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

## 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.

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).

## PDF

##### 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.