Consider the higher-order neutral delay differential equation (*) {{{d^n}} \over {d{t^n}}}\left( {x(t) + ∑\limits_{i = 1}^L {{p_i}x(t - {τ _i})} - ∑\limits_{j = 1}^M {{r_j}x(t - {p_j})} } \
ight) + ∑\limits_{k = 1}^N {{q_k}x(t - {u_k})} = 0 where the coefficients and the delays are nonnegative constants with n ≥ 1 odd. Then a necessary and sufficient condition for the oscillation of (*) is that the characteristic equation F(λ ) = {λ ^n} + {λ ^n}∑\limits_{i = 1}^L {{p_i}{e^{ - λ {p_i}}}} - {λ ^n}∑\limits_{j = 1}^M {{r_j}{e
- λ pj}} + ∑\limits_{k = 1}^N {{q_k}{e^{ - λ {u_k}}}} = 0 has no real roots.
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