On the semilocal convergence of Newton–type methods, when the derivative is not continuously invertible

Ioannis K. Argyros; Saïd Hilout

doi:10.4067/S0719-06462011000300001

Authors

Ioannis K. Argyros Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA.
Saïd Hilout Laboratoire de Mathématiques et Applications, Poitiers University, Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179 86962 Futuroscope Chasseneuil Cedex, France.

DOI:

https://doi.org/10.4067/S0719-06462011000300001

Keywords:

Newton–type methods, Banach space, small divisors, non–invertible operators, semilocal convergence, Newton–Kantorovich–type hypothesis

Abstract

We provide a semilocal convergence analysis for Newton–type methods to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The Fr´echet– derivative of the operator involved is not necessarily continuous invertible. This way we extend the applicability of Newton–type methods [1]–[12]. We also provide weaker sufficient convergence conditions, and finer error bound on the distances involved (under the same computational cost) than [1]–[12], in some intersting cases. Numerical examples are also provided in this study.