On some iterative methods for solving nonlinear operator equations

Let \(X,Y\) be two Banach spaces and \(P:X\rightarrow Y\) a nonlinear operator. For solving the equation \(P\left( x\right) =0\) we consider the following iterations: \begin{align*} x_{n+1} & =x_{n}-A_{n}P\left( x_{n}\right) ;\\ A_{n+1} & =A_{n}\left( 2E-P^{\prime}\left( x_{n-1}\right) A_{n}\right),\ \ n=0,1,…,\end{align*}\(E\) the identity operator. We obtain sufficient conditions such that the sequences \(\left( x_{n}\right)_{n\geq0}\), \(\left( A_{n}\right) _{n\geq0}\) to converge to the solution \(x^{\ast}\) of the equation \(P\left( x\right) =0\), respectively to \(P^{\prime}\left( x^{\ast}\right)^{-1}\). The above method has the advantage that if does not require the computation of \(F^{\prime}\left( x_{k}\right)\) at each iteration step, and also that it does not require the solving of a linear system at each step.

Adrian Diaconu, Ion Păvăloiu

Original title (in Romanian)

Asupra unor metode iterative pentru rezolvarea ecuaţiilor operaţionale neliniare

English translation of the title

On some iterative methods for solving nonlinear operator equations

semilocal convergence; nonlinear operator equation; iterative method; Schultz method

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Scanned paper.

A. Diaconu, I. Păvăloiu, Asupra unor metode iterative pentru rezolvarea ecuaţiilor operaţionale neliniare, Rev. Anal. Numer. Teoria Aproximaţiei, 2 (1973) no. 1, pp. 61-69 (in Romanian).

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