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

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