On the oscillation of second order nonlinear neutral delay differential equations
In this paper, the authors study the oscillatory behavior of solutions of the second-order neutral delay differential equation \[ (r(t)|z'(t)|^{\alpha -1} z'(t))'+f(t, x(\delta(t)))=0,\quad t\geq t_0 \] where \(z(t) = x(t) \pm p(t)x (\tau (t))\) and \(\alpha\) is a positive constant. Throughout the paper, the authors assume that \((A_1)\)\ \ \(r, p \in C([t_0, \infty), \mathbb{R})\), \(r(t)>0\) and \(0 \leq p(t) \leq 1\). \((A_2)\)\ \ \(\tau, \delta \in C^{1}([t_0, \infty), \mathbb{R})\), \(\tau(t) \leq t\), \(\delta(t) \leq t\) and \(\lim_{t \to \infty} \tau(t)=\lim_{t \to \infty}\delta(t)=\infty\). \((A_3)\)\ \ \(f \in C([t_0, \infty)\times \mathbb{R}, \mathbb{R})\), \(uf(u)\geq 0\) for all \(u \neq 0\). The authors first consider the case with negative neutral coefficients i.e. \[ (r(t)|z'(t)|^{\alpha -1} z'(t))'+f(t, x(\delta(t)))=0,\ \ t\geq t_0 \] where \(z(t) = x(t) - p(t)x (\tau (t))\). The main theorem stated in this paper is as follows: Theorem: Assume that \((A_1)-(A_3)\) hold and \(\int_{t_0}^{\infty}\frac{dt}{r^{1/\alpha}(t)}=\infty\). If there exists a function \(\gamma(t) \in C([t_0, \infty), (0, \infty))\) such that for any constant \(M>0\), \(1 < \beta \leq \alpha\) and \(\delta'(t)>0\), one has \[ \int_{t_0}^{\infty} \biggl[\gamma(t)q(t)-\frac{(\gamma'(t))^{2}}{4\gamma^{2}(t)\phi(t)}\biggr]dt=\infty, \] then all solutions of the equation with negative neutral coefficients are oscillatory or tend to zero as \(t \rightarrow \infty\).
scientific article; zbMATH DE number 7438427
scientific article