Generalized reciprocity for self-adjoint linear differential equations

Ondřej Došlý

Archivum Mathematicum (1995)

  • Volume: 031, Issue: 2, page 85-96
  • ISSN: 0044-8753

Abstract

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Let L ( y ) = y ( n ) + q n - 1 ( t ) y ( n - 1 ) + ⋯ + q 0 ( t ) y , t ∈ [ 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 * p ( t ) L ( y ) = w ( t ) y is nonoscillatory at b if and only if the equation L w - 1 ( t ) L * ( y ) = 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.

How to cite

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Došlý, Ondřej. "Generalized reciprocity for self-adjoint linear differential equations." Archivum Mathematicum 031.2 (1995): 85-96. <http://eudml.org/doc/247697>.

@article{Došlý1995,
abstract = {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.},
author = {Došlý, Ondřej},
journal = {Archivum Mathematicum},
keywords = {Self-adjoint equation; reciprocal equation; property BD; principal solution; minimal differential operator.Supported by the Grant No. 201/93/0452 of the Czech Grant Agency; selfadjoint equation; nonoscillatory},
language = {eng},
number = {2},
pages = {85-96},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Generalized reciprocity for self-adjoint linear differential equations},
url = {http://eudml.org/doc/247697},
volume = {031},
year = {1995},
}

TY - JOUR
AU - Došlý, Ondřej
TI - Generalized reciprocity for self-adjoint linear differential equations
JO - Archivum Mathematicum
PY - 1995
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 031
IS - 2
SP - 85
EP - 96
AB - 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.
LA - eng
KW - Self-adjoint equation; reciprocal equation; property BD; principal solution; minimal differential operator.Supported by the Grant No. 201/93/0452 of the Czech Grant Agency; selfadjoint equation; nonoscillatory
UR - http://eudml.org/doc/247697
ER -

References

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