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

We prove that $u^{\mathrm{\prime }}=Au+\varphi$ has on $\mathbb{R}$ a mild solution $u_{\phi}\in BUC (\Bbb{R},X)$(thatisboundedanduniformlycontinuous),where$A$ is the generator of a $C_0$-semigroup on the Banach space $X$ with resolvent satisfying $\mathrm{\mid }\mathrm{\mid }R(it,A)\mathrm{\mid }\mathrm{\mid }=O(\mathrm{\mid }t{\mathrm{\mid }}^{-\theta })$, $\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty }$, with some $\theta

1/2$,$\phi\in L^{\infty} (\Bbb{R},{X})$and$i,sp (\phi)\cap \sigma (A)=\emptyset$\mathrm{.}Asaconsequenceitisshownthatif${\Cal F}$ is the space of almost periodic, almost automorphic, bounded Levitan almost periodic or certain classes of recurrent functions and $\varphi$ as above is such that $M_h \phi:=(1/h)\int_0^h \phi (\cdot+s), ds \in \Cal {F}$foreach$h >0$,then$u_{\phi}\in \Cal {F}\cap BUC (\Bbb{R},X)$. These results seem new and strengthen several recent theorems.