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On the distance between zeroes

1975
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Proceedings of the American Mathematical Society
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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 e (a, 6) where \x(t)\ is maximized. As a corollary, it is shown that if the above equation is oscillatory and if q (t) £ L [0, °»), 1 < p < °°, then the distance between consecutive zeroes must become unbounded. Consider the following second order linear differential equation: continuous on

doi:10.1090/s0002-9939-1975-0379986-5
fatcat:26kafxor3jc6dj4xg3i5jaryom