Oscillation of neutral delay differential equations

Abstract

Consider the following neutral delay differential equationwhere P ∈ ℝ, T ∈ (0, ∞), σ ∈ (0, ∞) and Q ∈ C[(t0, ∞), [0, ∞)]. We obtain a sufficient condition for the oscillation of all solutions of Equation (*) with P = −1, which does not require thatBut, 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.