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Proceedings of the American Mathematical Society

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

 

 

Oscillation and comparison for second order differential equations


Author: Keith Schrader
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
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Abstract: Consider the equations

$\displaystyle (1)\quad x'' = f(t,x,x')$

and

$\displaystyle (2)\quad x'' = g(t,x,x')$

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,x')/x \leq g(t,x,x')/x$ for $ \vert x\vert > 0$ and $ (t,x,x')$ 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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DOI: https://doi.org/10.1090/S0002-9939-1975-0372336-X
Keywords: Oscillation, comparison, second order, nonlinear
Article copyright: © Copyright 1975 American Mathematical Society