Comparison theorems for the \(N^{th}\) order differential equations (Q2702975)

The author investigates oscillatory and asymptotic properties of solutions to the \(n\)th order linear nonhomogeneous differential equation NEWLINE\[NEWLINE y^{(n)}+\alpha_1(t)y^{(n-1)}+\dots+\alpha_n(t)y+p(t)y=r(t). \tag{*} NEWLINE\]NEWLINE In addition to the usual continuity assumptions it is supposed that \(\lim_{t\to \infty}\alpha_i(t)=A_i\), \(i=1,\dots,n,\) exist (finite) and that the functions \(p,r\) are in a certain sense bounded. The main result of the paper gives conditions on the functions \(\alpha_i,p,r\) such that any eventually positive solution \(y\) to (*) has the following property for large \(t\): given any \(l>0\), any interval \(I\) of the length \(l\) and any \(\lambda\in [0,1]\), there exists \(\varepsilon>0\) such that the Lebesgue measure of the points where \(y(t)\leq \varepsilon\) is less than \(\lambda\cdot l\). A related author's paper dealing with a similar problem is \textit{M. Kováčová} [CDDE 2000 Proceedings (Brno). Arch. Math. (Brno) 36 suppl., 487-498 (2000)].