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**Additional info for Almost periodic solutions of differential equations in Banach spaces**

**Example text**

From the uniqueness of y(·) (and then that of x(·)) we have x(t + 1) = x(t), ∀t. 1 this yields that 1 ∈ ρ(Q(0)), or in other words, eiµ ∈ ρ(P ) . From the arbitrary nature of µ, S 1 ∩ σ(P ) = . 4) is uniquely solvable in the function space AP (X) if and only if S 1 ∩ σ(P ) = . 1). Before applying the above results to study the exponential dichotomy of 1-periodic strongly continuous processes we recall that a given 1-periodic strongly continuous evolutionary process (U (t, s))t≥s is said to have an exponential dichotomy if there exist a family of projections Q(t), t ∈ R and positive constants M, α such that the following conditions are satisfied: i) For every fixed x ∈ X the map t → Q(t)x is continuous, ii) Q(t)U (t, s) = U (t, s)Q(s), ∀t ≥ s, iii) U (t, s)x ≤ M e−α(t−s) x , ∀t ≥ s, x ∈ ImQ(s), iv) U (t, s)y ≥ M −1 eα(t−s) y , ∀t ≥ s, y ∈ KerQ(s), v) U (t, s)|KerQ(s) is an isomorphism from KerQ(s) onto KerQ(t), ∀t ≥ s.

The following lemma will be needed in the sequel. 10 Let A be the generator of a C0 -semigroup and M be a closed translation invariant subspace of AAP (X) which satisfies condition H1. Then DM − AM = LM . Proof. Let us consider the semigroup (T h )h≥0 T h v(t) := ehA v(t − h), v ∈ M, h ≥ 0. By condition H1, clearly, (T h )h≥0 leaves M invariant. 1, since M ⊂ AAP (X) this semigroup is strongly continuous which has −LM as its generator. On the other hand, since (T h )h≥0 is the composition of two commuting and strongly continuous semigroups, by [163, p.

12 Let A be the generator of a C0 -semigroup and B be an autonomous functional operator. 38) on R if t u(t) = e(t−s)A u(s) + e(t−ξ)A [(Bu)(ξ) + f (ξ)]dξ, ∀t ≥ s. 38) in the case where the operator A generates a strongly continuous semigroup. 38). 38) is well posed. However, as shown below we can extend our approach to this case. Now we formulate the main result for this subsection. 8 Let A be the infinitesimal generator of an analytic strongly continuous semigroup, B be an autonomous functional operator on the function space BU C(R, X) and M be a closed translation invariant subspace of AAP (X) which satisfies condition H3.