Oscillatory behavior of solutions of dynamic equations of higher order on time scales

Abstract

We study the $n$-th-order nonlinear dynamic equations

$$x^{\left[n\right]}\left(t\right)+p\left(t\right){\varphi }_{{\alpha }_{n-1}}\left[{\left(x^{\left[n-2\right]}\left(t\right)\right)}^{\mathrm{\Delta }\sigma }\right]+q\left(t\right){\varphi }_{\gamma }\left(x\left(g\left(t\right)\right)\right)=0$$

on an unbounded time scale $\mathbb{𝕋}$, where $n\ge 2$ and for $i=1,\dots ,n-1$

$$x^{\left[i\right]}\left(t\right):=r_i\left(t\right){\varphi }_{{\alpha }_i}\left[{\left(x^{\left[i-1\right]}\left(t\right)\right)}^{\mathrm{\Delta }}\right],$$

with $r_n={\alpha }_n=1$ and $x^{\left[0\right]}=x$; here the constants ${\alpha }_i$ and the functions $r_i$, $i=1,\dots ,n-1$, are positive and $p$, $q$ are nonnegative functions. Criteria are established for the oscillation of solutions for both even- and odd-order cases. The results improve several known results in the literature on second-order, third-order, and higher-order linear and nonlinear dynamic equations. In particular our results can be applied when $g$ is not (delta) differentiable and the forward jump operator $\sigma$ and $g$ do not commute.