Conditional stability and periodicity of solutions to evolution equations

We propose a new approach toward the existence and uniqueness of periodic solutions to linear and semilinear evolution equations. Our approach is based on the connection of the conditional stability of evolution families (i.e., stability only in a subspace of the Banach space containing the initial data) with the choice of the initial data from which emanates the periodic solution. We also give applications to exponentially dichotomic evolution families as well as to nonautonomous damped wave equations.