On the local convergence of a two-step steffensen-type method for solving generalized equations
DOI:
https://doi.org/10.4067/S0716-09172008000300007Keywords:
Banach space, Steffensen’s method, generalized equation, Aubin continuity, Hölder continuity, radius of convergence, divided difference, set—valued map, espacio de Banach, método de Steffensen, ecuación generalizada, continuidad de Aubin.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.References
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[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, (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, (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, (2004).
[5] I. K. Argyros, On the approximation of strongly regular solutions for generalized equations, Comm. Appl. Nonlinear Anal., 12, pp. 97—107, (2005).
[6] I. K. Argyros, Approximate solution of operator equations with applications, World Scientific Publ. Comp., New Jersey, U. S. A., (2005).
[7] I. K. Argyros, An improved convergence analysis of a superquadratic method for solving generalized equations, Rev. Colombiana Math., 40, pp. 65—73, (2006).
[8] J. P. Aubin, H. Frankowska, Set—valued analysis, Birkhäuser, Boston, (1990).
[9] A. L. Dontchev, W. W. Hager, An inverse function theorem for set—valued maps, Proc. Amer. Math. Soc., 121, pp. 481—489, (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, (2006).
[11] M. A. Hernández, The Newton method for operators with Hölder continuous first derivative, J. Optim. Theory Appl., 109, pp. 631—648, (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, (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, (2007).
[14] S. Hilout, An uniparametric Newton—Steffensen—type methods for perturbed generalized equations, to appear in Advances in Nonlinear Variational Inequalities, (2007).
[15] S. Hilout, Convergence analysis of a family of Steffensen—type methods for solving generalized equations, submitted, (2007).
[16] S. Hilout, A. Piétrus, A semilocal convergence of a secant—type method for solving generalized equations, Positivity, 10, pp. 673—700, (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, (1994).
[18] S. M. Robinson, Generalized equations and their solutions, part I: basic theory, Math. Programming Study, 10, pp. 128—141, (1979).
[19] S. M. Robinson, Generalized equations and their solutions, part II: applications to nonlinear programming, Math. Programming Study, 19, pp. 200—221, (1982).
[20] R. T. Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Analysis 9, pp. 867—885, (1984).
[21] R. T. Rockafellar, R. J—B. Wets, Variational analysis, A Series of Comprehensives Studies in Mathematics, Springer, 317, (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).
[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, (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, (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, (2004).
[5] I. K. Argyros, On the approximation of strongly regular solutions for generalized equations, Comm. Appl. Nonlinear Anal., 12, pp. 97—107, (2005).
[6] I. K. Argyros, Approximate solution of operator equations with applications, World Scientific Publ. Comp., New Jersey, U. S. A., (2005).
[7] I. K. Argyros, An improved convergence analysis of a superquadratic method for solving generalized equations, Rev. Colombiana Math., 40, pp. 65—73, (2006).
[8] J. P. Aubin, H. Frankowska, Set—valued analysis, Birkhäuser, Boston, (1990).
[9] A. L. Dontchev, W. W. Hager, An inverse function theorem for set—valued maps, Proc. Amer. Math. Soc., 121, pp. 481—489, (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, (2006).
[11] M. A. Hernández, The Newton method for operators with Hölder continuous first derivative, J. Optim. Theory Appl., 109, pp. 631—648, (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, (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, (2007).
[14] S. Hilout, An uniparametric Newton—Steffensen—type methods for perturbed generalized equations, to appear in Advances in Nonlinear Variational Inequalities, (2007).
[15] S. Hilout, Convergence analysis of a family of Steffensen—type methods for solving generalized equations, submitted, (2007).
[16] S. Hilout, A. Piétrus, A semilocal convergence of a secant—type method for solving generalized equations, Positivity, 10, pp. 673—700, (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, (1994).
[18] S. M. Robinson, Generalized equations and their solutions, part I: basic theory, Math. Programming Study, 10, pp. 128—141, (1979).
[19] S. M. Robinson, Generalized equations and their solutions, part II: applications to nonlinear programming, Math. Programming Study, 19, pp. 200—221, (1982).
[20] R. T. Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Analysis 9, pp. 867—885, (1984).
[21] R. T. Rockafellar, R. J—B. Wets, Variational analysis, A Series of Comprehensives Studies in Mathematics, Springer, 317, (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).
Published
2017-04-06
How to Cite
[1]
I. K. Argyros and S. Hilout, “On the local convergence of a two-step steffensen-type method for solving generalized equations”, Proyecciones (Antofagasta, On line), vol. 27, no. 3, pp. 319-330, Apr. 2017.
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