Existence of bounded uniformly continuous mild solutions on $\mathbb{R}$ of evolution equations and some applications

We prove that there is $x_{\varphi }\in X$ for which (*)$\frac{d u(t)}{dt}= A u(t) + \phi (t) $,$u(0)=x$hason$\r$amildsolution$u\in C_{ub} (\r,X)$(thatisboundedanduniformlycontinuous)with$u(0)=x_{\phi}$,where$A$ is the generator of a holomorphic $C_0$-semigroup $(T(t){)}_{t\ge 0}$ on $X$ with sup ${}_{t\ge 0}\mathrm{\mid }\mathrm{\mid }T(t)\mathrm{\mid }\mathrm{\mid }<\mathrm{\infty }$, $\varphi \in L^{\mathrm{\infty }}(\mathring{,}X)$ and $i,sp (\phi)\cap \sigma (A)=\emptyset$\mathrm{.}Asaconsequenceitisshownthatif$\n$ is the space of almost periodic $AP$, almost automorphic $AA$, bounded Levitan almost periodic $LA{P}_b$, certain classes of recurrent functions $RE{C}_b$ and $\varphi \in L^{\mathrm{\infty }}(\mathring{,}X)$ such that $M_h \phi:=(1/h)\int_0^h \phi (\cdot+s), ds \in \n$foreach$h >0$,then$u\in \n\cap C_{ub}$. These results seem new and generalize and strengthen several recent Theorems.