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

DOI:
https://doi.org/10.1090/S0002-9939-02-06392-X

Published electronically:
March 15, 2002

MathSciNet review:
1908924

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Abstract | References | Similar Articles | Additional Information

Abstract: Conditions are given which guarantee that the least real eigenvalue is positive for certain boundary value problems for the vector-matrix equation . This leads to conditions which guarantee the stable boundedness, according to Krein, for solutions of with certain real values of . As a consequence, a result first stated by Krein is proven.

**1.**R. P. Agarwal and P. Y. H. Pang,*Opial Inequalities with Applications in Differential and Difference Equations*, Kluwer Academic Publishers, Dordrecht, 1995. MR**96h:34001****2.**C. Ahlbrandt and A. Peterson,*Discrete Hamiltonian Systems*, Kluwer Academic Publishers, Dordrecht, 1996. MR**98m:39043****3.**M. S. Ashbaugh and R. D. Benguria,*Eigenvalue ratios for Sturm-Liouville operators*, J. Diff. Eqs.**103**(1993), 205-219. MR**94c:34125****4.**C. Bandle,*Extremal problems for eigenvalues of the Sturm-Liouville type*, In*General Inequalities*, 5, Birkhäuser, Basel, 1987, 319-339. MR**90k:34020****5.**C. Bennewitz and E. J. M. Veling,*Optimal bounds for the spectrum of a one-dimensional operator*, In*General Inequalities*, 6, Birkhäuser, Basel, 1992, 257-268. MR**94c:34126****6.**R. C. Brown, D. B. Hinton and S. Schwabik,*Applications of a one-dimensional Sobolev inequality to eigenvalue problems*, Diff. and Integral Eqs.**9**(1996), 481-498. MR**96k:34180****7.**R. C. Brown, private communication, September, 2000.**8.**R. C. Brown, A. M. Fink, and D. B. Hinton,*Some Opial, Lyapunov, and De la Valée Poussin inequalities with nonhomogeneous boundary conditions*, J. of Inequal. and Appl.**5**(2000), 11-37. MR**2000m:34073****9.**S. Clark and D. Hinton,*A Liapunov inequality for linear Hamiltonian systems*, Math. Inequalities and Appl.**1**(1998), 201-209. MR**99c:34056****10.**W. A. Coppel,*Disconjugacy*, Lecture Notes in Mathematics**220**, Springer-Verlag, Berlin, 1971. MR**57:778****11.**E. B. Davies,*A hierarchical method for obtaining eigenvalue enclosures*, Math. of Computation,**69**(2000), 1435-1455. MR**2001a:34148****12.**M. S. P. Eastham,*The Spectral Theory of Periodic Differential Equations*, Scottish Academic Press, Edinburgh, 1973.**13.**F. Gesztesy, G. M. Graf and B. Simon,*The ground state energy of Schrödinger operators*, Commun. Math. Phys.**150**(1992), 375-384. MR**93j:47070****14.**P. Hartman,*Ordinary Differential Equations*, 2nd ed., Birkhäuser, Boston, 1982. MR**83e:34002****15.**M. G. Krein,*On tests for stable boundedness of solutions of periodic canonical systems*, Amer. Math. Soc. Transl. Ser. (2),**120**(1983), 71-110.**16.**M. G. Krein,*On certain problems of the maximum and minimum of characteristic values and on the Lyapunov zones of stability*, Amer. Math. Soc. Transl. Ser. (2),**1**(1955), 163-187. MR**17:484e****17.**D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink,*Inequalities Involving Functions and their Integrals and Derivatives*, Kluwer, Dordrecht, 1991. MR**93m:26036****18.**M. Plum,*Eigenvalue inclusions for second order ordinary differential operators by a numerical homotopy method*, Z. Angew. Math. Phys.**41**(1990), 205-226. MR**91d:65116****19.**J. D. Pryce,*Numerical Solutions of Sturm-Liouville Problems*, Clarendon Press, Oxford, 1993. MR**95h:65056****20.**W. T. Reid,*Ordinary Differential Equations*, Wiley, New York, 1971. MR**42:7963****21.**W. T. Reid,*Sturmian Theory for Ordinary Differential Equations*, Springer-Verlag, New York, 1980. MR**82f:34002**

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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:
https://doi.org/10.1090/S0002-9939-02-06392-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