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Oscillation of even-order neutral differential equations with retarded and advanced arguments. (English) Zbl 1487.34123

In this paper, the authors study the oscillatory behavior of even-order nonlinear neutral differential equations of the form \[ ( x(t) + ax(t - \tau_1) + bx(t + \tau_2))^{(n)} + p(t)x^{\alpha}(t - \sigma_1) + q(t)x^{\beta}(t + \sigma_2) = 0,\tag{1.1} \] for \( t\geq t_0>0\), where \(n \geq 4\) is an even integer, under the following conditions:
\((A_1)\)\( p, q \in C([t_0, \infty), \mathbb{R})\).
\((A_2)\)\(a, b, \tau_1, \tau_2, \sigma_1, \sigma_2 \geq 0.\)
\((A_3)\)\(\alpha\) and \(\beta\) are the ratios of odd positive integers.
The authors first introduce the following notation: \[ \begin{aligned} z(t) &= x(t) + ax(t - \tau_1) + bx(t + \tau_2),\\ P(t) &= \min \{p(t - \tau_1), p(t), p(t + \tau_2)\},\\ Q(t) &= \min \{q(t - \tau_1), q(t), q(t + \tau_2)\}. \end{aligned} \] The main reads as follows:
Assume \(\alpha < 1 < \beta\) and \(\sigma_1 > \tau_1\). If there exists a nondecreasing \(\delta \in C^{1}([t_0, \infty), (0, \infty)) \) such that \[ \lim_{t \to \infty} \int_{t+\tau_1-\sigma_1}^{t} A(s)(s-\sigma_1)^{n-1}ds >\frac{(n-1)!(1+d_1+d_2)}{\lambda_0 e} \] and for \(0 <l <1\) \[ \limsup_{j \to \infty}\int_{t_1}^j \bigg[\frac{\delta(t)}{(n-3)!}\int_{t}^{\infty}(s-t)^{n-3}A(s)\bigg(1-\frac{\sigma_1+\tau_2}{s}\bigg)^{\frac{1}{l}}ds-\frac{(\delta'(t))^{2}}{4\delta(t)}\bigg]dt=\infty, \] for some \(\lambda_0 \in (0, 1)\), then all solutions of Eq. (1.1) are oscillatory.

MSC:

34K11 Oscillation theory of functional-differential equations
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