By Hino Y., et al.
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Extra info for Almost periodic solutions of differential equations in Banach spaces
Various spectral criteria of the type σ(DM ) ∩ σ(A) = for the admissibility of the function space M and applications will be discussed. We now describe more detailedly our approach which is based on the notion of evolution semigroups and the method of sums of commuting operators. Let A be the infinitesimal generator of a C0 -semigroup (etA )t≥0 . 23) is defined by T h g(t) := ehA g(t − h), ∀t ∈ R, h ≥ 0, g ∈ M. Under certain conditions on M, the evolution semigroup (T h )h≥0 is strongly continuous.
Put v := ψ ∗ u, g := ψ ∗ f . 5] spF (g) ⊂ suppFψ ∩ spF (f ) ⊂ Λ. Thus g ∈ Λ(X) ∩ F. 32) has a unique solution in Λ(X) which should be v. 5 we can see that the function v should belong to Λ(X) ∩ F. We have in fact proved that λ0 ∈ spF (u). Hence the assertion of the theorem has been proved. 6 Let F be a B-class, σ(A) ∩ (iR)n be countable. 32) that 1 t→∞ t t e−iλs u(x + s)ds lim 0 exists for every λ ∈ spF (u) uniformly with respect to x ∈ R. Then u ∈ F. Proof. 6. 32). Other B-classes can be read in .
Moreover, assume that σ(DM ) ∩ σ(A + B) = . Then M is mildly admissible for Eq.