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Almost automorphic solutions of evolution equations. (English) Zbl 1053.34050

The authors consider evolution equations of the form

$\frac{du}{dt}=Au+f\left(t\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

in a complex Banach space $X$. A continuous function $f:ℝ\to X$ is almost automorphic if for any sequence of real numbers, there exists a subsequence $\left\{{s}_{n}\right\}$ such that

$\underset{m\to \infty }{lim}\underset{n\to \infty }{lim}f\left(t+{s}_{n}-{s}_{m}\right)=f\left(t\right)$

for all $t\in ℝ$. The uniform spectrum of a bounded, continuous function $f:ℝ\to X$, denoted by ${\text{sp}}_{u}\left(f\right)$, is defined and its properties are investigated. Let ${\Lambda }$ be a closed subset of $ℝ$ and let $A{A}_{{\Lambda }}\left(X\right)=\left\{f:f$ is almost automorphic and ${\text{sp}}_{u}\left(f\right)\subseteq {\Lambda }\right\}$. Assuming that $A$ is an infinitesimal generator of an analytic semigroup of linear operators on $X$ and $f\in A{A}_{{\Lambda }}\left(X\right)$, the existence and uniqueness of a mild solution in $A{A}_{{\Lambda }}\left(X\right)$ of (1) are proven if and only if $\sigma \left(A\right)\cap i{\Lambda }=\phi$, where $\sigma \left(A\right)$ denotes the spectrum of $A$. Letting ${\Lambda }={\text{sp}}_{u}\left(f\right)$, it follows that there exists a unique almost automorphic mild solution $w$ of (1) such that $s{p}_{u}\left(w\right)\subseteq {\text{sp}}_{u}\left(f\right)$.

##### MSC:
 34G10 Linear ODE in abstract spaces 43A60 Almost periodic functions on groups, etc.; almost automorphic functions