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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

(r(t)k 1 (x,x ' )) ' +p(t)k 2 (x,x ' )x ' +q(t)f(x)=0,(*)

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.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory