Eigenvalues of a Sturm-Liouville problem and inequalities of Lyapunov type
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- Proc. Amer. Math. Soc. 126 (1998), 3507-3511 Request permission
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.References
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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
- Received by editor(s): September 23, 1996
- Communicated by: Hal L. Smith
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3507-3511
- MSC (1991): Primary 34L15, 34L20
- DOI: https://doi.org/10.1090/S0002-9939-98-05010-2
- MathSciNet review: 1622758