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Revista Colombiana de Matemáticas
versão impressa ISSN 0034-7426
Rev.colomb.mat. v.39 n.2 Bogotá jul./dez. 2005
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.
FULL TEXT IN PDF
[1] J-P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87-111. [ Links ]
[2] J-P. Aubin & H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990. [ Links ]
[3] A. L. Dontchev, Local convergence of the newton method for generalized equations, C. R. Acad. Sc. 1 (1996), 327-329. [ Links ]
[4] A. L. Dontchev, Uniform convergence of the newton method for Aubin continuous maps, Serdica. Math. J. 22 (1996), 385-398. [ Links ]
[5] A. L. Dontchev & W. W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), 481-489. [ Links ]
[6] A. D. Ioffe & V. M. Tikhomirov, Theory of Extremal Problems, North Holland, Amsterdam, 1979. [ Links ]
[7] J. M. Ortega & W. C Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New-York and London, 1970. [ Links ]
[8] A. M Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Academic Press, New-York and London, 1970. [ Links ]
[9] A. Pietrus, Does Newton's method for set-valued maps converges uniformly in mild differentiability context?, Rev. Colombiana Mat. 32 (2000), 49-56. [ Links ]
[10] A. Pietrus, Generalized equations under mild differentiability conditions, Rev. S. Acad. Cienc. Exact. Fis. Nat. 94 no. 1 (2000), 15-18. [ Links ]
[11] R. T. Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Anal. 9 (1984) 867-885. [ Links ]
[12] R. T. Rockafellar & R. Wets, Variational analysis, Ser. Com. Stu. Math., Springer, 1998. [ Links ]
(Recibido en julio de 2005. Aceptado en agosto de 2005)