Abstract
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.
Authors
Ion Păvăloiu
Authors
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
Keywords
nonlinear operator equation; Banach space; iterative method;
References
[1] L.V. Kantorovici, O metodi Niutona Trudi Mat. Inst. V.A. Steklova 28, 104–144 (1949).
[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).
About this paper
Cite this paper as:
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).
Journal
Seminar on functional analysis and numerical methods,
Preprint
Publisher Name
“Babes-Bolyai” University,
Faculty of Mathematics,
Research Seminars
DOI
Not available yet.