Let $L(y)=y^{(n)}+q_{n-1}(t)y^{(n-1)}+\dots +q_0(t)y,\,t\in [a,b)$, be an $n$-th order differential operator, $L^*$ be its adjoint and $p,w$ be positive functions. It is proved that the self-adjoint equation $L^*\bigl (p(t)L(y)\bigr ) =w(t)y$ is nonoscillatory at $b$ if and only if the equation $L\bigl (w^{-1}(t)L^*(y)\bigr )=p^{-1}(t)y$ is nonoscillatory at $b$. Using this result a new necessary condition for property BD of the self-adjoint differential operators with middle terms is obtained.
@article{107529,
author = {Ond\v rej Do\v sl\'y},
title = {Generalized reciprocity for self-adjoint linear differential equations},
journal = {Archivum Mathematicum},
volume = {031},
year = {1995},
pages = {85-96},
zbl = {0841.34032},
mrnumber = {1357977},
language = {en},
url = {http://dml.mathdoc.fr/item/107529}
}
Došlý, Ondřej. Generalized reciprocity for self-adjoint linear differential equations. Archivum Mathematicum, Tome 031 (1995) pp. 85-96. http://gdmltest.u-ga.fr/item/107529/
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