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Zhang, Chao; Sun, Shurong

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

J. Appl. Math. 2012, Article ID 486230, 10 p. (2012).
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.

MSC:

34N05 Dynamic equations on time scales or measure chains
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\textit{C. Zhang} and \textit{S. Sun}, J. Appl. Math. 2012, Article ID 486230, 10 p. (2012; Zbl 1244.34117)
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References:

[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
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