×

Matrix representations of Sturm-Liouville problems with eigenparameter-dependent boundary conditions. (English) Zbl 1262.34031

Summary: We show that a class of regular self-adjoint Sturm-Liouville problems with eigenparameter-dependent boundary conditions are equivalent to a certain class of matrix problems. Equivalent here means that they have exactly the same eigenvalues.

MSC:

34B24 Sturm-Liouville theory
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atkinson, F. V., Discrete and Continuous Boundary Value Problems (1964), Academic Press: Academic Press New York/London · Zbl 0117.05806
[2] Kong, Q.; Wu, H.; Zettl, A., Sturm-Liouville problems with finite spectrum, J. Math. Anal. Appl., 263, 748-762 (2001) · Zbl 1001.34019
[3] Kong, Q.; Volkmer, H.; Zettl, A., Matrix representations of Sturm-Liouville problems with finite spectrum, Results Math., 54, 103-116 (2009) · Zbl 1185.34032
[4] Ao, J. J.; Sun, J.; Zhang, M. Z., The finite spectrum of Sturm-Liouville problems with transmission conditions, Appl. Math. Comput., 218, 1166-1173 (2011) · Zbl 1242.34042
[5] Ao, J. J.; Sun, J.; Zhang, M. Z., The matrix representations of Sturm-Liouville problems with transmission conditions, Comput. Math. Appl., 63, 1335-1348 (2012) · Zbl 1247.34039
[6] J.J. Ao, J. Sun, The finite spectrum of Sturm-Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions, Results Math., in press. · Zbl 1280.34029
[7] Kong, Q.; Zettl, A., The study of Jacobi and cyclic Jacobi matrix eigenvalue problems using Sturm-Liouville theory, Linear Algebra Appl., 434, 1648-1655 (2011) · Zbl 1210.15011
[8] Kong, Q.; Zettl, A., Inverse Sturm-Liouville problems with finite spectrum, J. Math. Anal. Appl., 386, 1-9 (2012) · Zbl 1232.34023
[9] Everitt, W. N.; Race, D., On necessary and sufficient conditions for the existence of Caratheodory solutions of ordinary differential equations, Quaest. Math., 3, 507-512 (1976) · Zbl 0392.34002
[10] Zettl, A., Sturm-Liouville Theory, Mathematical Surveys and Monographs, vol. 121 (2005), Amer. Math. Soc. · Zbl 1074.34030
[11] Fulton, C.; Pruess, S., Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions, J. Math. Anal. Appl., 71, 431-462 (1979) · Zbl 0464.65056
[12] Binding, P. A.; Browne, P. J.; Watson, B. A., Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, II, J. Comput. Appl. Math., 148, 147-168 (2002) · Zbl 1019.34028
[13] Binding, P. A.; Browne, P. J.; Watson, B. A., Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, J. London Math. Soc., 62, 1, 161-182 (2000) · Zbl 0960.34010
[14] Fulton, C., Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 77, 293-308 (1977) · Zbl 0376.34008
[15] Walter, J., Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z., 133, 301-312 (1973) · Zbl 0246.47058
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.