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

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, $a_1$ $b_1$ arbitrary. behavior its solutions for $t\rightarrow \infty is analyzed provided that the transcendental \lambda = -a_0\mathrm{e}^{-\lambda \tau }+b_0\mathrm{e}^{\lambda \sigma } has positive real root. An exponential-type function approximating solution searched to be used in proving existence semi-global solution. Moreover, lower upper estimates given such