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

Source: arXiv

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.

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