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Proceedings of the American Mathematical Society

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

 

 

On the location of zeros of second-order differential equations


Author: Vadim Komkov
Journal: Proc. Amer. Math. Soc. 35 (1972), 217-222
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1972-0298128-5
MathSciNet review: 0298128
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Abstract: The paper considers the location of zeros of the equation $ (\alpha (t)x')' + \gamma (t)x = 0,t \in [{t_0},{t_1}]$. The following theorem is proved. Let $ [a,a + T],T = na$ (n a positive integer), be a subset of $ [{t_0},{t_1}]$. Denote $ \omega = \pi /T$. Let the coefficient functions obey the inequality $ \smallint _a^{a + T}\{ \gamma (t) - {\omega ^2}\alpha (t){\sin ^2}(\omega t)\} dt > {\omega ^2}\smallint _a^{a + T}\{ \alpha \cos 2\omega t\} dt$. Then every solution of this equation will have a zero on $ [a,a + T]$. A more general form of this theorem is also proved.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0298128-5
Keywords: Zeros of second-order equations, Leighton variational theorem
Article copyright: © Copyright 1972 American Mathematical Society