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Motivated by thesmoothing properties of heat semigroups on unbounded domains and the conditional stability of hyperbolic semigroups, we develop a unified approach toward the problems on the existence of periodic solutions to the evolution equation. Our method is based on the analysis of-stability of the semigroupgenerated by  A , i.e.,  t &gt;0, for certain couple of Banach spacesand real-valued function  φ  satisfying. Our theory covers both cases corresponding tosmoothing properties and the conditional stability of hyperbolic semigroups as well as some other important situations relating to the polynomial or exponential stability of semigroups. As illustrations for our theory, we give applications to the existence and uniqueness of periodic solutions to Navier–Stokes and damped wave equations.

dynamical systems

(                     X                     ,                     Y                     ,                     φ                     )                  -Stable semigroups, periodic solutions, and applications