On the asymptotic behaviour of the solutions of a system of functional equations

Abstract

In this paper we study the asymptotic behaviour of the solutions of the functional equation x(t) = A(t)x(t - 1) + B(t)x(p(t)), where x(t) ∈ R n, A and B are n x n real matrix valued functions, p is a real function with p(t) < t - δ for some δ > 0 and lim t→∞ p(t) = ∞. In the first part of the paper we obtain asymptotic estimates for the rate of convergence of the solutions in the case when A(t) is a diagonal matrix. In the second part we prove results without assuming that A(t) is diagonal. © Akadémiai Kiadó.