## Abstract

We consider the solving of the equation \[x=\lambda D\left( x\right)+y,\] where \(E\) is a Banach space and \(D:E\rightarrow E\), \(\lambda\in \mathbb{R}\), \(y\in E\). We study the convergence of the iterations \[x_{n+1}=x_{n}-A\left( x_{n}\right)\left[ x_{n}-\lambda D\left( x_{n}\right) -y\right], \ n=0,1,…, \ x_{0}\in E,\] where \(A:E\rightarrow E\) is a linear mapping. We assume that the operator \(P\) given by \(P\left( x\right) =x-\lambda D\left( x\right) -y\) is two times Frechet differentiable, with \(P^{\prime}\left( x\right)=I-\lambda D^{\prime}\left( x\right)\), \(P^{\prime \prime}\left(x\right) =-\lambda D^{\prime \prime}\left( x\right) \). Under certain assumptions on boundedness of \(A\) and \(P\) we obtain convergence results for the considered sequences.

## Authors

Ion Păvăloiu

## Authors

### Original title (in French)

*La convergence de certaines méthodes itératives pour résoudre certaines equations operationnelles*

### English translation of the title

*The convergence of certain iterative methods for solving certain operator equations*

## Keywords

nonlinear operator equation; Banach space; iterative method;

## References

[1] L.V. Kantorovici, *O metodi Niutona* Trudi Mat. Inst. V.A. Steklova 28, 104–144 (1949).

[2] A. Diaconu, I. Pavaloiu, *Sur quelque methodes iteratives pour la resolution des equations op erationnelles,* Rev. Anal. Num´er. Theor. Approx., vol. 1, 45–61 (1972). (journal link )

[3] I. Pavaloiu, *Sur les procedes iteratifs a un ordre eleve de convergence*, Mathematica (Cluj), 12 (35) 1, 149–158 (1970).

## About this paper

##### Cite this paper as:

I. Păvăloiu, *La convergence de certaines méthodes itératives pour résoudre certaines equations operationnelles*, Seminar on functional analysis and numerical methods, Preprint no. 1 (1986), pp. 127-132 (in French).

##### Journal

Seminar on functional analysis and numerical methods,

Preprint

##### Publisher Name

“Babes-Bolyai” University,

Faculty of Mathematics,

Research Seminars

##### DOI

Not available yet.