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Proceedings of the American Mathematical Society

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

 

 

On the distance between zeroes


Author: William T. Patula
Journal: Proc. Amer. Math. Soc. 52 (1975), 247-251
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1975-0379986-5
MathSciNet review: 0379986
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Abstract: For the equation $ x'' + q(t)x = 0$, let $ x(t)$ be a solution with consecutive zeroes at $ t = a$ and $ t = b$. A simple inequality is proven that relates not only $ a$ and $ b$ to the integral of $ {q^ + }(t)$ but also any point $ c\epsilon (a,b)$ where $ \vert x(t)\vert$ is maximized. As a corollary, it is shown that if the above equation is oscillatory and if $ {q^ + }(t)\epsilon {L^p}[0,\infty ),1 \leqslant p < \infty $, then the distance between consecutive zeroes must become unbounded.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0379986-5
Article copyright: © Copyright 1975 American Mathematical Society