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Proceedings of the American Mathematical Society

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

 
 

 

Conjugate point properties for an even order linear differential equation


Author: George W. Johnson
Journal: Proc. Amer. Math. Soc. 45 (1974), 371-376
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1974-0348186-6
MathSciNet review: 0348186
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Abstract: Consider an even order equation $ {D_n}y = 0$, where $ {D_n}$ is defined recursively by $ {D_0}y = y,{a_{k,k + 1}}{D_k}y = \{ ({D_{k - 1}}y)' - \Sigma _{i = 1}^k{a_{ki}}{D_{i - 1}}y\} $, where $ {a_{ij}}$ is continuous on $ [0,\infty ),{a_{i,i + 1}} > 0,i = 1, \cdots ,n - 1,{a_{n,n + 1}} \equiv 1$, and $ {a_{ij}} \equiv 0$ for $ j > i + 1$. The $ k$th 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 exactly $ n - 1 + k$ zeros on $ [a,{\eta _k}(a)]$, and are shown to exist, to have odd 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)$ if $ {\eta _k}(a) < \infty $.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0348186-6
Keywords: Extremal solution, conjugate point, zero of multiplicity $ k$
Article copyright: © Copyright 1974 American Mathematical Society

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