## Abstract

Let \(\left( X,\rho \right)\) be a complete metric space, \(f:X\rightarrow X\) a nonlinear mapping. In order to solve the equation \(x=f\left( x\right) \) we consider a multistep method \[x_{n+k+1}=G(x_{n},x_{n+1},…,x_{n+k}), \quad n=1,2,… \] generated by a mapping \(G:X^{k+1}\rightarrow X\), whose diagonal restriction coincides with \(f\): \(G(x,…,x)=f(x)\). Under Lipschitz assumption on \(G\) we determine the algebraic equation whose unique positive solution leads to the convergence order of the iterations. We also study the case when the operator \(G\) replaced by an approximation of it.

## Authors

Ion Păvăloiu

## Title

### Original title (in French)

*Sur l’approximation des racines des equations dans une espace métrique*

### English translation of the title

*On approximating the solutions of equations in metric spaces*

## Keywords

multistep iterative methods; convergence; successive approximations

## References

[1] I. Pavaloiu, I., Serb, *Sur des methodes iteratives optimales*, Research Seminars, Seminar on Functional Analysis and Numerical Methods, Preprint Nr.1 (1983), 175–182.

[2] I.A. Rus, *An iterative method for the solution of the equation x = f (x, x, . . . , x)*, Anal. Num´er. Theor. Approx., 10 (1981), 95–100.

[3] Weinischke, J.H., *Uber eine klasse von Iterationverfahren*, Numerische Mathematik 6 (1964), 395–404.

Scanned paper.

## About this paper

##### Cite this paper as:

I. Păvăloiu, *Sur l’approximation des racines des equations dans une espace métrique*, Seminar on functional analysis and numerical methods, Preprint no. 1 (1989), pp. 95-104 (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.