appearing in elasticity theory. This equation pertains to the buckling of a thin shallow spherical shell clamped at the edge and under uniform external pressure.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 74K25
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 65R20
zbMATH DE Number: 1326394
zbMATH Keywords: inexact Newton-like methods
zbMATH Keywords: Banach spaces
zbMATH Keywords: Ptak-like conditions
zbMATH Keywords: convergence
zbMATH Keywords: nonlinear integral equation of Uryson type
reviewed by: Werner H. Schmidt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1326394