Asymptotic behavior of solutions of retarded differential equations

Abstract:

In this paper we obtain sufficient conditions under which every solution of the retarded differential equation \[ (1)\quad x’(t) + p(t)x(t - \tau ) = 0,\quad t \geqslant {t_0},\], where $\tau$ is a nonnegative constant, and $p(t) > 0$, is a continuous function, tends to zero as $t \to \infty$. Also, under milder conditions, we prove that every oscillatory solution of (1) tends to zero as $t \to \infty$. More precisely the following theorems have been established. Theorem 1. Assume that $\int _{{t_0}}^\infty {p(t)dt = + \infty }$ and ${\lim _{t \to \infty }}\int _{t - \tau }^t {p(s)ds < \pi /2}$ or $\lim {\sup _{t \to \infty }}\int _{t - \tau }^t {p(s)ds < 1}$. Then every solution of (1) tends to zero as $t \to \infty$. Theorem 2. Assume that $\lim {\sup _{t \to \infty }}\int _{t \to \tau }^t {p(s)ds < 1}$. Then every oscillatory solution of (1) tends to zero as $t \to \infty$.