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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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  • 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

Don Hinton
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996

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