Oscillation and nonoscillation criteria for delay differential equations

Abstract

Oscillation and nonoscillation criteria for the first-order delay differential equation \[ x ′ ( t ) + p ( t ) x ( τ ( t ) ) = 0 , t ≥ t 0 , τ ( t ) > t , x’(t) + p(t)x(\tau (t)) = 0,\quad t \geq {t_0},\tau (t) > t, \] are established in the case where \[ ∫ τ ( t ) t p ( s ) d s ≥ 1 e and lim t → ∞ ∫ τ ( t ) t p ( s ) d s = 1 e . \int _{\tau (t)}^t {p(s)ds \geq \frac {1}{e}\quad {\text {and}}\quad \lim \limits _{t \to \infty } \int _{\tau (t)}^t {p(s)ds = \frac {1}{e}.} } \]