| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2008-06-06T21:28:16Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107529 |
| . |
| 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 |
| . |
