Oscillations of First Order Neutral Differential Equations with Variable Coefficients

Consider the first-order neutral delay differential equation ddt[x(t)−P(t)x(t−τ)]+Q(t)x(t−δ(t))=0, t≥t0 where τ>0, P, Q, δ∈C([t0, ∞), R+), and limt→∞(t−δ(t))=∞. First we obtain a necessary and sufficient condition for the oscillation of all solutions of this equation. And then some applications of this result are listed. Our results can improve many known results in the literature.