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Lv, Haiyan; Shi, Yuming

Error estimate of eigenvalues of perturbed second-order discrete Sturm-Liouville problems. (English) Zbl 1172.39022

Linear Algebra Appl. 430, No. 8-9, 2389-2415 (2009).
The authors study the error estimates of eigenvalues of perturbed second-order discrete Sturm-Liouville problem under a certain non-singularity. They discuss two special perturbed problems: 1) the equation is perturbed and the boundary condition is invariant; 2) the coefficients of equation are perturbed and the weight function and the boundary condition are invariant. The error estimate will be simpler for these two special perturbed problems. Finally, an example to illustrate the necessity of the non-singularity condition is given.
Reviewer: Hussain A. El-Saify (Beni Suef)

Cited in 4 Documents

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34B24 Sturm-Liouville theory
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators

Keywords:

second-order vector difference equation; Sturm-Liouville problem; error estimate of eigenvalue; continuous dependence; perturbation

Software:

SLEIGN2
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Full Text: DOI

References:

[1] Ahlbrandt, C. D.; Peterson, A. C., Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations (1996), Kluwer Academic Publishers: Kluwer Academic Publishers The Netherlands · Zbl 0860.39001
[2] Atkinson, F. V., Discrete and Continuous Boundary Problems (1964), Academic Press Inc.: Academic Press Inc. New York · Zbl 0117.05806
[3] Bailey, P. B.; Everitt, W. N.; Zettl, A., Computing eigenvalues of singular Sturm-Liouville problems, Results Math., 20, 391-423 (1991) · Zbl 0755.65082
[4] Bailey, P. B.; Everitt, W. N.; Zettl, A., Regular and singular Sturm-Liouville problems with coupled boundary conditions, Proc. Roy. Soc. Edinburgh, 126A, 505-514 (1996) · Zbl 0855.34026
[5] P.B. Bailey, W.N. Everitt, A. Zettl, SLEIGN2: a Fortran code for the numerical approximation of eigenvalues, eigenfunctions and continuous spectrum of Sturm-Liouville problems, 1997, available on <www.math.niu.edu/Zettl/SL2>.; P.B. Bailey, W.N. Everitt, A. Zettl, SLEIGN2: a Fortran code for the numerical approximation of eigenvalues, eigenfunctions and continuous spectrum of Sturm-Liouville problems, 1997, available on <www.math.niu.edu/Zettl/SL2>.
[6] Bailey, P. B.; Everitt, W. N.; Zettl, A., The SLEIGN2 Sturm-Liouville code, ACM Trans. Math. Softw., 21, 143-192 (2001) · Zbl 1070.65576
[7] Bailey, P. B.; Gordon, M. K.; Shampine, L. F., Automatic solution of the Sturm-Liouville problem, ACM Trans. Math. Softw., 4, 193-208 (1978) · Zbl 0384.65045
[8] Bohner, M., Discrete linear Hamiltonian eigenvalue problems, Comput. Math. Appl., 36, 179-192 (1998) · Zbl 0933.39033
[9] Clark, S. L., A spectral analysis for self-adjoint operators generated by a class of second order difference equations, J. Math. Anal. Appl., 197, 267-285 (1996) · Zbl 0857.39008
[10] Hinton, D. B.; Lewis, R. T., Spectral analysis of second order difference equations, J. Math. Anal. Appl., 63, 421-438 (1978) · Zbl 0392.39001
[11] Horn, R. A.; Johnson, C. A., Matrix Analysis (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0576.15001
[12] Jirari, A., Second-order Sturm-Liouville difference equations and orthogonal polynomials, Mem. Amer. Math. Soc., 113 (1995) · Zbl 0817.39004
[13] Kong, Q.; Wu, H.; Zettl, A., Dependence of the \(n\) th Sturm-Liouville eigenvalue on the problem, J. Differential Equations, 156, 328-354 (1999) · Zbl 0932.34081
[14] Kong, Q.; Zettl, A., Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131, 1-19 (1996) · Zbl 0862.34020
[15] Shi, Y., Spectral theory of discrete linear Hamiltonian systems, J. Math. Anal. Appl., 289, 554-570 (2004) · Zbl 1047.39016
[16] Shi, Y., Weyl-Titchmarsh theory for a class of discrete linear Hamiltonian systems, Linear Algebra Appl., 416, 452-519 (2006) · Zbl 1100.39020
[17] Shi, Y.; Chen, S., Spectral theory of second-order vector difference equations, J. Math. Anal. Appl., 239, 195-212 (1999) · Zbl 0934.39002
[18] Wang, Y.; Shi, Y., Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions, J. Math. Anal. Appl., 309, 56-69 (2005) · Zbl 1083.39019
[19] Sun, H.; Shi, Y., Eigenvalue of second-order difference equations with coupled boundary conditions, Linear Algebra Appl., 414, 361-372 (2006) · Zbl 1092.39011
[20] S. Sun, Y. Shi, H. Wu, On discrete Sturm-Liouville problems, I. Structure on space of problems and their applications, submitted for publication.; S. Sun, Y. Shi, H. Wu, On discrete Sturm-Liouville problems, I. Structure on space of problems and their applications, submitted for publication.
[21] S. Sun, Y. Shi, H. Wu, On discrete Sturm-Liouville problems, II. Dependence of the \(n\) th eigenvalue on the problem, submitted for publication.; S. Sun, Y. Shi, H. Wu, On discrete Sturm-Liouville problems, II. Dependence of the \(n\) th eigenvalue on the problem, submitted for publication.
[22] A. Zettl, Strum-Liouville Theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Rhode, Island, 2005.; A. Zettl, Strum-Liouville Theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Rhode, Island, 2005. · Zbl 1103.34001
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