A sharp oscillation criterion for second-order half-linear advanced differential equations.

We study the half-linear advanced differential equation (r (t) | y ′ (t) | α - 1 y ′ (t)) ′ + q (t) | y | α - 1 (σ (t)) y (σ (t)) = 0 , t ≥ t 0 > 0 , where α > 0 , r (t) > 0 , q (t) > 0 , σ (t) ≥ t , and R (t) : = ∫ t 0 t r - 1 / α (s) d s → ∞ as t → ∞ . We prove that such an equation is oscillatory if λ ∗ : = lim inf t → ∞ R (σ (t)) R (t) < ∞ and lim inf t → ∞ r 1 / α (t) R α + 1 (t) q (t) > max { α m α (1 - m) λ ∗ - α m : 0 < m < 1 } or lim t → ∞ R (σ (t)) R (t) = ∞ and lim inf t → ∞ r 1 / α (t) R α + 1 (t) q (t) > 0. The obtained criteria can be regarded as a natural extension of the well-known Kneser oscillation criterion for half-linear ordinary differential equations. Our oscillation constant is optimal for the correponding half-linear Euler-type delay differential equation. [ABSTRACT FROM AUTHOR]