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 \(P:X\rightarrow Y\) a nonlinear operator and the equation \(P\left( x\right) =0\) with solution \(x^{\ast}\). Consider \(\Sigma:=\left( x_{n}\right) _{n\geq0}\) a sequence from \(X\) and define the convergence order of \(\Sigma\) with respect to the solution of equation \(P\left( x\right) =0\). We give a general result with sufficient conditions such that the sequence \(\Sigma\) converge to the solution \(x^{\ast}\) with a given convergence order.