# On approximating the solutions of equations in metric spaces

## 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.

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.

## PDF

##### 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.