Linear differential equations with unbounded delays and a forcing term

Abstract

The paper discusses the asymptotic behaviour of all solutions of the differential equation $\dot{y}\left(t\right)=-a\left(t\right)y\left(t\right)+{\sum }_{i=1}^n{b}_i\left(t\right)y\left({\tau }_i\left(t\right)\right)+f\left(t\right)$, $t\in I=\left[t_0,\infty \right)$, with a positive continuous function $a$, continuous functions $b_i$, $f$, 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.