# On some Steffensen-type iterative methods for a class of nonlinear equations

## Abstract

Consider the nonlinear equations $$H(x):=F(x)+G(x)=0$$, with $$F$$ differentiable and $$G$$ continuous, where $$F,G,H:X \rightarrow X$$ are nonlinear operators and $$X$$ is a Banach space.

The Newton method for solving the nonlinear equation $$H(x)=0$$ cannot be applied, and we propose an iterative method for solving this equation by combining the Newton method with the Steffensen method: $x_{k+1} = \big(F^\prime(x_k)+[x_k,\varphi(x_k);G]\big)^{-1}(F(x_k)+G(x_k)),$ where $$\varphi(x)=x-\lambda (F(x)+G(x))$$, $$\lambda >0$$ fixed.

The method is obtained by combining the Newton method for the differentiable part with the Steffensen method for the nondifferentiable part.

We show that the R-convergence order of this method is 2, the same as of the Newton method.

We provide some numerical examples and compare different methods for a nonlinear system in $$\mathbb{R}^2$$.

## Authors

E. Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)

## Keywords

nonlinear equation; Banach space; Newton method; Steffensen method; combined method; nondifferentiable mapping; nonsmooth mapping; r-convergence order.

## Cite this paper as:

E. Cătinaş, On some Steffensen-type iterative methods for a class of nonlinear equations, Rev. Anal. Numér. Théor. Approx., 24 (1995) nos. 1-2, pp. 37-43.

2457-6794

2501-059X

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