  We consider functional differential equations(FDEs) which are perturbations
of smooth ordinary differential equations(ODEs). The FDE can involve multiple
state-dependent delays or distributed delays (forward or backward). We show
that, under some mild assumptions, if the ODE has a nondegenerate periodic
orbit, then the FDE has a smooth periodic orbit. Moreover, we get smooth
dependence of the periodic orbit and its frequency on parameters with high
regularity.
  The result also applies to FDEs which are perturbations of some evolutionary
partial differential equations(PDEs).
  The proof consists in solving functional equations satisfied by the
parameterization of the periodic orbit and the frequency using a fixed point
approach. We do not need to consider the smoothness of the evolution or even
the phase space of the FDEs.
