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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 \[ (1)\quad x'' = f(t,x,xโ€™)\] 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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Keywords: Oscillation, comparison, second order, nonlinear
Article copyright: © Copyright 1975 American Mathematical Society