  The paper concerns boundary value problems for general nonautonomous first
order quasilinear hyperbolic systems in a strip. We construct small global
classical solutions, assuming that the right hand sides are small. In the case
that all data of the quasilinear problem are almost periodic, we prove that the
bounded solution is also almost periodic. For the nonhomogeneous version of a
linearized problem, we provide stable dissipativity conditions ensuring a
unique bounded continuous solution for any smooth right-hand sides. In the
autonomous case, this solution is two times continuously differentiable. In the
nonautonomous case, the continuous solution is differentiable under additional
dissipativity conditions, which are essential. A crucial ingredient of our
approach is a perturbation theorem for general linear hyperbolic systems. One
of the technical complications we overcome is the "loss of smoothness" property
of hyperbolic PDEs.
