| Title:
|
Some estimates for the first eigenvalue of the Sturm-Liouville problem with a weight integral condition (English) |
| Author:
|
Telnova, Maria |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
137 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
229-238 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\lambda _1(Q)$ be the first eigenvalue of the Sturm-Liouville problem $$ y''-Q(x)y+\lambda y=0,\quad y(0)=y(1)=0,\quad 0<x<1. $$ We give some estimates for $m_{\alpha ,\beta ,\gamma }=\inf _{Q\in T_{\alpha ,\beta ,\gamma }}\lambda _1(Q)$ and $M_{\alpha ,\beta ,\gamma }=\sup _{Q\in T_{\alpha ,\beta ,\gamma }}\lambda _1(Q)$, where $T_{\alpha ,\beta ,\gamma }$ is the set of real-valued measurable on $\left [0,1\right ]$ $x^\alpha (1-x)^\beta $-weighted $L_\gamma $-functions $Q$ with non-negative values such that $\int _0^1x^\alpha (1-x)^\beta Q^{\gamma }(x) {\rm d} x=1$ $(\alpha ,\beta ,\gamma \in \mathbb {R},\gamma \neq 0)$. (English) |
| Keyword:
|
first eigenvalue |
| Keyword:
|
Sturm-Liouville problem |
| Keyword:
|
weight integral condition |
| MSC:
|
34B24 |
| MSC:
|
34L15 |
| idZBL:
|
Zbl 1265.34313 |
| idMR:
|
MR2978268 |
| DOI:
|
10.21136/MB.2012.142868 |
| . |
| Date available:
|
2012-06-08T10:15:25Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142868 |
| . |
| Reference:
|
[1] Egorov, Yu. V., Kondrat'ev, V. A.: Estimates for the first eigenvalue in some Sturm-Liouville problems.Russ. Math. Surv. 51 (1996), translation from Usp. Math. Nauk 51 (1996), 73-144. Zbl 0883.34027, MR 1406051 |
| Reference:
|
[2] Kuralbaeva, K. Z.: On estimate of the first eigenvalue of a Sturm-Liouville operator.Differents. Uravn. 32 852-853 (1996). |
| Reference:
|
[3] Besov, O. V., Il'in, V. P., Nikol'skiy, S. M.: Integral Representations of Functions and Imbedding Theorems.Nauka, Moskva (1996), Russian. MR 1450401 |
| . |
