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A Note on Klein’s Oscillation Theorem for Periodic Boundary Conditions

Published online by Cambridge University Press:  20 November 2018

M. Faierman
M. Faierman*
Affiliation:
Loyola College, Montreal, Quebec
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Recently Howe [4] has considered the oscillation theory for the two-parameter eigenvalue problem

1a

1b

subjected to the boundary conditions

2a

2b

where for i = 1, 2, — ∞<ai<bi<∞, and qi are real-valued, continuous functions in [ai, bi], pi is positive in [ai, biz], and pi(ai)=pi(bi). Furthermore, it is also assumed that (A1B2—A2B1)≠0 for all values of x1 and x2 in their respective intervals.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 5 , February 1975 , pp. 749 - 755
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Atkinson, F. V., Multiparameter Eigenvalue Problems, Vol. I, Academic, New York, N.Y., 1972.Google Scholar
2. Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, McGraw-Hill, New York, N.Y., 1955.Google Scholar
3. Faierman, M., Asymptotic formulae for the eigenvalues of a two parameter system of ordinary differential equations of the second order, Canad. Math. Bull, (to appear).Google Scholar
4. Howe, A., Klein’s oscillation theorem for period boundary conditions, Canad. J. Math. 23 (1971), 699–703.Google Scholar
5. Richardson, R. G. D., Theorems of oscillation for two linear differential equations of the second order, Trans. Amer. Math. Soc. 13 (1912), 22–34.Google Scholar
6. Stewart, C. A., Advanced Calculus, Methuen, London, 1940.Google Scholar