On the local convergence of a Newton-type method in Banach spaces under a gamma—type condition


  • Ioannis K. Argyros Cameron University.
  • Saïd Hilout Poitiers University.




Banach space, Newton—type method, local convergence, gamma—type condition, Frechet—derivative, radius of convergence, espacio de Banach, método tipo Newton, convergencia local, condiciones tipo gamma, derivada de Frechet, radio de convergencia.


We provide a local convergence analysis for a Newton-type method to approximate a locally unique solution of an operator equation in Banach spaces. The local convergence of this method was studied in the elegant work by Werner in [11], using information on the domain of the operator. Here, we use information only at a point and a gamma-type condition [4], [10]. It turns out that our radius of convergence is larger, and more general than the corresponding one in [10]. More over the same can hold true when our radius is compared with the ones given in [9] and [11]. A numerical example is also provided.

Author Biographies

Ioannis K. Argyros, Cameron University.

Department of Mathematics Sciences.

Saïd Hilout, Poitiers University.

Laboratoire de Mathématiques et Applications.


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

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

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

[4] I.K. Argyros, Approximate solution of operator equations with applications, World Scientific Publ. Comp., New Jersey, U.S.A., (2005).

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

[6] J.M. Gutiérrez, M.A. Hernández, M.A. Salanova, Accessibility of solutions by Newton’s method, Inter. J. Comput. Math., 57, pp. 239—241, (1995).

[7] L.V. Kantorovich, G.P. Akilov, Functional analysis in normed spaces, Pergamon Press, Oxford, (1982).

[8] R.F. King, Tangent methods for nonlinear equations, Numer. Math., 18, pp. 298—304, (1972).

[9] W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Banach Center Publ., 3, pp. 129—142, (1975).

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

[11] W. Werner, Uber ein verfahren ordnung 1 + v2 zur Nullstellenbestimmung, Num. Math., 32, pp. 333—342, (1979).

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



How to Cite

I. K. Argyros and S. Hilout, “On the local convergence of a Newton-type method in Banach spaces under a gamma—type condition”, Proyecciones (Antofagasta, On line), vol. 27, no. 1, pp. 1-14, May 2017.




Most read articles by the same author(s)

1 2 > >>