## Oscillation criteria for nonlinear differential equations with damping.(English)Zbl 1034.34041

The author presents oscillation criteria for the damped nonlinear differential equation $(r(t)x')'+ p(t)x' + q(t)f(x)=0, \tag{*}$ where $$r,f$$ are continuously differentiable functions, $$r(t)>0$$, $$p,q$$ are continuous, $$xf(x)>0$$ for $$x\neq 0$$, and it is supposed that there exists $$\lambda>0$$ such that $$f'(x)>\lambda$$ for $$x\in \mathbb{R}$$. Under the last assumption on the derivative $$f'$$, equation (*) is, in a certain sense, the Sturmian majorant of the linear equation $(r(t)y')'+p(t)y+ \lambda q(t)y=0. \tag{**}$ Using a combination of the generalized Riccati substitution and the $$H$$-function averaging technique, conditions on the functions $$r,p,q$$ are given which guarantee that (**) is oscillatory and this, in turn, implies that (*) possesses no nonoscillatory solution which is extensible up to $$\infty$$. The obtained oscillation criteria are illustrated by a number of examples and corollaries.

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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### References:

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