Abstract
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.
Authors
Adrian Diaconu, Ion Păvăloiu
Title
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
Keywords
semilocal convergence; nonlinear operator equation; iterative method; Schultz method
References
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About this paper
Cite this paper as:
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).
Journal
Revista de Analiza Numerica si Teoria Aproximaţiei
Publisher Name
Academia Republicii S.R.
DOI
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