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September/October 2010 Periodic solutions of the forced relativistic pendulum
Haöim Brezis, Jean Mawhin
Differential Integral Equations 23(9/10): 801-810 (September/October 2010). DOI: 10.57262/die/1356019113


The existence of at least one classical T-periodic solution is proved for differential equations of the form \begin{eqnarray*} (\phi(u'))' - g(x,u) = h(x) \end{eqnarray*} when $\phi : (-a,a) \to {\mathbb R}$ is an increasing homeomorphism, $g$ is a Carathéodory function T-periodic with respect to $x$, $2\pi$-periodic with respect to $u$, of mean value zero with respect to $u,$ and $h \in L^1_{loc}(R)$ is T-periodic and has mean value zero. The problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space of T-periodic Lipschitz functions, and then to show, using variational inequalities techniques, that such a minimum solves the differential equation. A special case is the ``relativistic forced pendulum equation'' \begin{eqnarray*} \Big (\frac{u'}{\sqrt{1 - u'^2}} \Big )' + A \sin u = h(x). \end{eqnarray*}


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Haöim Brezis. Jean Mawhin. "Periodic solutions of the forced relativistic pendulum." Differential Integral Equations 23 (9/10) 801 - 810, September/October 2010.


Published: September/October 2010
First available in Project Euclid: 20 December 2012

zbMATH: 1240.34207
MathSciNet: MR2675583
Digital Object Identifier: 10.57262/die/1356019113

Primary: 34C25 , 49J40 , 58E30 , 58E35

Rights: Copyright © 2010 Khayyam Publishing, Inc.


Vol.23 • No. 9/10 • September/October 2010
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