Sur l'approximation des solutions des équations à l'aide des suites à éléments dans un espaces de Banach

On the approximation of solutions of equations using sequences with elements in a Banach space

Abstract

Let $X$ be a Banach space, $Y$ a normed space $P:X\to Y$ a nonlinear operator and the equation $P\left(x\right)=0$ with solution $x^{\ast }$. Consider $\mathrm{\Sigma }:={\left(x_n\right)}_{n\ge 0}$ a sequence from $X$ and define the convergence order of $\mathrm{\Sigma }$ with respect to the solution of equation $P\left(x\right)=0$. We give a general result with sufficient conditions such that the sequence $\mathrm{\Sigma }$ converge to the solution $x^{\ast }$ with a given convergence order.