# Solving the systems of operator equations by iterative methods

## Abstract

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

## Title

### 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

## Keywords

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

## References

[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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##### Cite this paper as:

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

Mathematica

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