New convergence criteria for the Newton-Kantorovich method and some applications to nonlinear integral equations
type: Item
instance of: scholarly article
title: New convergence criteria for the Newton-Kantorovich method and some applications to nonlinear integral equations
zbMATH Open document ID: 0923.65029
author: E. De Pascale
author: Peter P. Zabreĭko
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1999-10-15T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108458
review text: Starting from the fact that the Newton-Kantorovich method in the classical setting does not apply to Urysohn integral equations in some spaces (for instance, in \(L^p\)-spaces, for \(1<p<2\)), the main aim of the paper is to obtain convergence criteria for the Newton-Kantorovich method, by modifying the smoothness hypotheses in a suitable way. One merit of these new conditions is that they may be checked rather effectively for Urysohn integral equations. To emphasize this, the new convergence criteria are then systematically applied to nonlinear integral equations of Urysohn type, taking as the ambient Banach space \(X\) various particular spaces: the Chebyshev space \(C\), the Lebesgue space \(L^p\) and the Orlicz space \(LM\). It is also remarked that all the obtained results carry over the ideal spaces of measurable functions.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1279136
zbMATH Keywords: Newton-Kantorovich method
zbMATH Keywords: convergence
zbMATH Keywords: error estimates
zbMATH Keywords: nonlinear integral equations of Urysohn type
zbMATH Keywords: Banach space
reviewed by: Vasile Berinde
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1279136