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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 \(sp_u(w)\subseteq \text{sp}_u(f)\).

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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