2011/08/17 by Basit, Bolis, Günzler, Hans
#43A60 Secondary 43A99 #47A10 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 47D06
paper · doi:10.48550/arxiv.1108.3398
We prove that u'= A u + ϕ has on ℝ a mild solution uϕ∈ BUC (ℝ,X) (that is bounded and uniformly continuous), where A is the generator of a C0-semigroup on the Banach space X with resolvent satisfying ||R(it,A)||= O(|t|-θ), |t|→ ∞ , with some θ> 1/2, ϕ∈ L∞ (ℝ,X) and i sp (ϕ)∩ σ(A)=∅. As a consequence it is shown that if \Cal F is the space of almost periodic, almost automorphic, bounded Levitan almost periodic or certain classes of recurrent functions and ϕ as above is such that Mh ϕ:=(1/h)∫0h ϕ(⋅+s) ds ∈ \Cal F for each h >0, then uϕ∈ \Cal F∩ BUC (ℝ,X). These results seem new and strengthen several recent theorems.