  A computer-assisted argument is given, which provides existence proofs for
periodic orbits in state-dependent delayed perturbations of ordinary
differential equations (ODEs). Assuming that the unperturbed ODE has an
isolated periodic orbit, we introduce a set of polynomial inequalities whose
successful verification leads to the existence of periodic orbits in the
perturbed delay equation. We present a general algorithm, which describes a way
of computing the coefficients of the polynomials and optimizing their variables
so that the polynomial inequalities are satisfied. The algorithm uses the tools
of validated numerics together with Chebyshev series expansion to obtain the
periodic orbit of the ODE as well as the solution of the variational equations,
which are both used to compute rigorously the coefficients of the polynomials.
We apply our algorithm to prove the existence of periodic orbits in a
state-dependent delayed perturbation of the van der Pol equation.
