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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