Existence of bounded uniformly continuous mild solutions on of evolution equations and their asymptotic behaviour

We prove that u ′ = A u + ϕ has on R a mild solution u ϕ ∈ B U C ( R , X ) (that is, bounded and uniformly continuous), where A is the generator of a C 0 -semigroup on the Banach space X with resolvent satisfying ‖ R ( i t , A ) ‖ = O ( | t | − θ ) , | t | → ∞ , for some θ > 1 2 , ϕ ∈ L ∞ ( R , X ) and i s p ( ϕ ) ∩ σ ( A ) = 0̸ (sp=Beurlingspectrum). As a consequence it is shown that if F is the space of almost periodic, almost automorphic, bounded Levitan almost periodic or one of certain classes of recurrent functions and ϕ as above is such that only M h ϕ ≔ ( 1 / h ) ∫ 0 h ϕ ( ⋅ + s ) d s ∈ F for each h > 0 , then u ϕ ∈ F ∩ B U C ( R , X ) . These results seem new and strengthen several recent theorems.