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

Abstract

We prove that there is $x_{\phi}\in X$ for which (*)$\frac{d u(t)}{dt}= A u(t) + \phi (t) $, $u(0)=x$ has on $\r$ a mild solution $u\in C_{ub} (\r,X)$ (that is bounded and uniformly continuous) 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 $_{t\ge 0} \,||T(t)|| < \infty$, $\phi\in L^{\infty} (\r,{X})$ and $i\,sp (\phi)\cap \sigma (A)=\emptyset$. As a consequence it is shown that if $\n$ is the space of almost periodic $AP$, almost automorphic $AA$, bounded Levitan almost periodic $LAP_b$, certain classes of recurrent functions $REC_b$ and $\phi \in L^{\infty} (\r,{X})$ such that $M_h \phi:=(1/h)\int_0^h \phi (\cdot+s)\, ds \in \n$ for each $h >0$, then $u\in \n\cap C_{ub}$. These results seem new and generalize and strengthen several recent Theorems.