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

Abstract

We study the half-linear advanced differential equation

$$(r(t)| y'(t)|^{\alpha-1}y'(t))'+q(t)|y|^{\alpha-1}(\sigma(t))y(\sigma(t))= 0, \quad t \geq t_0 >0,$$

where $\alpha>0, r(t)>0, q(t) >0, \sigma(t)\geq t$, and $R(t):= \int_{t_0}^{t}r^{-1/\alpha}(s) \, {\rm d}s \to \infty$ as $t\to \infty$. We prove that such an equation is oscillatory if

$$ \lambda_*:= \liminf_{t\to \infty}\frac{R(\sigma(t))}{R(t)}<\infty$$

and

$$\liminf_{t\to \infty}r^{1/\alpha}(t)R^{\alpha+1}(t)q(t)> \max\{\alpha m^\alpha(1-m)\lambda_*^{-\alpha m}: 0<m<1 \}$$

or

$$\lim_{t\to \infty}\frac{R(\sigma(t))}{R(t)}=\infty\quad \text{and}\quad \liminf_{t\to \infty}r^{1/\alpha}(t)R^{\alpha+1}(t)q(t)> 0.$$

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.