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.
Ioannis Konstantinos Argyros, Yeol Je Cho, and Santhosh George
Cameron University, Gyeongsang National University, National Institute of Technology Karnataka
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
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