On a system of functional equations in a multi-dimensional domain

Abstract

We study the system of functional equations fi(x)=Σj=1nΣk=1maijk[x,fj(Sijk(x))]+g i(x) (1≤ i ≤ n) for x ∈ Ωi where Ωi are compact or non-compact domains of ℝp, gi : Ωi → R, Sijk : Ωi → Ωj, aijk : Ωi × ℝ → ℝ are given continuous functions and fi : Ωi → 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 aijk are linear and Sijk are affine functions. © Heldermann Verlag.