| Title:
|
Oscillatory properties of fourth order self-adjoint differential equations (English) |
| Author:
|
Fišnarová, Simona |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
40 |
| Issue:
|
4 |
| Year:
|
2004 |
| Pages:
|
457-469 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Oscillation and nonoscillation criteria for the self-adjoint linear differential equation \[ (t^\alpha y^{\prime \prime })^{\prime \prime }-\frac{\gamma _{2,\alpha }}{t^{4-\alpha }}y=q(t)y,\quad \alpha \notin \lbrace 1, 3\rbrace \,, \] where \[ \gamma _{2,\alpha }=\frac{(\alpha -1)^2(\alpha -3)^2}{16}\] and $q$ is a real and continuous function, are established. It is proved, using these criteria, that the equation \[\left(t^\alpha y^{\prime \prime }\right)^{\prime \prime }-\left(\frac{\gamma _{2,\alpha }}{t^{4-\alpha }} + \frac{\gamma }{t^{4-\alpha }\ln ^2 t}\right)y = 0\] is nonoscillatory if and only if $\gamma \le \frac{\alpha ^2-4\alpha +5}{8}$. (English) |
| Keyword:
|
self-adjoint differential equation |
| Keyword:
|
oscillation and nonoscillation criteria |
| Keyword:
|
variational method |
| Keyword:
|
conditional oscillation. |
| MSC:
|
34C10 |
| idZBL:
|
Zbl 1117.34038 |
| idMR:
|
MR2129965 |
| . |
| Date available:
|
2008-06-06T22:44:51Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107927 |
| . |
| Reference:
|
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| Reference:
|
[2] Došlý O.: Nehari-type oscillation criteria for self-adjoint linear equations.J. Math. Anal. Appl. 182 (1994), 69–89. MR 1265883 |
| Reference:
|
[3] Došlý O.: Oscillatory properties of fourth order Sturm-Liouville differential equations.Acta Univ. Palack. Olomuc. Fac. Rerum. Natur. Math. 41 (2002), 49–59. Zbl 1055.34065, MR 1967340 |
| Reference:
|
[4] Došlý O., Osička J.: Oscillation and nonoscillation of higher order self-adjoint differential equations.Czechoslovak Math. J. 52 (127) (2002), 833-849. MR 1940063 |
| Reference:
|
[5] Došlý O., Osička J.: Oscillatory properties of higher order Sturm-Liouville differential equations.Studies Univ. Žilina, Math. Ser. 15 (2002), 25–40. Zbl 1062.34034, MR 1980760 |
| Reference:
|
[6] Glazman I. M.: Direct Methods of Qualitative Anylysis of Singular Differential Operators.Davey, Jerusalem 1965. |
| Reference:
|
[7] 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. Zbl 0298.34018, MR 0367358 |
| . |
