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Oscillations of functional differential equations. (English) Zbl 0902.34062

The author establishes some oscillation criteria (without proofs) for linear delay differential equations \[ x'(t)+ p(t)x(\tau (t)) =0, \qquad \tau(t) <t,\quad t\geq t_0 \tag{*} \] and its discrete analogue \[ x_{n+1}-x_n+p_nx_{n-k}=0, \qquad k\in \mathbb{Z}^+, \tag{**} \] respectively, under some conditions which contain the following:
(i) \(\liminf_{t\to \infty} \int^t_{\tau(t)} p(s)ds \leq \frac 1e\) and \(\limsup_{t\to \infty} \int^t_{\tau(t)} p(s)ds <1\);
(ii) \(\liminf_{n\to \infty} \sum^{n-1}_{i=n-k} p_i \leq (\frac k{k+1})^{k+1} \) and \(\limsup_{n\to \infty}\sum^n_{i=n-k} p_i < 1\), respectively.
This work is related to L. H. Erbe and B. G. Zhang [Differ. Integral Equ. 1, No. 3, 305-314 (1988; Zbl 0723.34055); ibid. 2, No. 3, 300-309 (1989; Zbl 0723.39004)].

MSC:

34K11 Oscillation theory of functional-differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34K25 Asymptotic theory of functional-differential equations
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