## 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.

## Authors

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].

Scanned paper.

## About this paper

##### 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).

##### Journal

Mathematica

##### Publisher Name

Academia R.S. Romania

##### DOI

Not available yet.

##### Print ISBN

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##### Online ISBN

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