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Disconjugacy and the limit-point condition for fourth-order operators*
Published online by Cambridge University Press: 14 February 2012
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