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

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- by Steve Clark and Don Hinton PDF
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**130**(2002), 3005-3015 Request permission

## 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.## References

- Ravi P. Agarwal and Peter Y. H. Pang,
*Opial inequalities with applications in differential and difference equations*, Mathematics and its Applications, vol. 320, Kluwer Academic Publishers, Dordrecht, 1995. MR**1340422**, DOI 10.1007/978-94-015-8426-5 - Calvin D. Ahlbrandt and Allan C. Peterson,
*Discrete Hamiltonian systems*, Kluwer Texts in the Mathematical Sciences, vol. 16, Kluwer Academic Publishers Group, Dordrecht, 1996. Difference equations, continued fractions, and Riccati equations. MR**1423802**, DOI 10.1007/978-1-4757-2467-7 - Mark S. Ashbaugh and Rafael D. Benguria,
*Eigenvalue ratios for Sturm-Liouville operators*, J. Differential Equations**103**(1993), no. 1, 205–219. MR**1218744**, DOI 10.1006/jdeq.1993.1047 - Catherine Bandle,
*Extremal problems for eigenvalues of the Sturm-Liouville type*, General inequalities, 5 (Oberwolfach, 1986) Internat. Schriftenreihe Numer. Math., vol. 80, Birkhäuser, Basel, 1987, pp. 319–336. MR**1018157**, DOI 10.1007/978-3-0348-7192-1_{2}6 - C. Bennewitz and E. J. M. Veling,
*Optimal bounds for the spectrum of a one-dimensional Schrödinger operator*, General inequalities, 6 (Oberwolfach, 1990) Internat. Ser. Numer. Math., vol. 103, Birkhäuser, Basel, 1992, pp. 257–268. MR**1213012**, DOI 10.1007/978-3-0348-7565-3_{2}2 - R. C. Brown, D. B. Hinton, and Š. Schwabik,
*Applications of a one-dimensional Sobolev inequality to eigenvalue problems*, Differential Integral Equations**9**(1996), no. 3, 481–498. MR**1371703** - R. C. Brown, private communication, September, 2000.
- Richard C. Brown, A. M. Fink, and Don B. Hinton,
*Some Opial, Lyapunov, and De la Valée Poussin inequalities with nonhomogeneous boundary conditions*, J. Inequal. Appl.**5**(2000), no. 1, 11–37. MR**1740210**, DOI 10.1155/S1025583400000023 - Steve Clark and Don Hinton,
*A Liapunov inequality for linear Hamiltonian systems*, Math. Inequal. Appl.**1**(1998), no. 2, 201–209. MR**1613448**, DOI 10.7153/mia-01-18 - W. A. Coppel,
*Disconjugacy*, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR**0460785**, DOI 10.1007/BFb0058618 - E. B. Davies,
*A hierarchical method for obtaining eigenvalue enclosures*, Math. Comp.**69**(2000), no. 232, 1435–1455. MR**1710648**, DOI 10.1090/S0025-5718-00-01238-2 - M. S. P. Eastham,
*The Spectral Theory of Periodic Differential Equations*, Scottish Academic Press, Edinburgh, 1973. - F. Gesztesy, G. M. Graf, and B. Simon,
*The ground state energy of Schrödinger operators*, Comm. Math. Phys.**150**(1992), no. 2, 375–384. MR**1194022**, DOI 10.1007/BF02096665 - Philip Hartman,
*Ordinary differential equations*, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR**658490** - M. G. Krein,
*On tests for stable boundedness of solutions of periodic canonical systems*, Amer. Math. Soc. Transl. Ser. (2),**120**(1983), 71–110. - Saunders MacLane,
*Steinitz field towers for modular fields*, Trans. Amer. Math. Soc.**46**(1939), 23–45. MR**17**, DOI 10.1090/S0002-9947-1939-0000017-3 - D. S. Mitrinović, J. E. Pečarić, and A. M. Fink,
*Inequalities involving functions and their integrals and derivatives*, Mathematics and its Applications (East European Series), vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1991. MR**1190927**, DOI 10.1007/978-94-011-3562-7 - Michael Plum,
*Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method*, Z. Angew. Math. Phys.**41**(1990), no. 2, 205–226. MR**1045812**, DOI 10.1007/BF00945108 - John D. Pryce,
*Numerical solution of Sturm-Liouville problems*, Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR**1283388** - William T. Reid,
*Ordinary differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0273082** - William T. Reid,
*Sturmian theory for ordinary differential equations*, Applied Mathematical Sciences, vol. 31, Springer-Verlag, New York-Berlin, 1980. With a preface by John Burns. MR**606199**, DOI 10.1007/978-1-4612-6110-0

## 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
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1908924