AOC preprint number 03/06 :

On some iterative method for solving non linear equations under mild differentiability conditions

G. Mophou, A. Piétrus

Laboratoire AOC, Département de Mathématiques et Informatique, Université des Antilles et de la Guyane, Campus de Fouillole, F-97159 Pointe-à-Pitre Cedex, France; e-mail: Gisele.Mophou@univ-ag.fr, apietrus@univ-ag.fr

Keywords : Divided differences of operators, Hölder condition, Newton's method, the Secant method.

Abstract : In this paper, we show the convergence of the sequence $x_{n+1} = x_n - \bigl (f'(x_n) + [x_{n-1},x_n;g]\bigr )^{-1}(f(x_n) + g(x_n)), x_0,x_1 \in X, n=1,2, \ldots$ to a solution $x^{*}$ of $f(x)+g(x)=0$ where $X$ is a Banach space, $f,g : X \longrightarrow X$, $f$ is differentiable and $g$ is continuous when $f'$ satisfy a Hölder condition.

AMS classification scheme numbers : 47J25, 65J15.