JournalszaaVol. 19, No. 4pp. 1017–1034

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

  • Nguyen Thanh Long

    Polytechnic University of Ho Chi Minh City, Vietnam
  • Nguyen Hoi Nghia

    National University of Ho Chi Minh City, Vietnam
On a System of Functional Equations in a Multi-Dimensional Domain cover
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Abstract

We study the system of functional equations

fi(x)=j=1nk=1maijk[x,fi(Sijk(x))]+gi(x)  (lin)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 xΩix \in \Omega_i where Ωi\Omega_i are compact or non-compact domains of Rp,gi:ΩiR,Sijk:ΩiΩj,aijk:Ωi×RR\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 fi:ΩiRf_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 aijka_{ijk} are linear and SijkS_{ijk} are affine functions.

Cite this article

Nguyen Thanh Long, Nguyen Hoi Nghia, On a System of Functional Equations in a Multi-Dimensional Domain. Z. Anal. Anwend. 19 (2000), no. 4, pp. 1017–1034

DOI 10.4171/ZAA/995