The convergence of certain iterative methods for solving certain operator equations

We consider the solving of the equation \[x=\lambda D\left( x\right)+y,\] where \(E\) is a Banach space and \(D:E\rightarrow E\), \(\lambda\in \mathbb{R}\), \(y\in E\). We study the convergence of the iterations \[x_{n+1}=x_{n}-A\left( x_{n}\right)\left[ x_{n}-\lambda D\left( x_{n}\right) -y\right],  \ n=0,1,…, \ x_{0}\in E,\] where \(A:E\rightarrow E\) is a linear mapping. We assume that the operator \(P\) given by \(P\left( x\right) =x-\lambda D\left( x\right) -y\) is two times Frechet differentiable, with \(P^{\prime}\left( x\right)=I-\lambda D^{\prime}\left( x\right)\), \(P^{\prime \prime}\left(x\right) =-\lambda D^{\prime \prime}\left( x\right) \). Under certain assumptions on boundedness of \(A\) and \(P\) we obtain convergence results for the considered sequences.

Ion Păvăloiu

Original title (in French)

La convergence de certaines méthodes itératives pour résoudre certaines equations operationnelles

English translation of the title

The convergence of certain iterative methods for solving certain operator equations

nonlinear operator equation; Banach space; iterative method;

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[2] A. Diaconu, I. Pavaloiu, Sur quelque methodes iteratives pour la resolution des equations op erationnelles, Rev. Anal. Num´er. Theor. Approx., vol. 1, 45–61 (1972). (journal link )

[3] I. Pavaloiu, Sur les procedes iteratifs a un ordre eleve de convergence, Mathematica (Cluj), 12 (35) 1, 149–158 (1970).

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I. Păvăloiu, La convergence de certaines méthodes itératives pour résoudre certaines equations operationnelles, Seminar on functional analysis and numerical methods, Preprint no. 1 (1986), pp. 127-132 (in French).

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