A. Raouf Chouikha
The Period Function of Second Order Differential Equations
(261K, Postscript)

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 $0$.