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Title: Generalized reciprocity for self-adjoint linear differential equations (English)
Author: Došlý, Ondřej
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 31
Issue: 2
Year: 1995
Pages: 85-96
Summary lang: English
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Category: math
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Summary: 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. (English)
Keyword: Self-adjoint equation
Keyword: reciprocal equation
Keyword: property BD
Keyword: principal solution
Keyword: minimal differential operator.Supported by the Grant No. 201/93/0452 of the Czech Grant Agency
MSC: 34B05
MSC: 34C10
MSC: 34L05
idZBL: Zbl 0841.34032
idMR: MR1357977
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Date available: 2008-06-06T21:28:16Z
Last updated: 2012-05-10
Stable URL: http://hdl.handle.net/10338.dmlcz/107529
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Reference: [1] Ahlbrandt, C. D.: Principal and antiprincipal solutions of selfadjoint diferential systems and their reciprocals.Rocky Mountain J. Math. 2 (1972), 169–189. MR 0296388
Reference: [2] Ahlbrandt, C. D.: Equivalent boundary value problems for self-adjoint differential systems.J. Diff. Equations 9 (1971), 420–435. Zbl 0218.34020, MR 0284636
Reference: [3] Ahlbrandt, C. D., Hinton, D. B., Lewis, R. T.: The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory.J. Math. Anal. Appl. 81 (1981), 234–277. MR 0618771
Reference: [4] Ahlbrandt, C. D., Hinton, D. B., Lewis, R. T.: Necessary and sufficient conditions for the discreteness of the spectrum of certain singular differential operators.Canad J. Math. 33 (1981), 229–246. MR 0608867
Reference: [5] Coppel, W. A.: Disconjugacy.Lectures Notes in Math., No. 220, Springer Verlag, Berlin-Heidelberg 1971. Zbl 0224.34003, MR 0460785
Reference: [6] Došlý, O.: On transformation of self-adjoint linear diferential systems and their reciprocals.Annal. Pol. Math. 50 (1990), 223–234.
Reference: [7] Došlý, O.: Oscillation criteria and the discreteness of the spectrum of self-adjoint, even order, differential operators.Proc. Roy. Soc. Edinburgh 119A (1991), 219–232.
Reference: [8] Došlý, O.: Transformations of linear Hamiltonian systems preserving oscillatory behaviour.Arch. Math. 27 (1991), 211–219. MR 1189218
Reference: [9] Došlý, O.: Principal solutions and transformations of linear Hamiltonian systems.Arch. Math. 28 (1992), 113–120. MR 1201872
Reference: [10] Došlý, O., Osička, J.: Kneser type oscillation criteria for self-adjoint differential equations.Georgian Math. J. 2 (1995), 241–258. MR 1334880
Reference: [11] Dunford, N., Schwartz, J. T.: Linear Operators II, Spectral Theory.Interscience, New York 1982.
Reference: [12] Evans, W. D., Kwong, M. K., Zettl, A.: Lower bounds for spectrum of ordinary differential operators.J. Diff. Equations 48 (1983), 123–155. MR 0692847
Reference: [13] Glazman, I. M.: Direct Methods of Qualitative Analysis of Singular Differential Operators.Jerusalem 1965.
Reference: [14] Hinton, D. B., Lewis, R. T.: Discrete spectra criteria for singular differential operators with middle terms.Math. Proc. Cambridge Philos. Soc. 77 (1975), 337–347. MR 0367358
Reference: [15] Lewis, R. T.: The discreteness of the spectrum of self-adjoint, even order, differential operators.Proc. Amer. Mat. Soc. 42 (1974), 480–482. MR 0330608
Reference: [16] Müller-Pfeiffer, E.: Spectral Theory of Ordinary Differential Operators.Chelsea, 1981. MR 0606197
Reference: [17] Naimark, M. A.: Linear Differential Operators,.Part II, Ungar, New York, 1968. Zbl 0227.34020
Reference: [18] Reid, W. T.: Sturmian Theory for Ordinary Differential Equations.Springer Verlag, New York 1980. Zbl 0459.34001, MR 0606199
Reference: [19] Rasmussen, C. H.: Oscillation and asymptotic behaviour of systems of ordinary linear differential equations.Trans. Amer. Math. Soc. 256 (1979), 1–48. MR 0546906
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