On a Steffensen type method for solving nonlinear operator equations

Let \(X\) be a Banach space, \(Y\) a normed space and the nonlinear operator equation \(P\left( x\right) =0\), where \(P:X\rightarrow Y\). We consider two operators \(Q_{1},Q_{2}:X\rightarrow X\) attached to \(P\) and we study the convergence of the Steffensen type method \[x_{n+1}=Q_1(x_n)-[Q_1( x_n), Q_2( x_n);P]^{-1}P(Q_1(x_n)). \] We give some conditions ensuring the convergence of this sequence to the solution and we obtain the convergence order of the sequence in terms of the convergence orders of \(Q_{1}\) and \(Q_{2}\).

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Original title (in French)

Sur une méthode de type Steffensen utilisée pour la résolution des equations operationnelles non-linéaires

English translation of the title

On a Steffensen type method for solving nonlinear operator equations

Steffensen type method; Banach space; iterative method; convergence order

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PDF-LaTeX version of the paper.

I. Păvăloiu, Sur une méthode de type Steffensen utilisée pour la résolution des equations operationnelles non-linéaires, Seminar on functional analysis and numerical methods, Preprint no. 1 (1989), pp. 105-110 (in French).

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