  We apply a Lyapunov function to obtain conditions for the existence and
uniqueness of small classical time-periodic solutions to first order
quasilinear 1D hyperbolic systems with (nonlinear) nonlocal boundary conditions
in a strip. The boundary conditions cover different types of reflections from
the boundary as well as integral operators with delays. In the first step we
use a Lyapunov approach to derive sufficient conditions for the robust
exponential stability of the boundary value problems for a linear(ized)
homogeneous problem. Under those conditions and a number of non-resonance
conditions, in the second step we prove the existence and uniqueness of smooth
time-periodic solutions to the corresponding linear nonhomogeneous problems. In
the third step, we prove a perturbation theorem stating that the periodic
solutions survive under small perturbations of all coefficients of the
hyperbolic system. In the last step, we apply the linear results to construct
small and smooth time-periodic solutions to the quasilinear problems.
