Existence of bounded uniformly continuous mild solutions on $\Bbb{R}$ of evolution equations and their asymptotic behaviour

Abstract

We prove that $u'= A u + ϕ$ has on $\Bbb{R}$ a mild solution $u_ϕ\in BUC (\Bbb{R},X)$ (that is bounded and uniformly continuous), where $A$ is the generator of a $C_0$-semigroup on the Banach space ${X}$ with resolvent satisfying $||R(it,A)||= O(|t|^{-θ})$, $|t|\to \infty $, with some $θ> 1/2$, $ϕ\in L^{\infty} (\Bbb{R},{X})$ and $i\,sp (ϕ)\cap σ(A)=\emptyset$. 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 $M_h ϕ:=(1/h)\int_0^h ϕ(\cdot+s)\, ds \in \Cal {F}$ for each $h >0$, then $u_ϕ\in \Cal {F}\cap BUC (\Bbb{R},X)$. These results seem new and strengthen several recent theorems.