On the oscillation of third-order quasi-linear delay differential equations

Abstract

The aim of this work is to study asymptotic properties of the
third-order quasi-linear delay differential equation
\begin{equation*}\label{E}
\left[a(t)\left(x''(t)\right)^\alpha\right]'+q(t)x^\alpha(\tau(t))=0,
 \tag{$E$}
\end{equation*}%
where $\alpha>0$, $\int_{t_0}^\infty\frac{1}{a^{1/\alpha}(t)}{\rm
d}t<\infty$ and $\tau(t)\leq t$. We establish a new
 condition which guarantees that every solution of $(E)$ is either oscillatory
 or converges to zero. These results improve some known results in the literature. An example is given to illustrate the
 main results.