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.

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.

PDF

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.

Google Scholar Profile

Related Posts

Menu