01-175 A. Raouf Chouikha
The Period Function of Second Order Differential Equations (261K, Postscript) 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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