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

The authors consider evolution equations of the form

du dt=Au+f(t)(1)

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

lim m lim n f(t+s n -s m )=f(t)

for all t. The uniform spectrum of a bounded, continuous function f:X, denoted by sp u (f), is defined and its properties are investigated. Let Λ be a closed subset of and let AA Λ (X)={f:f is almost automorphic and sp u (f)Λ}. Assuming that A is an infinitesimal generator of an analytic semigroup of linear operators on X and fAA Λ (X), the existence and uniqueness of a mild solution in AA Λ (X) of (1) are proven if and only if σ(A)iΛ=ϕ, where σ(A) denotes the spectrum of A. Letting Λ=sp u (f), it follows that there exists a unique almost automorphic mild solution w of (1) such that sp u (w)sp u (f).


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