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Oscillation theorems for nonlinear second-order differential equations with damping. (English) Zbl 0973.34021

The authors deal with the second-order nonlinear differential equation \[ \big (a(t)\Psi(x(t))k(x'(t))\big)' + p(t)k(x'(t))+q(t)f(x(t))=0, \quad t\geq t_{0}. \] They present four theorems where several conditions are given to guarantee the oscillatory character of the solutions. The most important role in the proofs plays an auxiliary function of the form \(w(t)=\Phi(t) \big [\frac{a(t)\Psi(x(t))k(x'(t))}{x(t)}+a(t)R(t)+\frac{\gamma_{1}}{2}p(t)\big ]\) for some functions \(\Phi\) and \(R\).

MSC:

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