Periodic solutions of the forced relativistic pendulum

Abstract

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*}