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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 19, Issue 4, 2000, pp. 1017–1034**

**DOI: 10.4171/ZAA/995**

Published online: 2000-12-31

On a System of Functional Equations in a Multi-Dimensional Domain

Nguyen Thanh Long^{[1]}and Nguyen Hoi Nghia

^{[2]}(1) Polytechnic University of Ho Chi Minh City, Vietnam

(2) National University of Ho Chi Minh City, Vietnam

We study the system of functional equations $$f_i(x) = \sum^n_{j=1} \sum^m_{k=1} a_{ijk} [x, f_i (S_{ijk} (x))] + g_i (x) \ \ (l ≤ i ≤n)$$ for $x \in \Omega_i$ where $\Omega_i$ are compact or non-compact domains of $\mathbb R^p, g_i : \Omega_i \to R, S_{ijk} : \Omega_i \to \Omega_j, a_{ijk} : \Omega_i \times \mathbb R \to \mathbb R$ are given continuous functions and $f_i : \Omega_i \to \mathbb R$ are unknown functions. The paper consists of two mains parts. In the first part we give some results on existence, uniqueness and stability of the solutions of such systems and study sufficient conditions to obtain quadratic convergence. In the second part we obtain the Maclaurin expansion and approximation of solution in the case that $a_{ijk}$ are linear and $S_{ijk}$ are affine functions.

*Keywords: *Systems of functional equations, Maclaurin expansion, convergence in square mean

Long Nguyen Thanh, Nghia Nguyen Hoi: On a System of Functional Equations in a Multi-Dimensional Domain. *Z. Anal. Anwend.* 19 (2000), 1017-1034. doi: 10.4171/ZAA/995