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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:
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


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