Some estimates for the first eigenvalue of the Sturm-Liouville problem with a weight integral condition

Maria Telnova

Mathematica Bohemica (2012)

  • Volume: 137, Issue: 2, page 229-238
  • ISSN: 0862-7959

Abstract

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Let λ 1 ( Q ) be the first eigenvalue of the Sturm-Liouville problem y ' ' - Q ( x ) y + λ y = 0 , y ( 0 ) = y ( 1 ) = 0 , 0 < x < 1 . We give some estimates for m α , β , γ = inf Q ∈ T α , β , γ λ 1 ( Q ) and M α , β , γ = sup Q ∈ T α , β , γ λ 1 ( Q ) , where T α , β , γ is the set of real-valued measurable on 0 , 1 x α ( 1 - x ) β -weighted L γ -functions Q with non-negative values such that ∫ 0 1 x α ( 1 - x ) β Q γ ( x ) d x = 1 ( α , β , γ ∈ ℝ , γ ≠ 0 ) .

How to cite

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Telnova, Maria. "Some estimates for the first eigenvalue of the Sturm-Liouville problem with a weight integral condition." Mathematica Bohemica 137.2 (2012): 229-238. <http://eudml.org/doc/246694>.

@article{Telnova2012,
abstract = {Let $\lambda _1(Q)$ be the first eigenvalue of the Sturm-Liouville problem \[ y^\{\prime \prime \}-Q(x)y+\lambda y=0,\quad y(0)=y(1)=0,\quad 0<x<1. \] 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 \ne 0)$.},
author = {Telnova, Maria},
journal = {Mathematica Bohemica},
keywords = {first eigenvalue; Sturm-Liouville problem; weight integral condition; first eigenvalue; Sturm-Liouville problem; weight integral condition},
language = {eng},
number = {2},
pages = {229-238},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some estimates for the first eigenvalue of the Sturm-Liouville problem with a weight integral condition},
url = {http://eudml.org/doc/246694},
volume = {137},
year = {2012},
}

TY - JOUR
AU - Telnova, Maria
TI - Some estimates for the first eigenvalue of the Sturm-Liouville problem with a weight integral condition
JO - Mathematica Bohemica
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 137
IS - 2
SP - 229
EP - 238
AB - Let $\lambda _1(Q)$ be the first eigenvalue of the Sturm-Liouville problem \[ y^{\prime \prime }-Q(x)y+\lambda y=0,\quad y(0)=y(1)=0,\quad 0<x<1. \] 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 \ne 0)$.
LA - eng
KW - first eigenvalue; Sturm-Liouville problem; weight integral condition; first eigenvalue; Sturm-Liouville problem; weight integral condition
UR - http://eudml.org/doc/246694
ER -

References

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  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. (1996) Zbl0883.34027MR1406051
  2. Kuralbaeva, K. Z., On estimate of the first eigenvalue of a Sturm-Liouville operator, Differents. Uravn. 32 852-853 (1996). (1996) 
  3. Besov, O. V., Il'in, V. P., Nikol'skiy, S. M., Integral Representations of Functions and Imbedding Theorems, Nauka, Moskva (1996), Russian. (1996) MR1450401

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