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Application of two parameter eigencurves to Sturm–Liouville problems with eigenparameter-dependent boundary conditions

Published online by Cambridge University Press:  14 November 2011

P. A. Binding  and
Patrick J. Browne
P. A. Binding
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, CanadaT2N 1N4
Patrick J. Browne
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, CanadaS7N 0W0
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Abstract

Oscillation, comparison and asymptotic theory for the Sturm-Liouville problem

with 1/p, q, r ε L1 ([0, 1]), p, r > 0, are studied subject to eigenvalue-dependent boundary conditions

This continues previous work on cases with (− 1)j δj ≦ 0 where δj = ajdj − bjcj. We now consider the remaining sign conditions for δj, exploiting the interplay between the graph of cot θ− (λ, 1), for a modified Prüfer angle θ−, and the eigencurves of a related two-parameter problem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 6 , 1995 , pp. 1205 - 1218
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Atkinson, F. V.. Discrete and Continuous Boundary Problems (New York: Academic Press, 1964).Google Scholar
2Binding, P. A. and Browne, P. J.. Applications of two-parameter spectral theory to symmetric generalized eigenvalue problems. Appl. Anal. 29 (1988), 107–42.CrossRefGoogle Scholar
3Binding, P. A. and Browne, P. J.. Spectral properties of two parameter eigenvalue problems II. Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), 39–51. Corrigendum (with R. H. Picard) ibid. 115 (1990), 87-90.CrossRefGoogle Scholar
4Binding, P. A. and Ye, Q.. Variational principles without definiteness conditions. S1AM J. Math. Anal. 22(1991), 1575–83.Google Scholar
5Binding, P. A. and Ye, Q.. Variational principles for indefinite eigenvalue problems Lin. Alg. Appl, 218(1995), 251–62.CrossRefGoogle Scholar
6Binding, P. A., Browne, P. J. and Seddighi, K.. Sturm-Liouville problems with eigenparameter dependent boundary conditions. Proc. Edinburgh Math. Soc. 37 (1993), 57–72.CrossRefGoogle Scholar
7Dijksma, A.. Eigenfunction expansion for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 1–27.CrossRefGoogle Scholar
8Dijksma, A., Langer, H. and Snoo, H. de. Symmetric Sturm-Liouville operators with eigenvalue depending boundary conditions. Oscillation, Bifurcation, Chaos, CMS Conference Proceedings 8, 87–116 (Providence, R.I.: American Mathematical Society, 1987).Google Scholar
9Fulton, C. T.. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 11 (1977), 293–308.CrossRefGoogle Scholar
10Lancaster, P., Shkalikov, A. and Ye, Q.. Strongly definitizable linear pencils in Hilbert space. Integral Equations Operator Theory 17 (1993), 338–60.CrossRefGoogle Scholar
11Langer, H. and Schneider, A.. On spectral properties of regular quasidefinite pencils F – AG. Results Math. 19(1991), 89–109.CrossRefGoogle Scholar
12Reid, W. T.. Sturmian Theory for Ordinary Differential Equations (Berlin: Springer, 1980).CrossRefGoogle Scholar
13Walter, J.. Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 133(1973), 301–12.CrossRefGoogle Scholar