A sharp oscillation criterion for second-order half-linear advanced differential equations (Q2036577)

Considered in this paper is the second-order half-linear advanced differential equation \[(r(t)\phi(y^{\prime}(t)))^{\prime}+q(t)\phi(y(\sigma(t)))=0\] with \(\phi(u):=\vert u\vert ^{{\alpha}-1}u\), and \(\int_{t_0}^tr^{-1/{\alpha}} (s) ds\to \infty\) as \(t\to \infty\). Using the notation \(R(t)=\int_{t_0}^tr^{-1/{\alpha}} (s) ds\) and \({\lambda}^*=\liminf R(\sigma (t))/R(t)\), the authors obtain oscillation criteria for the cases where \({\lambda}^*<\infty\) and \({\lambda}^*=\infty\), respectively. The criteria can be regarded as a natural extension of the Kneser oscillation criterion for half-linear ordinary differential equations. Two examples are given to show that the results are optimal for certain half-linear Euler-type advanced differential equations.