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Motivated by the L p − L q smoothing 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 u ˙ = A u (t) + B g (u) (t). Our method is based on the analysis of (X , Y , φ) -stability of the semigroup (T (t)) t ≥ 0 generated by A, i.e. ‖ T (t) x ‖ Y ⩽ φ (t) ‖ x ‖ X , t>0, for certain couple of Banach spaces (X , Y) and real-valued function φ satisfying lim t → ∞ φ (t) = 0. Our theory covers both cases corresponding to L p − L q smoothing 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. [ABSTRACT FROM AUTHOR]