On conjugacy of high-order linear ordinary differential equations

Abstract

It is shown that the differential equation

$$u^{(n)} = p(t)u$$

wheren≥2 andp: [a, b] →R is a summable function, is not conjugate in the segment [a, b], if for somel∈{1,...,n−1}, α∈]a,b[, and β∈]α,b[ the inequalities

$$\begin{gathered} n \geqslant 2 + \frac{1}{2}(1 + ( - 1)^{n - 1} ), ( - 1)^{n - 1} p(t) \geqslant 0 for t \in [a,b], \hfill \\ \int_\alpha ^\beta {(t - a)^{n - 2} (b - t)^{n - 2} \left| {p(t)} \right|} dt \geqslant l!(n - l)!\frac{{(b - a)^{n - 1} }}{{(b - \beta )(\alpha - a)}} \hfill \\ \end{gathered}$$

, hold.