OSCILLATION OF NEUTRAL DELAY DIFFERENTIAL-EQUATIONS

Consider the following neutral delay differential equation (*) d/dt[x(t) + Px(t - tau)] + Q(t)x(t - sigma) = 0, t greater-than-or-equal-to t0 where P is-an-element-of R, tau is-an-element-of (0, infinity), sigma is-an-element-of [0, infinity) and Q is-an-element-of C[[t0, infinity), [0, infinity)]. We obtain a sufficient condition for the oscillation of all solutions of Equation (*) with P = -1, which does not require that (**) integral-infinity/t0 Q(s)ds = infinity. But, for the cases -1 < P < 0 and P < -1, we show that (**) is a necessary condition for the oscillation of all solutions of Equation (*). These new results solve some open problems in the literature.