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We prove that $u'= A u + \phi $ has on $\Bbb{R}$ a mild solution $u_{\phi}\inBUC (\Bbb{R},X)$ (that is bounded and uniformly continuous), where $A$ is thegenerator of a $C_0$-semigroup on the Banach space ${X}$ with resolventsatisfying $||R(it,A)||= O(|t|^{-\theta})$, $|t|\to \infty $, with some $\theta> 1/2$, $\phi\in L^{\infty} (\Bbb{R},{X})$ and $i\,sp (\phi)\cap \sigma(A)=\emptyset$. As a consequence it is shown that if ${\Cal F}$ is the space ofalmost periodic, almost automorphic, bounded Levitan almost periodic or certainclasses of recurrent functions and $\phi$ as above is such that $M_h\phi:=(1/h)\int_0^h \phi (\cdot+s)\, ds \in \Cal {F}$ for each $h >0$, then$u_{\phi}\in \Cal {F}\cap BUC (\Bbb{R},X)$. These results seem new andstrengthen several recent theorems.

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