Oscillation and asymptotic behavior of second order neutral differential equations

Abstract

Consider the second- order neutral delay differential equation $$\frac{{d^2 }}{{dt^2 }}[y(t) + P(t - \tau )] + Q(t)y(t - \sigma ) = 0,t \geqq t_0 ,$$ (1) where P, Q e C([t0g, ∞), R) and the delays τ and σ are nonnegative real numbers. We examined the asymptotic behavior of the nonoscillatory solutions of eq. (1) and obtained sufficient conditions for the oscillation of (i) all solutions of eq. (1); (ii) all bounded solutions of eq. (1); (iii) all unbounded solutions of eq. (1).