On some iterative methods for solving operator equations

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}

Adrian Diaconu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Originnal title (in French)

Sur quelques méthods itératives pour la résolution des equations operationnelles

English translation of the title

On some iterative methods for solving operator equations

Newton method; chord method; Steffensen method; Schultz method; semilocal convergence

Scanned version (on the journal website).

PDF-Latex version of the paper.

A. Diaconu, I. Păvăloiu, Sur quelques méthods itératives pour la résolution des equations operationnelles, Rev. Anal. Numér. Théor. Approx., 1 (1972), pp. 45-61, https://doi.org/10.33993/jnaat11-3 (in French).

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