| Title:
|
Conjugacy criteria and principal solutions of self-adjoint differential equations (English) |
| Author:
|
Došlý, Ondřej |
| Author:
|
Komenda, Jan |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
31 |
| Issue:
|
3 |
| Year:
|
1995 |
| Pages:
|
217-238 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Oscillation properties of the self-adjoint, two term, differential equation \[(-1)^n(p(x)y^{(n)})^{(n)}+q(x)y=0\qquad \mathrm {(*)}\] are investigated. Using the variational method and the concept of the principal system of solutions it is proved that (*) is conjugate on $R=(-\infty ,\infty )$ if there exist an integer $m\in \lbrace 0,1,\dots ,n-1\rbrace $ and $c_0,\dots ,c_m\in R$ such that \[\int _\infty ^0 x^{2(n-m-1)}p^{-1}(x)\,dx=\infty =\int _0^\infty x^{2(n-m-1)}p^{-1}(x)\,dx\] and \[\limsup _{x_1\downarrow -\infty ,x_2\uparrow \infty }\int _{x_1}^{x_2}q(x)(c_0+c_1x+\dots + c_mx^m)^2\,dx\le 0,\quad q(x)\lnot \equiv 0.\] Some extensions of this criterion are suggested. (English) |
| Keyword:
|
conjugate points |
| Keyword:
|
principal system of solutions |
| Keyword:
|
variational method |
| Keyword:
|
conjugacy criteria |
| MSC:
|
34C10 |
| idZBL:
|
Zbl 0841.34033 |
| idMR:
|
MR1368260 |
| . |
| Date available:
|
2008-06-06T21:29:03Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107542 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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