## On the distance between zeroes

HTML articles powered by AMS MathViewer

- by William T. Patula
- Proc. Amer. Math. Soc.
**52**(1975), 247-251 - DOI: https://doi.org/10.1090/S0002-9939-1975-0379986-5
- PDF | Request permission

## 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.## References

- Earl A. Coddington and Norman Levinson,
*Theory of ordinary differential equations*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR**0069338** - W. N. Everitt, M. Giertz, and J. Weidmann,
*Some remarks on a separation and limit-point criterion of second-order, ordinary differential expressions*, Math. Ann.**200**(1973), 335–346. MR**326047**, DOI 10.1007/BF01428264 - Philip Hartman,
*Ordinary differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0171038** - W. T. Patula and J. S. W. Wong,
*An $L^{p}$-analogue of the Weyl alternative*, Math. Ann.**197**(1972), 9–28. MR**299865**, DOI 10.1007/BF01427949 - Aurel Wintner,
*A criterion of oscillatory stability*, Quart. Appl. Math.**7**(1949), 115–117. MR**28499**, DOI 10.1090/S0033-569X-1949-28499-6

## Bibliographic Information

- © Copyright 1975 American Mathematical Society
- 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