Oscillation properties of the self-adjoint, two term, differential equation \[(-1)^n(p(x)y^{(n)})^{(n)}+q(x)y=0\qquad \mathrm {(*)}\] are investigated. Using the variational method and the concept of the principal system of solutions it is proved that (*) is conjugate on $R=(-\infty ,\infty )$ if there exist an integer $m\in \lbrace 0,1,\dots ,n-1\rbrace $ and $c_0,\dots ,c_m\in R$ such that \[\int _\infty ^0 x^{2(n-m-1)}p^{-1}(x)\,dx=\infty =\int _0^\infty x^{2(n-m-1)}p^{-1}(x)\,dx\] and \[\limsup _{x_1\downarrow -\infty ,x_2\uparrow \infty }\int _{x_1}^{x_2}q(x)(c_0+c_1x+\dots + c_mx^m)^2\,dx\le 0,\quad q(x)\lnot \equiv 0.\] Some extensions of this criterion are suggested.
@article{107542,
author = {Ond\v rej Do\v sl\'y and Jan Komenda},
title = {Conjugacy criteria and principal solutions of self-adjoint differential equations},
journal = {Archivum Mathematicum},
volume = {031},
year = {1995},
pages = {217-238},
zbl = {0841.34033},
mrnumber = {1368260},
language = {en},
url = {http://dml.mathdoc.fr/item/107542}
}
Došlý, Ondřej; Komenda, Jan. Conjugacy criteria and principal solutions of self-adjoint differential equations. Archivum Mathematicum, Tome 031 (1995) pp. 217-238. http://gdmltest.u-ga.fr/item/107542/
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