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Oscillation and nonoscillation for a class of neutral differential equations. (English) Zbl 0873.34056
The neutral differential equations $[x(t)-x(\tau(t))]'= \sum_{i=1}^m Q_i(t)f_i(x(\delta_i(t)))=0, \qquad t\geq t_0,\tag{1}$ where $$Q_i\in C([t_0,\infty, \mathbb{R}^+)$$, $$\tau,\delta_i\in C([t_0,\infty), \mathbb{R})$$, $$\tau(t)<t$$, $$\lim_{t\to\infty} \tau(t)=\infty$$ and $\lim_{t\to\infty} \delta_i(t)=\infty, \qquad i=1,\dots,m; \tag{2}$
$f_i\in C(\mathbb{R},\mathbb{R}) \quad\text{and}\quad xf_i(x)>0\text{ for }x\neq 0, \qquad i=1,\dots,m, \tag{3}$ are considered. A necessary and sufficient condition for (1) to have a bounded nonoscillatory solution is obtained. An oscillation criterion is obtained.
Theorem 1. If (2), (3) holds, $$\tau(t)$$ is increasing, $$t-M\leq \tau(t)<t$$ for $$t\geq t_0$$, $$M>0$$, $$f_i(x)$$ is nondecreasing, $$i=1,\dots,m$$, then (1) has a bounded nonoscillatory solution if and only if $$\int_{t_0}^\infty t\sum_{i=1}^m Q_i(t)dt<\infty$$. Theorem 2. If (2), (3) holds, $$\tau(t)= t-\sigma$$, $$\sigma>0$$, $$f_i$$ is nondecreasing, $$i=1,\dots, m$$, and for some $$T>t_0$$, $\int_T^\infty \sum_{i=1}^m Q_i(t) \Biggl|f_i(c\delta_i(t)) \int_{\delta_i(t)}^\infty \sum_{i=1}^m Q_i(s)ds \Biggr|dt=\infty, \qquad \text{for any } c\neq 0,$ then every solution of (1) is oscillatory.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
##### Keywords:
oscillation; nonoscillation; neutral differential equations