CUBO A Mathematical Journal Vol.13, N° 01, (45-60). March 2011

CONTENTS

 

On the solution of generalized equations and variational inequalities

 

Ioannis K. Argyros and Saïd Hilout

Cameron University, Department of Mathematics Sciences, Universidad Nacional Autonoma de Mexico, Lawton, OK 73505, USA. email: iargyros@cameron.edu

Poitiers University, Laboratoire de Mathématiques et Applications, Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179, 86962 Futuroscope Chasseneuil Cedex, France. email: said.hilout@math.univ-poitiers.fr

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ABSTRACT

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.

Keywords: Generalized equation, variational inequality, nonlinear complementarity problem, nonlinear operator equation, Kantorovich theorem, generalized Newton's method, center-Lipschitz condition.

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RESUMEN

Uko 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.

AMS Subject Classification: 65K10, 65J99, 49M15, 49J53, 47J20, 47H04, 90C30, 90C33.

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Received: October 2009. Revised: November 2009.