Une variante de la méthode de Newton

A variant of the Newton method

Abstract

Let $X$ be a Banach space, $Y$ a normed space, $P:X\to Y$ a nonlinear operator and the equation $P\left(x\right)=0$. The first order derivative is approximated by the partial derivative of an operator of two variables, whose diagonal restrictive coincides with $P$. We give a semilocal convergence result for the considered method.