Oscillations of first-order neutral delay differential equations

AbstractConsider the neutral delay differential equation (∗) (ddt)[y(t) + py(t − τ)] + qy(t − σ) = 0, t ⩾ t0, where τ, q, and σ are positive constants, while p ϵ (−∞, −1) ∪ (0, + ∞). (For the case p ϵ [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (∗) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and q(σ − τ)(1 + p) > (1e). Then every solution of Eq. (∗) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (∗) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (∗) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, andq(σ − τ)(1 + p) > (1e). Then every solution of Eq. (∗) oscillates. Extensions of these results to equations with variable coefficients are also obtained