Philos' inequality on time scales and its application in the oscillation theory

Abstract

In [Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), no.~7-8, 367--370], Philos proved the following result: Let $f:[t_0,\infty {)}_{\mathbb{R}}\to \mathbb{R}$ be an $n$-times differentiable function such that $f^{(n)}(t)\le 0$ ($\not\equiv 0$) and $f(t)>0$ for all $t\ge t_0$. If $f$ is unbounded, then $f(t)\ge \frac{\lambda t^{n-1}}{(n-1)!}{f}^{(n-1)}(t)$ for all sufficiently large $t$, where $\lambda \in (0,1{)}_{\mathbb{R}}$. In this work, we first present time scales unification of this result. Then, by using it, we provide sufficient conditions for oscillation and asymptotic behaviour of solutions to higher-order neutral dynamic equations.