On a~comparison theorem for linear differential equations

T. A. Chanturiya

Izvestiya. Mathematics , Tome 10 (1976) no. 5, pp. 1075-1088

Résumé

It is proved in the paper that the equation $u^{(n)}=a(t)u$ has property $\mathrm B$ (i.e. each solution of it, in the case of even $n$, either is oscillating or satisfies the condition $|u^{(i)}(t)|\downarrow0$ for $t\to+\infty$ ($i=0,\dots, n-1$) or satisfies the condition $|u^{(i)}(t)|\uparrow+\infty$ for $t\to+\infty$ ($i=0,\dots,n-1$), and in the case of odd $n$, either is oscillating or satisfies the condition $|u^{(i)}(t)|\uparrow+\infty$ for $t\to+\infty$ ($i=0,\dots,n-1$)) if the equation $u^{(n)}=b(t)$ has the property $\mathrm B$ and $a(t)\geqslant b(t)\geqslant0$ for $t\in[0,+\infty)$. Bibliography: 8 titles.