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On the existence of oscillatory solutions to \(n\)th-order differential equations with quasiderivatives. (English) Zbl 0914.34029

The author studies the existence of an oscillatory solution to the \(n\)th-order nonlinear differential equation \[ y^{[n]}=f(t,y^{[0]},\dots , y^{[n-1]}) \] on \([0,\infty)\), where \(n\geq 3\), \(y^{[i]}\) are the quasiderivatives of \(y\) defined by \[ y^{[0]}=y,\;y^{[i]}=\frac {1}{a_i(t)}\bigl (y^{[i-1]} \bigr)', \;i=1,\dots , n-1, \;y^{[n]}=\bigl (y^{[n-1]} \bigr)', \] \(f\) fulfills the local Carathéodory conditions and \(f(x_1,\dots ,x_n)x_1\leq 0\).
The result (theorem 4) is applied to the third-order differential equation \[ y'''+q(t)y'+r(t)g(y)=0 \] where \(q\) may change sign, with \(r(t)>0\) and \(| g(x)| \geq | x| ^\lambda \) for large \(| x| \) and for \(\lambda \in [0,1]\).
Reviewer: Z.Došlá (Brno)

MSC:

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