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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

$\left[r\left(t\right)|{u}^{\text{'}}\left(t\right){|}^{\alpha -1}{u}^{\text{'}}\left(t\right)+p\left(t\right)|u\left(\tau \left(t\right)\right){|}^{\beta -1}u\left(\tau \left(t\right)\right)\right]=0\phantom{\rule{2.em}{0ex}}\left(*\right)$

with $\alpha ,\beta >0$, $r\left(t\right)>0$, $p\left(t\right)\ge 0$, $\tau \left(t\right)\le t$ and ${lim}_{t\to \infty }\tau \left(t\right)=\infty$. The results deal mostly with the case $\alpha =\beta$ and extend some earlier criteria for linear equations ${u}^{\text{'}\text{'}}+p\left(\tau \left(t\right)\right)=0$ 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 $r\left(t\right)\equiv 1$ is oscillatory provided one of the following conditions holds:

$\underset{t\to \infty }{lim}{t}^{\alpha }{\int }_{t}^{\infty }p\left(s\right){\left(\frac{\tau \left(s\right)}{s}\right)}^{\alpha }ds>1,\phantom{\rule{1.em}{0ex}}\text{or}\phantom{\rule{1.em}{0ex}}\underset{t\to \infty }{lim sup}{t}^{\alpha }{\int }_{\gamma \left(t\right)}^{\infty }p\left(s\right)ds>1,$

with $\gamma \left(t\right)=sup\left\{s:\tau \left(s\right)\le t\right\}$.

Reviewer: O.Došlý (Brno)
##### MSC:
 34K11 Oscillation theory of functional-differential equations 34C15 Nonlinear oscillations, coupled oscillators (ODE)