Geodesic
Generalized reciprocity for self-adjoint linear differential equations
Archivum mathematicum, Tome 31 (1995) no. 2, pp. 85-96 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 34B05, 34C10, 34L05
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 
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     title = {Generalized reciprocity for self-adjoint linear differential equations},
     journal = {Archivum mathematicum},
     pages = {85--96},
     year = {1995},
     volume = {31},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1995_31_2_a0/}
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Došlý, Ondřej. Generalized reciprocity for self-adjoint linear differential equations. Archivum mathematicum, Tome 31 (1995) no. 2, pp. 85-96. http://geodesic.mathdoc.fr/item/ARM_1995_31_2_a0/

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