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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The $ k$th conjugate point function for an even order linear differential equation


Author: George W. Johnson
Journal: Proc. Amer. Math. Soc. 42 (1974), 563-568
MSC: Primary 34C10
MathSciNet review: 0333340
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Abstract: For an even order, two term equation $ {L_n}y = p(x)y,p(x) > 0$, x in $ [0,\infty )$, the kth conjugate point function $ {\eta _k}(a)$ is defined and is shown to be a strictly increasing continuous function with domain [0, b) or $ [0,\infty )$. Extremal solutions are defined as nontrivial solutions with $ n - 1 + k$ zeros on $ [a,{\eta _k}(a)]$, and are shown to have exactly $ n - 1 + k$ zeros, with even order zeros at a and $ {\eta _k}(a)$ and exactly $ k - 1$ odd order zeros in $ (a,{\eta _k}(a))$, thus establishing that $ {\eta _k}(a) < {\eta _{k + 1}}(a)$.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0333340-X
Keywords: Extremal solution, conjugate point, zero of multiplicity k
Article copyright: © Copyright 1974 American Mathematical Society