On the Newton–Kantorovich hypothesis for solving equations

“…An important extension of the Kantorovich theorem was obtained recently by Argyros [1], [2], who used a combination of Lipschitz and center-Lipschitz conditions in place of the Lipschitz conditions used by Kantorovich. In the present paper, we will formulate and prove an extension of the Kantorovich theorem for the generalized equation (1.3). The depth and scope of this theorem is such that when we specialize it to nonlinear operator equations we get results that are weaker than the Kantorovich theorem.…”

Section: Introductionmentioning

confidence: 99%

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On the solution of generalized equations and variational inequalities

Argyros

Hilout

2011

Cubo

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Uko and Argyros provided in [18] 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 [18] 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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“…In this work, we use a similar one as that used by Argyros in [6] for the Lipschitz condition on the first derivative F , which consists of noticing that, as a consequence of the last condition being satisfied in Ω, we have, for the starting point x 0 , that:…”

Section: Introductionmentioning

confidence: 99%