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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 $$f:\mathbb{R}\to X$$ 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 $$t\in\mathbb{R}$$. The uniform spectrum of a bounded, continuous function $$f:\mathbb{R}\to X$$, denoted by $$\text{sp}_u(f)$$, is defined and its properties are investigated. Let $$\Lambda$$ be a closed subset of $$\mathbb{R}$$ 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 $$sp_u(w)\subseteq \text{sp}_u(f)$$.

##### MSC:
 34G10 Linear differential equations in abstract spaces 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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