Periodic solutions of the forced relativistic pendulum

The existence of at least one classical T-periodic solution is proved for di erential equations of the form (φ(u'))' - g(x, u) = h(x) when φ : (-a, a) → R is an increasing homeomorphism, g is a Carathéodory function T-periodic with respect to x, 2π-periodic with respect to u, of mean value zero with respect to u, and h ∈ L¹ _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 di erential equation. A special case is the relativistic forced pendulum equation" {equation presented}.