Retarded functional differential equations on manifolds and applications to motion problems for forced constrained systems

We prove an existence result for T-periodic retarded functional differential equations of the type x'(t) = f(t,x_t), where f is a T-periodic functional tangent vector field on a smooth manifold. As an application we show that any constrained system acted on by a periodic force, possibly with delay, admits a forced oscillation provided that the constraint is a topologically nontrivial compact manifold and the frictional coefficient is nonzero. We conjecture that the same assertion holds true even in the frictionless case.