Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2014; 51(2): 251-266

Printed March 1, 2014

https://doi.org/10.4134/JKMS.2014.51.2.251

Copyright © The Korean Mathematical Society.

On the "terra incognita'' for the Newton-Kantrovich method with applications

Ioannis Konstantinos Argyros, Yeol Je Cho, and Santhosh George

Cameron University, Gyeongsang National University, National Institute of Technology Karnataka

Abstract

In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before \cite{628-1}-\cite{628-12}, in some interesting cases, provided that the Fr\'echet-derivative of the operator involved is $p$-H\"older continuous ($p \in (0,1]$). Numerical examples involving two boundary value problems are also provided.

Keywords: Newton's method, Banach space, recurrent functions, H\"older continuity, Lipschitz continuity, semilocal convergence, Newton-Kantorovich hypothesis, differential equation

MSC numbers: 65H10, 65G99, 65J15, 47H17, 49M15