On the distance between zeroes

Patula, William T.

doi:10.1090/S0002-9939-1975-0379986-5

On the distance between zeroes
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by William T. Patula

Proc. Amer. Math. Soc. 52 (1975), 247-251

DOI: https://doi.org/10.1090/S0002-9939-1975-0379986-5

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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 $|x(t)|$ 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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