A general semilocal convergence result for Newton's method under centered conditions for the second derivative
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, p. 149-167

From Kantorovich's theory we present a semilocal convergence result for Newton's method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newton's method. We illustrate this study with applications to nonlinear integral equations of mixed Hammerstein type.

DOI : https://doi.org/10.1051/m2an/2012026
Classification:  45G10,  47H99,  65J15
Keywords: Newton's method, the Newton-Kantorovich theorem, semilocal convergence, majorizing sequence, a priori error estimates, Hammerstein's integral equation
@article{M2AN_2013__47_1_149_0,
author = {Ezquerro, Jos\'e Antonio and Gonz\'alez, Daniel and Hern\'andez, Miguel \'Angel},
title = {A general semilocal convergence result for Newton's method under centered conditions for the second derivative},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {47},
number = {1},
year = {2013},
pages = {149-167},
doi = {10.1051/m2an/2012026},
zbl = {1271.65092},
mrnumber = {2968699},
language = {en},
url = {http://www.numdam.org/item/M2AN_2013__47_1_149_0}
}

Ezquerro, José Antonio; González, Daniel; Hernández, Miguel Ángel. A general semilocal convergence result for Newton's method under centered conditions for the second derivative. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, pp. 149-167. doi : 10.1051/m2an/2012026. http://www.numdam.org/item/M2AN_2013__47_1_149_0/

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