# On the convergency of a Steffensen-type method

## Abstract

Let $$X_{1},X_{2}$$ be two Banach spaces, $$f:X_{1}\rightarrow X_{2}$$ a nonlinear mapping. We study the convergence of the Steffensen method for solving $$f\left( x\right) =0$$: $x_{n+1}=x_n-[x_{n},g(x_{n});f]^{-1}f(x_n), \quad n=1,…$ Under some simple Holder type conditions on the divided differences of order one of $$f$$, of the form $\|[y, u; f] − [x, y; f]\| ≤ l_1 \|x − u\| ^p + l_2 \|x − y\|^p + l_3 \|y − u\|^p$ we give some error estimations and we determine the convergence order.

Ion Păvăloiu

## Keywords

Steffensen type method; Holder conditions on divided differences; nonlinear equations in Banach spaces; iterative methods

## References

[1] Argyros, I.K., The secant method and fixed points of nonlinear operators, Mh. Math. 106, 85–94 (1988).

[2] Pavaloiu, I., Sur la methode de Steffensen pour la resolution des equations operationnelles non lineaires, Revue Roumaine des Mathematiques pures et appliquees, 1, XIII, 149–158 (1968).

[3] Pavaloiu, I., Introduction in the Theory of Approximation of Equations Solutions, Dacia Ed., Cluj-Napoca 1976 (in Romanian).

[4] Pavaloiu, I., Remarks on the secant method for the solution of nonlinear operatorial equations, Research Seminars, Seminar on Mathematical Analysis, Preprint no. 7, (1991), pp. 127 132.

[5] Ul’m, S., Ob obobscenie metod Steffensen dlea resenia nelineinih operatornih urnavnenii, Journal Vicisl., mat. i mat.-fiz. 4, 6 (1964).

## PDF

##### Cite this paper as:

I. Păvăloiu, On the convergency of a Steffensen-type method, Research Seminars, Seminar of Mathematical Analysis, Preprint no. 7 (1991), pp. 121-126.

##### Journal

Seminar on mathematical analysis,
Preprint

##### Publisher Name

“Babes-Bolyai” University,
Faculty of Mathematics,
Research Seminars

##### DOI

Not available yet.