ON THE EIGENVALUES AND THE NODAL POINTS OF THE EIGENFUNCTIONS OF SOME EIGENVALUE PROBLEMS WITH EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS
Consider the following two eigenvalue problems: (0.1) \begin{cases}\label{eqn:1abs}y"(x)+[\lambda^2-q(x)]y(x)=0, 0 \leq x \leq \pi,\\[3pt] y(0)=0, ay'(\pi)+\lambda y(\pi)=0, \end{cases} and (0.2) \begin{cases} z"(x)+[\mu^2-q(x)]z(x)=0, 0 \leq x \leq \pi,\\[3pt] z'(0)=0, az'(...
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| Vydáno v: | Glasgow mathematical journal Ročník 63; číslo 1; s. 158 - 178 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cambridge, UK
Cambridge University Press
01.01.2021
|
| Témata: | |
| ISSN: | 0017-0895, 1469-509X |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Consider the following two eigenvalue problems:
(0.1)
\begin{cases}\label{eqn:1abs}y"(x)+[\lambda^2-q(x)]y(x)=0, 0 \leq x \leq \pi,\\[3pt] y(0)=0, ay'(\pi)+\lambda y(\pi)=0, \end{cases}
and
(0.2)
\begin{cases} z"(x)+[\mu^2-q(x)]z(x)=0, 0 \leq x \leq \pi,\\[3pt] z'(0)=0, az'(\pi)+\mu z(\pi)=0, \end{cases}
where
$q(x)$
is real-valued and integrable on [0,
$\pi$
]. Let
$\{\lambda_n\}_{n\in \mathbb{Z}\setminus \{0\}}$
and
$\{\mu_n\}_{n\in \mathbb{Z}\setminus \{0\}}$
denote the eigenvalues of equations (0.1) and (0.2), respectively. Then
\[\cdots\lt\mu_{-3}\lt\lambda_{-2}\lt\mu_{-2}\lt\lambda_{-1}\lt\mu_{-1}\lt\mu_1\lt\lambda_1\lt\mu_2\lt\lambda_2\lt\mu_3\lt\cdots.\]
Moreover, the number of zeros of the eigenfunctions of (0.1) ((0.2), respectively) corresponding to
$\lambda_n$
(
$\mu_n$
, respectively) in (0,
$\pi$
) is equal to
$|n|-1$
. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0017-0895 1469-509X |
| DOI: | 10.1017/S0017089520000087 |