| Title:
|
Eventual disconjugacy of $y^{(n)} + \mu p(x) y = 0$ for every $\mu $ (English) |
| Author:
|
Elias, Uri |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
40 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
193-200 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The work characterizes when is the equation $ y^{ (n) } + \mu p(x) y = 0 $ eventually disconjugate for every value of $ \mu $ and gives an explicit necessary and sufficient integral criterion for it. For suitable integers $ q $, the eventually disconjugate (and disfocal) equation has 2-dimensional subspaces of solutions $ y $ such that $ y^{ (i) } > 0 $, $ i = 0, \ldots , q-1 $, $ (-1)^{i-q} y^{ (i) } > 0 $, $ i = q, \ldots , n $. We characterize the “smallest” of such solutions and conjecture the shape of the “largest” one. Examples demonstrate that the estimates are sharp. (English) |
| Keyword:
|
eventual disconjugacy |
| MSC:
|
34C10 |
| idZBL:
|
Zbl 1116.34317 |
| idMR:
|
MR2068690 |
| . |
| Date available:
|
2008-06-06T22:43:27Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107900 |
| . |
| Reference:
|
[Ea] Eastham, M. S. P.: The asymptotic solution of linear differential systems.University Press, Oxford, 1989. Zbl 0674.34045, MR 1006434 |
| Reference:
|
[E1] Elias, U.: Oscillation theory of two-term differential equations.Kluwer Academic Publishers, Dordrecht, 1997. Zbl 0878.34022, MR 1445292 |
| Reference:
|
[E2] Elias, U.: Comparison theorems for disfocality and disconjugacy of differential equations.SIAM J. Math. Anal. 15 (1984), 922–931. Zbl 0554.34021, MR 0755852 |
| Reference:
|
[KC] Kiguradze, I. T., and Chanturia, T. A.: Asymptotic properties of solutions of nonautonomous ordinary differential equations.Kluwer Academic Publishers, Dordrecht, 1993. MR 1220223 |
| Reference:
|
[Ki] Kim, W. J.: Asymptotic properties of nonoscillatory solutions of higher order differential equations.Pacific J. Math. 93 (1981), 107–114. Zbl 0488.34046, MR 0621601 |
| Reference:
|
[PSz] Pólya, G., and Szegö, G.: Problems and theorems in analysis.Springer-Verlag, Berlin, 1972. |
| . |
