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Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 115-141, 2013
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Extending the applicability of Newton's method using nondiscrete induction

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Ioannis K. Argyros, Saïd Hilout

* Ioannis K. Argyros*, Cameron University, Department of Mathematics Sciences, Lawton, OK 73505, USA, e-mail: ` iargyros@cameron.edu`; * Saïd Hilout*, Poitiers University, Laboratoire de Mathématiques et Applications, Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179, 86962 Futuroscope Chasseneuil Cedex, France, e-mail: ` said.hilout@math.univ-poitiers.fr`

**Abstract:** We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Ptak. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63-72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.

**Keywords:** Newton's method, Banach space, rate of convergence, semilocal convergence, nondiscrete mathematical induction, estimate function

**Classification (MSC 2010):** 65H10, 65G99, 49M15

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