Proceedings of the American Mathematical Society

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

Author(s): Steve Clark; Don Hinton
Journal: Proc. Amer. Math. Soc. 130 (2002), 3005-3015.
MSC (2000): Primary 34C10, 34L15; Secondary 34B24, 34D10
Posted: March 15, 2002
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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.

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

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34C10, 34L15, 34B24, 34D10

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

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