## Print version ISSN 0034-7426

### Rev.colomb.mat. vol.39 no.2 Bogotá July/Dec. 2005

A variant of Newton's method for generalized equations

Jean-Alexis Célia1 - Pietrus Alain2

1Université des Antilles et de la Guyane, France

e-mail: celia.jean-alexis@univ-ag.fr

2Laboratoire Analyse, Optimisation, Contrôle,Département de Mathématiques et Informatique. Université des Antilles et de la Guyane Campus de Fouillole, F-97159 Pointe-à-Pitre. France

e-mail: apietrus@univ-ag.fr

Abstract. In this article, we study a variant of Newton's method of the following form

0 ε f(xk) + hΔf(xkk)(xk+1 - xk) + F(xk+1)

where f is a function whose Frechet derivative is K-lipschitz, F is a set-valued map between two Banach spaces X and Y and h is a constant. We prove that this method is locally convergent to x* a solution of

0 ε f(x) + F(x),

if the set-valued map [f(x*) + hΔf(x*)(.- x*) + F(.)]-1 is Aubin continuous at (0, x*) and we also prove the stability of this method.

Keywords and phrases. Set-valued mapping, generalized equation, linear convergence, Aubin continuity.

2000 Mathematics Subject Classification. Primary: 49J53, 47H04. Secondary: 65K10.

Resumen. En este artículo estudiamos una variante del método de Newton de la forma

0 ε f(xk) + hΔf(xkk)(xk+1 - xk) + F(xk+1)

donde, f es una función cuya derivada de Frechet es K-lipschitz, F es una función entre dos espacios de Banach X y Y cuyos valores son conjuntos y h es una constante. Probamos que este método converge localmente a x*, una solución de

0 ε f(x) + F(x),

si la aplicación [f(x*) + hΔf(x*)(.- x*) + F(.)]-1 es Aubin continua en (0, x*). También probamos la estabilidad del método.

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(Recibido en julio de 2005. Aceptado en agosto de 2005)