On the local convergence of a midpoint method in banach spaces under a gamma-type condition

Ioannis K. Argyros


In this study we are concerned with the problem of approximating a locally unique solution of an operator equation in a Banach space setting using the midpoint method, introduced by us in [5], [6]. Here, we use gamma-type condition to provide a local convergence analysis. Our results compare favorably with the relevant ones in [9], [11], [12]-[14]- In particular our radius of convergence is larger. Numerical examples are also provided.

Palabras clave

Midpoint method ; Banach space ; Gamma—type condition ; Radius of convergence ; Local convergence ; Fréchet—derivative.

Texto completo:



Allgower, F. L., Böhmer, K., Potra, F. A. and Rheinboldt, W.C. A mesh independence principle for operator equations and their discretizations, SIAM J. Numer. Anal. 23, pp. 160—169, (1986).

Amat, S. and Busquier, S. Convergence and numerical analysis of a family of two—step Steffensen’s method, Comput. and Math. with appl., 49, pp. 13—22, (2005).

Argyros, I. K. A unifying local—semilocal convergence analysis and applicastions for two—point Newton—like methods in Banach space, J. Math. Anal. Appl., 298, pp. 374—397; (2004).

Argyros, I. K. Approximate solution of operator equations with applications, World Scientific Publ. Co. Ptl. Ltd., Hackensack, N. J., U. S. A., (2005).

Argyros, I. K. and Chen, D. On the midpoint methods for solving equations in Banach spaces, Appl. Math. Letter, Vol. 5, No.4, pp. 7—9, (1992).

Argyros, I. K., and Chen, D. On the midpoint iterative method for solving nonlinear operator equations and applications to the solution of integral equations, Revue d’Analyse Numérique et de Théorie de l’Approximation, Tome 23, fasc. 2, pp. 139—152, (1994).

Brown, P. N. A local convergence theory for combined inexact— Newton/finite—difference projection methods, SIAM J. Numer. Anal. 24, pp. 407—434, (1987).

Gutierrez, J. M., Hernandez, M. A. and Salanova, M.A. Accessibility of solutions by Newton’s method, J. Comput. Math. 57, pp. 239—247, (1995).

Homeir, H. H. H. A modified Newton method for root finding with cubic convergence, J. Comput. Appl. Math., 157, pp. 227—230, (2003).

Kantorovich, L. V., and Akilov, G. P. Functional Analysis in normed spaces, Pergamon Press, Oxford, (1982).

Ozban, A. Y. Some new variants of Newton’s method, Appl. Math. Letters, 17, pp. 677—682, (2004).

Rheinboldt, W. C. An adaptive continuation process for solving systems of nonlinear equations, Banach Center Publ. 3, pp. 129—142, (1977).

Ypma, T. J. Local convergence of inexact Newton Methods,SIAM J. Numer, Anal., 21, pp. 583—590, (1984).

Zhao, F. and Wang, D. The theory of Smale’s point estimation and its applications, J. Comput. Appl. Math., 60, pp. 253—269, (1995).

DOI: http://dx.doi.org/10.4067/S0716-09172009000200005

Enlaces refback

  • No hay ningún enlace refback.