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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Exponential solutions of $ y\sp{''}+(r-q)y=0$ and the least eigenvalue of Hill's equation


Author: Thomas T. Read
Journal: Proc. Amer. Math. Soc. 50 (1975), 273-280
MSC: Primary 34C10
MathSciNet review: 0377184
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Abstract: It is shown that if $ q$ is a nonnegative continuous function on $ [0,\infty )$ such that for some positive constants $ A$ and $ L$,

$\displaystyle \mathop {\lim \inf }\limits_{x \to \infty } \int_x^{x + A} {{q^{1/2}}(t)dt} > AL,$

then $ y'' + (r - q)y = 0$ has an exponentially increasing solution and an exponentially decreasing solution whenever the uniform norm of the continuous function $ r$ satisfies $ \vert\vert r\vert{\vert _\infty } < {[L/(AL + 1)]^2}$. A refinement of the proof is used to show that for all sufficiently large values of $ k$ the least eigenvalue $ \lambda (k)$ of the two parameter Hill equation $ y'' + (\lambda - kp)y = 0$ satisfies an inequality of the form $ \lambda (k) \geqslant Pk + {B_\beta }\vert k{\vert^\beta }$ where $ P = \min p$ if $ k > 0$, $ P = \max p$ if $ k < 0$, and $ \beta $ is a constant between 0 and 1 that depends on the periodic function $ p$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0377184-2
PII: S 0002-9939(1975)0377184-2
Keywords: Exponential solution, nonoscillation, Hill's equation, periodic solution
Article copyright: © Copyright 1975 American Mathematical Society