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)