By Hino Y., et al.

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

**Example text**

S(−1)|M(f ) ). σ(S(−1)|M(f ) )\{0} which follows from the commutativeness of the operator PˆM(f ) with the operator S(−1)|M(f ) , the above inclusion implies that 1 ∈ σ(T 1 |M(f ) ). 2. Unique solvability of nonlinearly perturbed equations Let us consider the semilinear equation t x(t) = U (t, s)x(s) + U (t, ξ)g(ξ, x(ξ))dξ. 19) CHAPTER 2. 19) for a larger class of the forcing term g. We shall show that the generator of evolutionary semigroup is still useful in studying the perturbation theory in the critical case in which the spectrum of the monodromy operator P may intersect the unit circle.

36) and the above corollary in the case n = 1 various criteria for stability for C0 -semigroups can be established (see [26]). 32). , the following equation d2 u = Au + f (t). 37) dt2 It turns out that for higher order equations conditions on A are much weaker than for the first order ones. Indeed, we have CHAPTER 2. 7 Let A be a linear operator on X such that there are positive constants R, θ and Σ(θ, R) ⊂ ρ(A) and sup |λ| R(λ, A) < ∞. λ∈Σ(θ,R) Furthermore, let M be a translation invariant closed subspace of BU C(R, X) which satisfies condition H1 such that 2 σ(DM ) ∩ σ(A) = .

35) for all t ≥ 0. By reversing the above argument we can easily show the converse. Hence, the lemma is proved. 32) and φ ∈ L1 (R) such that its Fourier transform has compact support. 32) with forcing term φ ∗ f . Proof. Let us define t U1 (t) = t f (s)ds, t ∈ R, u(s)ds, F1 (t) = 0 0 t Uk (t) = t Fk−1 (s)ds, t ∈ R, k ∈ N. Uk−1 (s)ds, Fk (t) = 0 0 Then, by definition, we have u(t) = Pn (t) + A(Un (t)) + Fn (t), t ∈ R, where Pn is a polynomial of order of n − 1. From the closedness of A, we have u ∗ φ(t) = Pn ∗ φ(t) + A(Un ∗ φ(t)) + Fn ∗ φ(t), t ∈ R.