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

 

Positive eigenvalues of second order boundary value problems and a theorem of M. G. Krein


Authors: Steve Clark and Don Hinton
Journal: Proc. Amer. Math. Soc. 130 (2002), 3005-3015
MSC (2000): Primary 34C10, 34L15; Secondary 34B24, 34D10
Published electronically: March 15, 2002
MathSciNet review: 1908924
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Abstract: Conditions are given which guarantee that the least real eigenvalue is positive for certain boundary value problems for the vector-matrix equation $-y^{\prime\prime}+p(x)y=\gamma w(x)y$. This leads to conditions which guarantee the stable boundedness, according to Krein, for solutions of $y^{\prime\prime}+\lambda p(x)y=0$ with certain real values of $\lambda$. As a consequence, a result first stated by Krein is proven.


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Additional Information

Steve Clark
Affiliation: Department of Mathematics, University of Missouri-Rolla, Rolla, Missouri 65409
Email: sclark@umr.edu

Don Hinton
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: hinton@math.utk.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06392-X
PII: S 0002-9939(02)06392-X
Keywords: Stable boundedness, positive eigenvalues, Opial inequality
Received by editor(s): September 8, 2000
Received by editor(s) in revised form: May 14, 2001
Published electronically: March 15, 2002
Communicated by: Carmen Chicone
Article copyright: © Copyright 2002 American Mathematical Society