| Title:
|
Singular eigenvalue problems for second order linear ordinary differential equations (English) |
| Author:
|
Elbert, Árpád |
| Author:
|
Kusano, Takaŝi |
| Author:
|
Naito, Manabu |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
59-72 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider linear differential equations of the form \[ (p(t)x^{\prime })^{\prime }+\lambda q(t)x=0~~~(p(t)>0,~q(t)>0) \qquad \mathrm {(A)}\] on an infinite interval $[a,\infty )$ and study the problem of finding those values of $\lambda $ for which () has principal solutions $x_{0}(t;\lambda )$ vanishing at $t = a$. This problem may well be called a singular eigenvalue problem, since requiring $x_{0}(t;\lambda )$ to be a principal solution can be considered as a boundary condition at $t=\infty $. Similarly to the regular eigenvalue problems for () on compact intervals, we can prove a theorem asserting that there exists a sequence $\lbrace \lambda _{n}\rbrace $ of eigenvalues such that $\displaystyle 0<\lambda _{0}<\lambda _{1}<\cdots <\lambda _{n}<\cdots $, $\displaystyle \lim _{n\rightarrow \infty }\lambda _{n}=\infty $, and the eigenfunction $x_{0}(t;\lambda _{n})$ corresponding to $\lambda = \lambda _{n}$ has exactly $n$ zeros in $(a,\infty ),~n=0,1,2,\dots $. We also show that a similar situation holds for nonprincipal solutions of () under stronger assumptions on $p(t)$ and $q(t)$. (English) |
| Keyword:
|
Singular eigenvalue problem |
| Keyword:
|
Sturm-Liouville equation |
| Keyword:
|
zeros of nonoscillatory solutions |
| MSC:
|
34B05 |
| MSC:
|
34B24 |
| MSC:
|
34B40 |
| MSC:
|
34C10 |
| idZBL:
|
Zbl 0914.34021 |
| idMR:
|
MR1629660 |
| . |
| Date available:
|
2009-02-17T10:10:18Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107633 |
| . |
| Reference:
|
[1] P. Hartman: Ordinary Differential Equations.Wiley, New York, 1964. Zbl 0125.32102, MR 0171038 |
| Reference:
|
[2] P. Hartman: Boundary value problems for second order ordinary differential equations involving a parameter.J. Differential Equations 12 (1972), 194–212. Zbl 0255.34012, MR 0335927 |
| Reference:
|
[3] E. Hille: Lectures on Ordinary Differential Equations.Addison-Wesley, Reading, Massachusetts, 1969. Zbl 0179.40301, MR 0249698 |
| Reference:
|
[4] Y. Kabeya: Uniqueness of nodal fast-decaying radial solutions to a linear elliptic equations on $\mathbb{R}^n$.preprint. |
| Reference:
|
[5] M. Naito: Radial entire solutions of the linear equation $\Delta u + \lambda p(|x|)u = 0$.Hiroshima Math. J. 19 (1989), 431–439. Zbl 0716.35002, MR 1027944 |
| Reference:
|
[6] Z. Nehari: Oscillation criteria for second-order linear differential equations.Trans. Amer. Math. Soc. 85 (1957), 428–445. Zbl 0078.07602, MR 0087816 |
| . |
