Sur quelques méthodes itératives pour la résolution des équations opérationnelles

On some iterative methods for solving operational equations

Abstract

Let \(X,Y\) be two Banach spaces and \(P:X\rightarrow Y\) a nonlinear operator. We study the semilocal convergence of the Newton, chord and Steffensen methods for which the derivative \(P^{\prime}\left( x\right) \) or the divided differences from each iteration step are approximated by a sequence of operators obtained with the Schultz method:
\begin{equation}
\left\{
\begin{array}
[c]{l}%
x_{n+1}=x_{n}-A_{n}P\left( x_{n}\right) \\
A_{n+1}=A_{n}\left( 2E-\left[ x_{n},x_{n+1};P\right] A_{n}\right)
,\qquad n=0,1,\ldots
\end{array}
\right. \label{f.1.6}%
\end{equation}
and considering the Steffensen method:%
\begin{equation}
\left\{
\begin{array}
[c]{l}%
x_{n+1}=x_{n}-A_{n}P\left( x_{n}\right) \\
A_{n+1}=A_{n}\left( 2E-\left[ x_{n+1},Q\left( x_{n+1}\right) ;P\right]
A_{n}\right) ,\qquad n=0,1,\ldots
\end{array}
\right. \label{f.1.7}%
\end{equation}

MR0381294 (52 #2191)

Zbl 0363.65046