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

We prove that u′=Au+ϕu′=Au+ϕ has on RR a mild solution uϕ∈BUC(R,X)uϕ∈BUC(R,X) (that is, bounded and uniformly continuous), where AA is the generator of a C0C0-semigroup on the Banach space XX with resolvent satisfying ‖R(it,A)‖=O(|t|−θ)‖R(it,A)‖=O(|t|−θ), |t|→∞|t|→∞, for some θ>12, ϕ∈L∞(R,X)ϕ∈L∞(R,X) and isp(ϕ)∩σ(A)=0̸ (sp=Beurlingspectrum). As a consequence it is shown that if FF is the space of almost periodic, almost automorphic, bounded Levitan almost periodic or one of certain classes of recurrent functions and ϕϕ as above is such that only Mhϕ≔(1/h)∫0hϕ(⋅+s)ds∈F for each h>0h>0, then uϕ∈F∩BUC(R,X)uϕ∈F∩BUC(R,X). These results seem new and strengthen several recent theorems.