We provide a local as well as a semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F (x) + G(x) = 0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton-Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton-Kantorovich theorem. 2004 Elsevier Inc. All rights reserved.
A new technique, using the contraction mapping theorem, for solving quadratic equations in Banach space is introduced.The results are then applied to solve Chandrasekhar's integral equation and related equations without the usual positivity assumptions.
IntroductionConsider the equationin a Banach space X over the field S of real numbers, where B : X*X •*• X is a bounded bilinear operator and y e X is fixed. We prove a consequence of the contraction mapping principle which can be used to prove existence and uniqueness of solutions of (1). For the special cases of Chandrasekhar's equation :
Uko and Argyros provided in  a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form f (u)+g(u) ∋ 0, where f is a Fréchet-differentiable function, and g is a maximal monotone operator defined on a Hilbert space. The sufficient convergence conditions are weaker than the corresponding ones given in the literature for the Kantorovich theorem on a Hilbert space. However, the convergence was shown to be only linear.In this study, we show under the same conditions, the quadratic instead of the linear convergenve of the generalized Newton iteration involved.
RESUMENUko y Argyros estudian en  un teorema tipo-Kantorovich en el existencia y unicidad de la solución de una ecuación generalizada de la forma f (u) + g(u) ∋ 0, donde f es una función Fréchet-diferenciable, y g es un operador monotono máximo definido en un espacio de Hilbert. Las condiciones de convergencia suficientes son más débiles que los correspondientemente dadas en la literatura para el teorema de Kantorovich en un espacio de Hilbert. Sin embargo, la convergencia ha demostrado ser sólo lineal.En este estudio, mostramos en las mismas condiciones, la ecuación cuadrática en lugar de la lineal convergente de la iteración generalizada de Newton involucradas.
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