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

Abstract

In [Bull. Acad. Polon. Sci. S\'{e}r. 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 $f^{(n)}(t)\leq0$ ($\not\equiv0$) and $f(t)>0$ for all $t\geq{}t_{0}$. If $f$ is unbounded, then $f(t)\geq\frac{\lambda{}t^{n-1}}{(n-1)!}f^{(n-1)}(t)$ 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.