On an approach to the analysis of asymptotic properties of solutions of first-order ordinary delay differential equations (Q944432)

The authors study oscillation, nonoscillation and asymptotic properties of the following scalar delay differential equation \[ \dot{u}(t)(t)+p(t)u(\tau(t))=0, \quad p(t)\geq 0,\;\tau(t)\leq t.\tag{1} \] They obtained several interesting results, including Theorem 3.1. Suppose that there exists a function \(\mu (t)\geq 0\) such that \[ \frac{1}{\mu(t)}\int_{\tau(t)}^t \exp\{\mu(s)\}p(s)\,ds\leq 1. \] Then, for some \(t_1\), there exists a solution \(u(t)>0, t\geq t_1\) of equation (1) satisfying the condition \[ \exp\left(-\int_{t_1}^t\exp\{\mu(s)\}p(s)\,ds\right)\leq u(t)\leq \exp\left(-\int_{t_1}^t p(s)\,ds\right),\quad t\geq t_1. \]