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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 estimates for solutions of $ y\sp{''}-q\sp{2}y=0$


Author: T. T. Read
Journal: Proc. Amer. Math. Soc. 45 (1974), 332-338
MSC: Primary 34D05
MathSciNet review: 0344611
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Abstract: It is shown for any nonnegative continuous function $ q$ on $ [0,\infty )$ and any $ c < 1$ that any positive increasing solution $ y$ of $ y'' - {q^2}y = 0$ satisfies $ y(x) \geq y(0)\exp (c\int_0^x {q(t)dt)} $ on the complement of a set of finite Lebesgue measure. It is also shown that if $ \lim \inf (\int_0^x {q(t)dt/x) > 0} $ then the equation has an exponentially increasing solution and an exponentially decreasing solution.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0344611-5
PII: S 0002-9939(1974)0344611-5
Keywords: Exponential dichotomy, monotonie solution, Riccati equation
Article copyright: © Copyright 1974 American Mathematical Society