Proyecciones Journal of Mathematics
Vol. 27, Nº 3, pp. 319-330, December 2008.
Universidad Católica del Norte
Antofagasta - Chile


ON THE LOCAL CONVERGENCE OF A TWO-STEP STEFFENSEN-TYPE METHOD FOR SOLVING GENERALIZED EQUATIONS


IOANNIS K. ARGYROS ¹
SAÏD HILOUT <sup>2

1</sup> Cameron university, U. S. A.
² University Morocco, Morocco.


Correspondencia a:

Abstract
We use a two-step Steffensen-type method [1], [2], [4], [6], [13]-[16] to solve a generalized equation in a Banach space setting under Hölder-type conditions introduced by us in [2], [6] for nonlinear equations. Using some ideas given in [4], [6] for nonlinear equations, we provide a local convergence analysis with the following advantages over related [13]-[16]: finer error bounds on the distances involved, and a larger radius of convergence. An application is also provided.


Key words : Banach space, Steffensen´s method, generalized equation, Aubin continuity, Hölder continuity, radius of convergence, divided difference, set- valued map.

AMS Subject Classification : 65K10, 65G99, 47H04, 49M15.

REFERENCES
[1] S. Amat, S. Busquier, Convergence and numerical analysis of a family of two- step Steffensen´s methods, Comput. and Math. with Appl., 49, pp. 13-22,         [ Links ] (2005).
[2] I. K. Argyros, A new convergence theorem for Steffensen´s method on Banach spaces and applications, Southwest J. of Pure and Appl. Math., 01, pp. 23-29,         [ Links ] (1997).
[3] I. K. Argyros, On the solution of generalized equations using m (m = 2) Fréchet differential operators, Comm. Appl. Nonlinear Anal., 09, pp. 85-89,         [ Links ] (2002).
[4] I. K. Argyros, A unifying local-semilocal convergence analysis and applications for Newton-like methods, J. Math. Anal. Appl., 298, pp. 374-397,         [ Links ] (2004).
[5] I. K. Argyros, On the approximation of strongly regular solutions for generalized equations, Comm. Appl. Nonlinear Anal., 12, pp. 97-107,         [ Links ] (2005).
[6] I. K. Argyros, Approximate solution of operator equations with applications, World Scientific Publ. Comp., New Jersey, U. S. A.,         [ Links ] (2005).
[7] I. K. Argyros, An improved convergence analysis of a superquadratic method for solving generalized equations, Rev. Colombiana Math., 40, pp. 65-73,         [ Links ] (2006).
[8] J. P. Aubin, H. Frankowska, Set-valued analysis, Birkhäuser, Boston,         [ Links ] (1990).
[9] A. L. Dontchev, W. W. Hager, An inverse function theorem for set-valued maps, Proc. Amer. Math. Soc., 121, pp. 481-489,         [ Links ] (1994).
[10] M. H. Geoffroy, S. Hilout, A. Piétrus, Stability of a cubically convergent method for generalized equations, Set-Valued Anal., 14, pp. 41-54,         [ Links ] (2006).
[11] M. A. Hernández, The Newton method for operators with Hölder continuous first derivative, J. Optim. Theory Appl., 109, pp. 631-648,         [ Links ] (2001).
[12] M. A. Hernández, M. J. Rubio, Semilocal convergence of the secant method under mild convergence conditions of differentiability, Comput. and Math. with Appl., 44, pp. 277-285,         [ Links ] (2002).
[13] S. Hilout, Superlinear convergence of a family of two-step Steffensen-type method for generalized equations, to appear in International Journal of Pure and Applied Mathematics,         [ Links ] (2007).
[14] S. Hilout, An uniparametric Newton-Steffensen-type methods for perturbed generalized equations, to appear in Advances in Nonlinear Variational Inequalities,         [ Links ] (2007).
[15] S. Hilout, Convergence analysis of a family of Steffensen-type methods for solving generalized equations, submitted,         [ Links ] (2007).
[16] S. Hilout, A. Pi´etrus, A semilocal convergence of a secant-type method for solving generalized equations, Positivity, 10, pp. 673-700,         [ Links ] (2006).
[17] B. S. Mordukhovich, Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, Trans. Amer. Math. Soc., 343, pp. 609-657,         [ Links ] (1994).
[18] S. M. Robinson, Generalized equations and their solutions, part I: basic theory, Math. Programming Study, 10, pp. 128-141,         [ Links ] (1979).
[19] S. M. Robinson, Generalized equations and their solutions, part II: applications to nonlinear programming, Math. Programming Study, 19, pp. 200-221,         [ Links ] (1982).
[20] R. T. Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Analysis 9, pp. 867-885,         [ Links ] (1984).
[21] R. T. Rockafellar, R. J-B. Wets, Variational analysis, A Series of Comprehensives Studies in Mathematics, Springer, 317,         [ Links ] (1998).
[22] J. D. Wu, J. W. Luo, S. J. Lu, A unified convergence theorem, Acta Mathematica Sinica, English Series, Vol. 21, (2), pp. 315-322, (2005).        [ Links ]

IOANNIS K. ARGYROS
Cameron university
Department of Mathematics Sciences
Lawton, OK 73505
U. S. A.
e-mail : ioannisa@cameron.edu

SAÏD HILOUT
Faculty of Science & Technics of Béni-Mellal
Department of Applied Mathematics & Computation
B. P. 523, Béni-Mellal 23000,
Morocco
e-mail : said_hilout@yahoo.fr

Received : October 2008. Accepted : November 2008