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Oscillation criteria for second-order retarded differential equations. (English) Zbl 0902.34061
The authors investigate oscillatory properties of second-order quasilinear equations $$[r(t)| u'(t)|^{\alpha-1}u'(t)+p(t)| u(\tau(t))|^{\beta-1}u(\tau(t))]=0 \tag*$$ with $\alpha,\beta>0$, $r(t)>0$, $p(t)\geq 0$, $\tau(t)\leq t$ and $\lim_{t\to \infty}\tau(t)=\infty$. The results deal mostly with the case $\alpha=\beta$ and extend some earlier criteria for linear equations $u''+p(\tau(t))=0$ given by {\it L. Erbe} [Canadian Math. Bull. 16, 49-56 (1973; Zbl 0272.34095)] and {\it J. Ohriska} [Czech. Math. J. 34, 107-112 (1984; Zbl 0543.34054)]. A typical result is the following oscillation criterion: Equation (*) with $\alpha=\beta$ and $r(t)\equiv 1$ is oscillatory provided one of the following conditions holds: $$\lim_{t\to\infty}t^{\alpha}\int_t^\infty p(s) \left({\tau(s)\over s}\right)^{\alpha} ds>1,\quad\text{or}\quad \limsup_{t\to \infty}t^{\alpha}\int_{\gamma(t)}^\infty p(s) ds>1,$$ with $\gamma(t)=\sup\{s: \tau(s)\leq t\}$.
Reviewer: O.Došlý (Brno)

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34C15 Nonlinear oscillations, coupled oscillators (ODE)
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##### References:
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