Asymptotic dichotomy in a class of third-order neutral differential equations (Q2892441)

The authors consider the third-order neutral differential equation NEWLINE\[CARRIAGE_RETURNNEWLINE \frac{d^3}{dt^3}\left\{x(t)-p(t)x(\gamma(t))\right\}+q(t)x(\tau(t))=0,\, t\geq t_0,\tag{1}CARRIAGE_RETURNNEWLINE\]NEWLINE where \(p\) and \(q\) are real continuous functions on \([t_0,\infty)\) such that \(p(t)>0, q(t)>0;\gamma\) and \(\tau\) are real continuous and strictly increasing functions on \((t_0, \infty)\) such that \(\lim\limits_{t\rightarrow\infty}\gamma(t)=+\infty\) and \(\lim\limits_{t\rightarrow\infty}\tau(t)=+\infty\). Then, they establish dichotomous criteria that guarantee solutions of (1) that are either oscillatory or tend to \(\pm\infty\) as \(t\rightarrow +\infty\).