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.


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