 01175 A. Raouf Chouikha
 The Period Function of Second Order Differential Equations
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May 11, 01

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Abstract. We are interested in the behaviour of the period function for
equations of the type $u'' + g(u) = 0$ and $u'' + f(u)u' + g(u) = 0$
with a center at the origin $0$. $g$ is a function of class $C^k$.
For the conservative case, if $k \geq 2$ one shows that the Opial
criterion is the better one among those for which these the necessary
condition $g''(0) = 0$ holds. In the case where $f$ is of class $C^1$
and $k \geq 3$ , the Lienard equations \ $ u'' + f(u) u' + g(u) = 0$
\ may have a monotonic period function if
$g'(0) g^{(3)}(0)  \frac{5}{3} {g''}^{2}(0)  \frac{2}{3} {f'}^{2}(0)
g'(0) \neq 0$ in a neighborhood of $0$.
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