  Let the motion of a rarefied gas between two parallel infinite plates of the
same temperature be governed by the Boltzmann equation with diffuse reflection
boundaries, where the left plate is at rest and the right one oscillates in its
normal direction periodically in time. For such boundary-value problem, we
establish the existence of a time-periodic solution with the same period,
provided that the amplitude of the right boundary is suitably small. The
positivity of the solution is also proved basing on the study of its large-time
asymptotic stability for the corresponding initial-boundary value problem. For
the proof of existence, we develop uniform estimates on the approximate
solutions in the time-periodic setting and make a bootstrap argument by
reducing the coefficient of the extra penalty term from a large enough constant
to zero.
