Periodic Solutions of the Forced Relativistic Pendulum

The existence of at least one classical T-periodic solution is proved for differential equations of the form (phi(u'))' - g(x, u) = h(x) when phi : (-a, a) -> R is an increasing homeomorphism, g is a Carathedory function T-periodic with respect to x, 2 pi-periodic with respect to u, of mean value zero with respect to u, and h is an element of L-loc(1)(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" (u'/root 1-u'(2))' + A sin u = h(x)