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

Canonical forms and principal systems for general disconjugate equations


Author: William F. Trench
Journal: Trans. Amer. Math. Soc. 189 (1974), 319-327
MSC: Primary 34C10
MathSciNet review: 0330632
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Abstract: It is shown that the disconjugate equation (1) $ Lx \equiv (1/{\beta _n})(d/dt) \cdot (1/{\beta _{n - 1}}) \cdots (d/dt)(1/{\beta _1})(d/dt)(x/{\beta _0}) = 0$ , a $ < t < b$, where $ {\beta _i} > 0$, and (2) $ {\beta _i} \in C(a,b)$, can be written in essentially unique canonical forms so that $ {\smallint ^b}{\beta _i}dt = \infty ({\smallint _a}{\beta _i}dt = \infty )$ for $ 1 \leq i \leq n - 1$. From this it follows easily that (1) has solutions $ {x_1}, \ldots ,{x_n}$ which are positive in (a, b) near $ b(a)$ and satisfy $ {\lim _{t \to b}} - {x_i}(t)/{x_j}(t) = 0({\lim _{t \to a}} + {x_i}(t)/{x_j}(t) = \infty )$ for $ 1 \leq i < j \leq n$. Necessary and sufficient conditions are given for (1) to have solutions $ {y_1}, \ldots ,{y_n}$ such that $ {\lim _{t \to b}} - {y_i}(t)/{y_j}(t) = {\lim _{t \to a}} + {y_j}(t)/{y_i}(t) = 0$ for $ 1 \leq i < j \leq n$. Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition $ (2'){\beta _i} \in {C^{(n - i)}}(a,b)$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0330632-X
PII: S 0002-9947(1974)0330632-X
Keywords: Disconjugacy, principal systems
Article copyright: © Copyright 1974 American Mathematical Society