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

Abstract

We study the system of functional equations

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.