Linear differential equations with unbounded delays and a forcing term

Abstract

The paper discusses the asymptotic behaviour of all solutions of the differential equation y˙(t)=−a(t)y(t)+∑i=1nbi(t)y(τi(t))+f(t), t∈I=[t0,∞), with a positive continuous function a, continuous functions bi, f, and n continuously differentiable unbounded lags. We establish conditions under which any solution y of this equation can be estimated by means of a solution of an auxiliary functional equation with one unbounded lag. Moreover, some related questions concerning functional equations are discussed as well.