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Oscillation theorems for second-order equations with damping. (English) Zbl 0972.34022

Here, the oscillation of the nonlinear second-order differential equation with damping \[ [r(t)x'(t)]'+p(t)x'(t)+q(t)f(x(t))=0, \tag{1} \] with \(r\in C^1([t_0,\infty);(0,\infty))\),\(p,q\in C([t_0,\infty);(-\infty,\infty))\), \(f\in C((-\infty,\infty);(-\infty,\infty))\) and \(xf(x)>0\) for \(x\neq 0\) is investigated. Some new oscillation criteria are given. They generalize several earlier oscillation results of similar type and can be applied in some cases in which other known oscillation theorems are not applicable. Also a theorem on the asymptotic behavior of solutions to the forced equation \([r(t)x'(t)]'+p(t)x'(t)+q(t)f(x(t))=e(t)\) is presented.

MSC:

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