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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.

34K11 Oscillation theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations