On the oscillation of even-order nonlinear differential equations with mixed neutral terms (Q2240750)

This paper deals with the oscillation problem of the even-order nonlinear differential equation with mixed neutral terms \[ \left(r(t)\left[y^{(n-1)}(t)\right]^{\alpha}\right)'+q(t)x^{\gamma}(\tau_{1}(t))=0 \tag{\(1\)} \] under the assumption that \[ y(t)=x(t)+p_{1}(t)x^{\beta}(\tau_{2}(t))-p_{2}(t)x^{\delta}(\tau_{2}(t)). \] Here, \(n>0\) is an even integer; \begin{itemize} \item[(i)] \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) are the ratios of two positive odd integers with \(\alpha\ge 1\);\\ \item[(ii)] \(p_{1}\), \(p_{2}\), \(q:[t_0, \infty)\to \mathbb{R}^{+}\) are continuous functions;\\ \item[(iii)] \(\tau_{k}:[t_0, \infty)\to \mathbb{R}\) are continuous functions; \(\tau_{k}(t)\le t\) and \(\tau_{k} \to \infty\) as \(t\to \infty\) for \(k=1,2\);\\ \item[(iv)] \(h(t)=\tau_{2}^{-1}(\tau_{1}(t))\le t\), and \(h(t)\to \infty\) as \(t\to \infty\). \end{itemize} In this paper, the authors give sufficient conditions which guarantee that equation \((1)\) with \(\beta<1\) and \(\delta>1\) is oscillatory or \(\lim_{t\to\infty}x(t)=\infty\) (see Theorem 2). Theorem 2 is their main result. To prove the main result, the authors use the oscillation of first-order delay differential equations. As a concrete example, the authors discuss the oscillation problem for the following equations \[ \left(e^{-t}\left(x(t)+\frac{1}{t}x^{1/3}(t/2)-x^{3}(t/2)\right)'\right)'+\left(\frac{3}{4}-\left(\frac{5}{36t}+\frac{1}{2t^2}+\frac{2}{t^3}\right)e^{-4t/3}\right)x(t/2)=0 \] and \[ \left(e^{-t}\left(x(t)+\frac{1}{t}x^{1/3}(t/2)-x^{3}(t/2)\right)^{(n-1)}\right)'+\left(\frac{1}{t}e^{-t/2}\right)x(t/2)=0. \]