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Oscillation results for a second order damped differential equation with nonmonotonous nonlinearity. (English) Zbl 1008.34029

The authors investigate oscillatory properties of the nonlinear second-order damped differential equation \[ (r(t)\psi(x')x')'+p(t)x'+q(t)f(x)=0, \tag{*} \] where the coefficients \(r,p,q\) and nonlinearities \(\psi,f\) satisfy certain (natural) restrictions. The results given in the paper can be regarded as a continuation of the research initiated by the second author in [Nonlinear Anal., Theory Methods Appl. 41A, No. 7-8, 1005-1028 (2000; Zbl 0972.34022)], where the case \(\psi\equiv 1\) is treated under more restrictive assumptions on the nonlinearity \(f\) and the damping term \(p\). In particular, no monotonicity restriction on the function \(f\) is imposed in the reviewed paper. The oscillation criteria presented here are proved using the Riccati method combined with the so-called \(H\)-function averaging technique introduced for second-order linear differential equations by Ch. G. Philos [Arch. Math. 53, No. 5, 483-492 (1989; Zbl 0661.34030)].

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C29 Averaging method for ordinary differential equations
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