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Oscillation theorems for differential equations with a nonlinear damping term. (English) Zbl 1027.34040

The authors investigate oscillatory properties of the nonlinear second-order differential equation

${\left(r\left(t\right){k}_{1}\left(x,{x}^{\text{'}}\right)\right)}^{\text{'}}+p\left(t\right){k}_{2}\left(x,{x}^{\text{'}}\right){x}^{\text{'}}+q\left(t\right)f\left(x\right)=0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

under natural restrictions on the functions $r,{k}_{1},{k}_{2},p,q$ and $f$. Using a modified $H$-function averaging technique introduced for linear equations by Ch. G. Philos [Arch. Math. 53, 482-492 (1989; Zbl 0661.34030)], additional conditions are found which guarantee that (*) possesses no nonoscillatory solution. The results of the paper are illustrated by a number of examples showing that the obtained criteria apply in situations where known criteria fail.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory