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Zeitschrift für Analysis und ihre Anwendungen

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Volume 35, Issue 3, 2016, pp. 309–331
DOI: 10.4171/ZAA/1567

Published online: 2016-09-13

A Modification of the Lipschitz Condition in the Newton–Kantorovich Theorem

José Antonio Ezquerro[1] and Miguel Ángel Hernández-Verón[2]

(1) Universidad de la Rioja, Logrono, Spain
(2) Universidad de la Rioja, Logrono - La Rioja, Spain

We analyse the semilocal convergence of Newton’s method in Banach spaces under a modification of the classic Lipschitz condition on the first derivative of the operator involved in Kantorovich’s theory. For this, we use a technique based on recurrence relations instead of the well-known majorant principle of Kantorovich. We illustrate this analysis with an application where a Hammerstein nonlinear integral equation of the second kind is involved.

Keywords: Newton’s method, semilocal convergence, the Newton-Kantorovich theorem, recurrence relations, error estimates, order of convergence, nonlinear integral equation

Ezquerro José Antonio, Hernández-Verón Miguel Ángel: A Modification of the Lipschitz Condition in the Newton–Kantorovich Theorem. Z. Anal. Anwend. 35 (2016), 309-331. doi: 10.4171/ZAA/1567