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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Eigenvalues of a Sturm-Liouville problem
and inequalities of Lyapunov type


Author: Chung-Wei Ha
Journal: Proc. Amer. Math. Soc. 126 (1998), 3507-3511
MSC (1991): Primary 34L15, 34L20
MathSciNet review: 1622758
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Abstract: We consider the eigenvalue problem $u''+\lambda u +p(x)u=0$ in $(0,\pi )$, $u(0)=u(\pi )=0$, where $p\in L^{1}(0,\pi )$ keeps a fixed sign and $\|p\|_{L^{1}}> 0$, and we obtain some lower and upper bounds for $\|p\|_{L^{1}}$ in terms of its nonnegative eigenvalues $\lambda $. Two typical results are: (1) $\|p\|_{L^{1}}>\sqrt {\lambda }\,|\sin {\sqrt {\lambda }\,\pi }|$ if $\lambda > 1 $ and is not the square of a positive integer; (2) $\|p\|_{L^{1}}\le 16/\pi $ if $\lambda =0$ is the smallest eigenvalue.


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Additional Information

Chung-Wei Ha
Affiliation: Department of Mathematics, National Tsing Hua University, Hsin Chu, Taiwan
Email: cwha@math.nthu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-98-05010-2
PII: S 0002-9939(98)05010-2
Received by editor(s): September 23, 1996
Communicated by: Hal L. Smith
Article copyright: © Copyright 1998 American Mathematical Society