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), 30053015
MSC (2000):
Primary 34C10, 34L15; Secondary 34B24, 34D10
Published electronically:
March 15, 2002
MathSciNet review:
1908924
Fulltext PDF Free Access
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 vectormatrix 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 SturmLiouville operators, J. Diff. Eqs. 103 (1993), 205219. MR 94c:34125
 4.
C. Bandle, Extremal problems for eigenvalues of the SturmLiouville type, In General Inequalities, 5, Birkhäuser, Basel, 1987, 319339. MR 90k:34020
 5.
C. Bennewitz and E. J. M. Veling, Optimal bounds for the spectrum of a onedimensional operator, In General Inequalities, 6, Birkhäuser, Basel, 1992, 257268. MR 94c:34126
 6.
R. C. Brown, D. B. Hinton and S. Schwabik, Applications of a onedimensional Sobolev inequality to eigenvalue problems, Diff. and Integral Eqs. 9 (1996), 481498. 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), 1137. MR 2000m:34073
 9.
S. Clark and D. Hinton, A Liapunov inequality for linear Hamiltonian systems, Math. Inequalities and Appl. 1 (1998), 201209. MR 99c:34056
 10.
W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics 220, SpringerVerlag, Berlin, 1971. MR 57:778
 11.
E. B. Davies, A hierarchical method for obtaining eigenvalue enclosures, Math. of Computation, 69 (2000), 14351455. 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), 375384. 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), 71110.
 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), 163187. 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), 205226. MR 91d:65116
 19.
J. D. Pryce, Numerical Solutions of SturmLiouville 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, SpringerVerlag, New York, 1980. MR 82f:34002
 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 SturmLiouville operators, J. Diff. Eqs. 103 (1993), 205219. MR 94c:34125
 4.
 C. Bandle, Extremal problems for eigenvalues of the SturmLiouville type, In General Inequalities, 5, Birkhäuser, Basel, 1987, 319339. MR 90k:34020
 5.
 C. Bennewitz and E. J. M. Veling, Optimal bounds for the spectrum of a onedimensional operator, In General Inequalities, 6, Birkhäuser, Basel, 1992, 257268. MR 94c:34126
 6.
 R. C. Brown, D. B. Hinton and S. Schwabik, Applications of a onedimensional Sobolev inequality to eigenvalue problems, Diff. and Integral Eqs. 9 (1996), 481498. 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), 1137. MR 2000m:34073
 9.
 S. Clark and D. Hinton, A Liapunov inequality for linear Hamiltonian systems, Math. Inequalities and Appl. 1 (1998), 201209. MR 99c:34056
 10.
 W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics 220, SpringerVerlag, Berlin, 1971. MR 57:778
 11.
 E. B. Davies, A hierarchical method for obtaining eigenvalue enclosures, Math. of Computation, 69 (2000), 14351455. 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), 375384. 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), 71110.
 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), 163187. 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), 205226. MR 91d:65116
 19.
 J. D. Pryce, Numerical Solutions of SturmLiouville 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, SpringerVerlag, New York, 1980. MR 82f:34002
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Additional Information
Steve Clark
Affiliation:
Department of Mathematics, University of MissouriRolla, 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/S000299390206392X
PII:
S 00029939(02)06392X
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
