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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 L. Erbe [Canadian Math. Bull. 16, 49-56 (1973; Zbl 0272.34095)] and 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 and coupled oscillators for ordinary differential equations
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##### References:
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