On oscillating differential equations of third order (Q1596162)

The equation \[ y^{(n)} = p(t)y \] with \( p(t) \in L_{\text{loc}}(\mathbb{R}_+) \) possesses the property A, if every nontrivial solution to this equation is oscillating for even \(n\) and is either oscillating or satisfies \[ y(t)y'(t)<0\quad\text{for}\quad t\geq t_0, \] for odd \(n\). The author sets out the following assertion. Let \(p(t) \in L_{\text{loc}}(\mathbb{R}_+)\), \(p(t)\leq 0\) for \(t\in \mathbb{R}_+ \) and \[ \int_1^{+\infty} t^2\biggl(p(t) + \frac{2\sqrt{3}}{9} t^{-3}\biggr) dt = -\infty. \] Then the equation \(y''' = p(t)y\) possesses the property~\(A\).