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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Focal points for a linear differential equation whose coefficients are of constant signs


Author: Uri Elias
Journal: Trans. Amer. Math. Soc. 249 (1979), 187-202
MSC: Primary 34C10; Secondary 34A30, 34B05
MathSciNet review: 526317
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Abstract: The differential equation considered is $ {y^{(n)}} + \Sigma {{p_i}(x){y^{(i)}}} = 0$, where $ {\sigma _i}{p_i}(x) \geqslant 0,i = 0,\ldots,n - 1,{\sigma _i} = \pm 1$. The focal point $ \zeta (a)$ is defined as the least value of s, $ s > a$, such that there exists a nontrivial solution y which satisfies $ {y^{(i)}}(a) = 0,{\sigma _i}{\sigma _{i + 1}} > 0$ and $ {y^{(i)}}(s) = 0$, $ {\sigma _i}{\sigma _{i + 1}} < 0$. Our method is based on a characterization of $ \zeta (a)$ by solutions which satisfy $ {\sigma _i}{y^{(i)}} > 0,i = 0,\ldots,n - 1$, on $ [a,b]$, $ b < \zeta (a)$. We study the behavior of the function $ \zeta $ and the dependence of $ \zeta (a)$ on $ {p_0},\ldots,{p_{n - 1}}$ when at least a certain $ {p_i}(x)$ does not vanish identically near a or near $ \zeta (a)$. As an application we prove the existence of an eigenvalue of a related boundary value problem.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0526317-1
PII: S 0002-9947(1979)0526317-1
Article copyright: © Copyright 1979 American Mathematical Society