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.
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José Antonio Ezquerro, Miguel Ángel Hernández-Verón, A Modification of the Lipschitz Condition in the Newton–Kantorovich Theorem. Z. Anal. Anwend. 35 (2016), no. 3, pp. 309–331DOI 10.4171/ZAA/1567