Asymptotic behaviour of oscillatory solutions to \(n\)th order differential equations with quasiderivatives. (English) Zbl 0930.34023

Bartušek, Miroslav

Czech. Math. J. 47, No. 2, 245-259 (1997).

Nonlinear differential equations with quasiderivatives of the form \[ y^{[n]}=f(t,y^{[0]}, \dots y^{[n-1]}), \] \(n\geq 3\) are considered in \(D= [0, \infty) \times \mathbb{R}^n\) with \(f: D \to \mathbb{R}\) fulfilling the Carathéodory conditions locally: \(y^{[0]}= \frac {y}{\alpha_0(t)}\), \( y^{[i]}= \frac {1}{\alpha_i(t)} (y^{[i-1]})'\), \(i=1,\dots , n-1 \), \(y^{[n]}= (y^{[n-1]})'\), \(\alpha_i:[0, \infty) \to (0, \infty)\). Sufficient conditions are presented under which the absolute values of the local extrema of the quasiderivatives \(y^{[i]}, i\in \{1,\dots , n-2\}\) are increasing and tending to \(\infty \). The existence of proper, oscillatory and unbounded solutions is proved.

Reviewer: Š.Schwabik (Praha)

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations

Keywords:

differential equation with quasiderivatives; proper solution; oscillatory solution; unbounded solution

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