Solutions of an advance-delay differential equation and their asymptotic behaviour

Abstract

The paper considers a scalar differential equation of an advance-delay type \begin{equation*} \dot{y}(t)= -\left(a_0+\frac{a_1}{t}\right)y(t-\tau )+\left(b_0+\frac{b_1}{t}\right)y(t+\sigma )\,, \end{equation*} where constants $a_0$, $b_0$, $\tau $ and $\sigma $ are positive, and $a_1$ and $b_1$ are arbitrary. The behavior of its solutions for $t\rightarrow \infty $ is analyzed provided that the transcendental equation \begin{equation*} \lambda = -a_0\mathrm{e}^{-\lambda \tau }+b_0\mathrm{e}^{\lambda \sigma } \end{equation*} has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.