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We prove an existence result for T-periodic retarded functional differential equations of the type xʹ(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.

Retarded Functional Differential Equations on Manifolds and Applications to Motion Problems for Forced Constrained Systems