Eigenvalue comparisons for second-order linear equations with boundary value conditions on time scales. (English) Zbl 1244.34117

Summary: We study the eigenvalue comparisons for second-order linear equations with boundary conditions on time scales. Using results from matrix algebras, the existence and comparison results concerning eigenvalues are obtained.


34N05 Dynamic equations on time scales or measure chains
Full Text: DOI


[1] C. C. Travis, “Comparison of eigenvalues for linear differential equations of order 2n,” Transactions of the American Mathematical Society, vol. 177, pp. 363-374, 1973. · Zbl 0272.34033
[2] J. M. Davis, P. W. Eloe, and J. Henderson, “Comparison of eigenvalues for discrete Lidstone boundary value problems,” Dynamic Systems and Applications, vol. 8, no. 3-4, pp. 381-388, 1999. · Zbl 0941.39009
[3] J. Diaz and A. Peterson, “Comparison theorems for a right disfocal eigenvalue problem,” in Inequalities and Applications, vol. 3 of World Scientific Series in Applicable Analysis, pp. 149-177, World Scientific Publishing, River Edge, NJ, USA, 1994. · Zbl 0882.34079
[4] D. Hankerson and J. Henderson, “Comparison of eigenvalues for n-point boundary value problems for difference equations,” in Differential Equations (Colorado Springs, CO, 1989), vol. 127 of Lecture Notes in Pure and Applied Mathematics, pp. 203-208, Dekker, New York, NY, USA, 1991. · Zbl 0713.39001
[5] D. Hankerson and A. Peterson, “Comparison of eigenvalues for focal point problems for nth order difference equations,” Differential and Integral Equations, vol. 3, no. 2, pp. 363-380, 1990. · Zbl 0733.39002
[6] D. Hankerson and A. Peterson, “A positivity result applied to difference equations,” Journal of Approximation Theory, vol. 59, no. 1, pp. 76-86, 1989. · Zbl 0695.39004
[7] D. Hankerson and A. Peterson, “Comparison theorems for eigenvalue problems for nth order differential equations,” Proceedings of the American Mathematical Society, vol. 104, no. 4, pp. 1204-1211, 1988. · Zbl 0692.34020
[8] J. Henderson and K. R. Prasad, “Comparison of eigenvalues for Lidstone boundary value problems on a measure chain,” Computers & Mathematics with Applications, vol. 38, no. 11-12, pp. 55-62, 1999. · Zbl 1010.34079
[9] E. R. Kaufmann, “Comparison of eigenvalues for eigenvalue problems of a right disfocal operator,” Panamerican Mathematical Journal, vol. 4, no. 4, pp. 103-124, 1994. · Zbl 0851.47016
[10] T. Ando, “Majorization, doubly stochastic matrices, and comparison of eigenvalues,” Linear Algebra and its Applications, vol. 118, pp. 163-248, 1989. · Zbl 0673.15011
[11] C. A. Akemann and N. Weaver, “Minimal upper bounds of commuting operators,” Proceedings of the American Mathematical Society, vol. 124, no. 11, pp. 3469-3476, 1996. · Zbl 0863.46037
[12] W. Arveson, “On groups of automorphisms of operator algebras,” Journal of Functional Analysis, vol. 15, pp. 217-243, 1974. · Zbl 0296.46064
[13] T. Kato, “Spectral order and a matrix limit theorem,” Linear and Multilinear Algebra, vol. 8, no. 1, pp. 15-19, 1979/80. · Zbl 0424.47018
[14] M. P. Olson, “The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice,” Proceedings of the American Mathematical Society, vol. 28, pp. 537-544, 1971. · Zbl 0215.20504
[15] J. Hamhalter, “Spectral order of operators and range projections,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1122-1134, 2007. · Zbl 1120.46040
[16] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001, An introduction with Applications. · Zbl 0978.39001
[17] V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, vol. 370 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996. · Zbl 0869.34039
[18] R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in Mathematics, vol. 35, no. 1-2, pp. 3-22, 1999. · Zbl 0927.39003
[19] J. Sherman and W. J. Morrison, “Adjustment of an inverse matrix corresponding to a change in one element of a given matrix,” Annals of Mathematical Statistics, vol. 21, pp. 124-127, 1950. · Zbl 0037.00901
[20] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, 1962. · Zbl 0133.08602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.