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Almost automorphic solutions of evolution equations. (English) Zbl 1053.34050
The authors consider evolution equations of the form $$\frac{du}{dt}= Au+f(t)\tag 1$$ in a complex Banach space $X$. A continuous function $f:\bbfR\to X$ is almost automorphic if for any sequence of real numbers, there exists a subsequence $\{s_n\}$ such that $$\lim_{m\to \infty}\lim_{n\to\infty} f(t + s_n - s_m) = f(t)$$ for all $t\in\bbfR$. The uniform spectrum of a bounded, continuous function $f:\bbfR\to X$, denoted by $\text{sp}_u(f)$, is defined and its properties are investigated. Let $\Lambda$ be a closed subset of $\bbfR$ and let $AA_\Lambda(X) =\{f: f$ is almost automorphic and $\text{sp}_u(f)\subseteq \Lambda\}$. Assuming that $A$ is an infinitesimal generator of an analytic semigroup of linear operators on $X$ and $f\in AA_\Lambda(X)$, the existence and uniqueness of a mild solution in $AA_\Lambda(X)$ of (1) are proven if and only if $\sigma(A)\cap i\Lambda=\phi$, where $\sigma(A)$ denotes the spectrum of $A$. Letting $\Lambda =\text{sp}_u(f)$, it follows that there exists a unique almost automorphic mild solution $w$ of (1) such that $sp_u(w)\subseteq \text{sp}_u(f)$.

34G10Linear ODE in abstract spaces
43A60Almost periodic functions on groups, etc.; almost automorphic functions
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