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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Interpolation between consecutive conjugate points of an $ n$th order linear differential equation


Author: G. B. Gustafson
Journal: Trans. Amer. Math. Soc. 177 (1973), 237-255
MSC: Primary 34B10
MathSciNet review: 0320419
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Abstract: The interpolation problem $ {x^{(n)}} + {P_{n - 1}}{x^{(n - 1)}} + \cdots + {P_0}x = 0$, $ {x^{(i)}}({t_j}) = 0,i = 0, \cdots ,{k_j} - 1,j = 0, \cdots ,m$, is studied on the conjugate interval $ [a,{\eta _1}(a)]$. The main result is that there exists an essentially unique nontrivial solution of the problem almost everywhere, provided $ {k_1} + \cdots + {k_m} \geq n$, and cer tain other inequalities are satisfied, with $ a = {t_0} < {t_1} < \cdots < {t_m} = {\eta _1}(a)$.

In particular, this paper corrects the results of Azbelev and Caljuk (Mat. Sb. 51 (93) (1960), 475-486; English transl., Amer. Math. Soc. Transl. (2) 42 (1964), 233-245) on third order equations, and shows that their results are correct almost everywhere.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0320419-5
Keywords: Disconjugacy, interpolation, first conjugate point
Article copyright: © Copyright 1973 American Mathematical Society