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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Oscillation and comparison for second order differential equations
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by Keith Schrader PDF
Proc. Amer. Math. Soc. 51 (1975), 131-136 Request permission

Abstract:

Consider the equations \[ (1)\quad x'' = f(t,x,xโ€™)\] and \[ (2)\quad x'' = g(t,x,xโ€™)\] where $f,g:[a, + \infty ) \times {R^2} \to R$ are continuous. Assume that solutions of initial value problems for (1) and for (2) are unique and extend to $[a, + \infty )$. Let $f(t,0,0) = 0 = g(t,0,0)$ for $t \epsilon [a, + \infty )$ and $f(t,x,xโ€™)/x \leq g(t,x,xโ€™)/x$ for $|x| > 0$ and $(t,x,xโ€™)$ in the domain of $f$ and $g$. Under these hypotheses it can be shown that if every solution of (2) has a zero on an interval $I \subset [a, + \infty )$ then it follows that every solution of (1) has a zero on $I$. In particular this shows that under these hypotheses (2) is oscillatory (every solution has a zero on $[a + n, + \infty )$ for each positive integer $n$) implies (1) is oscillatory.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 51 (1975), 131-136
  • MSC: Primary 34C10
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0372336-X
  • MathSciNet review: 0372336