Asymptotic formulas for a scalar linear delay differential equation

Abstract

The linear delay differential equation $$ x'(t)=p(t)x(t-r) $$ is considered, where $r>0$ and the coefficient $p:[t_0,\infty)\to\mathbb{R}$ is a continuous function such that $p(t)\to0$ as $t\to\infty$. In a recent paper [M. Pituk, G. Röst, Bound. Value Probl. 2014:114] an asymptotic description of the solutions has been given in terms of a special solution of the associated formal adjoint equation and the initial data. In this paper, we give a representation of the special solution of the formal adjoint equation. Under some additional conditions, the representation theorem yields explicit asymptotic formulas for the solutions as $t\to\infty$.

Keywords

• delay differential equation
• formal adjoint equation
• asymptotic formulas