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Disconjugacy and the limit-point condition for fourth-order operators*

Published online by Cambridge University Press:  14 February 2012

Robert M. Kauffman
Robert M. Kauffman
Affiliation:
Department of Mathematics, University of Dundee
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Synopsis

Let L denote the ordinary differential operator given by Lf = (pf″)″ + (qf′)′ + rf, with p″, q′ and r continuous functions on [0,∞), and with p>0, q ≦ 0, and r ≧ 0. It is proved that if the equation Lg = 0 possesses a non-oscillatory solution, then any non-trivial solution f to Lf = 0 such that f(0) = f′(0) = 0 is eventually bounded away from zero.

This theorem is used to prove that, for a general class of functions q and r containing the polynomials as a very special case, the equation Lg = 0 has at most two linearly independent square integrable solutions, when p is identically one, q ≦ 0 and r ≧ 0.

Finally, the main theorem is applied to show that certain sixth-order operators are limit-3.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 76 , Issue 1 , 1976 , pp. 77 - 80
Copyright
Copyright © Royal Society of Edinburgh 1976

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References

1Coppel, W. A.. Discondugacy. Lecture Notes in Mathematics 220 (Berlin: Springer, 1971).Google Scholar
2Eastham, M. S. P.. The limit-2 case of fourth order differential equations. Quart. J. Math. Oxford Ser. 22 (1971), 131–134.CrossRefGoogle Scholar
3Everitt, W. N.. Some positive definite differential operators. J. London Math. Soc. 43 (1968), 465–473.CrossRefGoogle Scholar
4Nehari, Z.. Disconjugate linear differential operators. Trans. Amer. Math. Soc. 129 (1967), 500–516.CrossRefGoogle Scholar
5Schneider, L. J.. Oscillation properties of the 2-2 disconjugate fourth order selfadjoint differential equation. Proc. Amer. Math. Soc. 28 (1971), 545–550.CrossRefGoogle Scholar