Solving the systems of operator equations by iterative methods

Let \(X,Y\) be two Banach spaces and \(Z=X\times Y\). We consider the system of nonlinear equations \[x=\varphi \left( x,y\right),\\ y=\psi \left(x,y\right),\] where \(\varphi:Z\rightarrow X\), \(\psi:Z\rightarrow Y\). Assuming that \(\varphi\) and \(\psi \ \) satisfy Lipschitz conditions we study the convergence of the Gauss-Seidel type method \[x_{n}=\varphi \left(x_{n-1},y_{n-1}\right), \\ y_{n}=\psi \left( x_{n},y_{n-1}\right) .\] The obtained result is applied to the solving of a linear system, for which the matrix is splitted in four submatrices. We illustrate the obtained results for some numerical examples.

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Original title (in French)

La résolution des systèmes d’équations opérationnelles à l’aide des méthodes itératives

English translation of the title

Solving the systems of operator equations by iterative methods

Gauss-Seidel method, system of equations in Banach spaces, linear systems

I. Păvăloiu, La résolution des systèmes d’équations opérationnelles à l’aide des méthodes itératives, Mathematica, 11(34) (1969), pp. 137-141 (in French).

Scanned paper (in French).

PDF-Latex version of the paper. (English translation)

[1] I. Pavaloiu, Observatii asupra rezolvarii sistemelor de ecuatii cu ajutorul procedeelor iterative, Studii si Cercetari Matematice, 19 (1967) no. 9, 1289–1298 (in Romanian) [English translation of the title: Remarks on solving the systems of equations by iterative methods].

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