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Additional resources for Almost periodic solutions of differential equations in Banach spaces, Edition: draft
4 Let (A + α) be of class Σ(θ + π/2, R) for some real α and M be a translation - invariant subspace of BU C(R, X). Moreover, let σ(A) ∩ σ(DM ) = . 24). 24). 24). Proof. Note that under the theorem’s assumption the operators A + α and D satisfy condition P for some real α. In fact, we can check only that sup λR(λ, DM ) < ∞, λ∈Σ(π/2−ε,R) where 0 < ε < π/2. Since λ ∈ Σ(π/2 − ε, R) with 0 < ε < π/2 CHAPTER 2. SPECTRAL CRITERIA 55 ∞ λR(λ, DM )f e−λt f (· + t)dt = |λ| 0 ∞ e−Reλt dt f ≤ |λ| 0 |λ| f Reλ ≤ M f , ≤ where M is a constant independent of f .
G. [137, p. 88]). 8, σ(AM∩Λ(X) ) ⊂ σ(A). 6 and the translation invariance and closedness of M, we observe that for every f ∈ M ∩ Λ(X) and λ ∈ ρ(DΛ(X) ), Reλ > 0 ∞ e−λt S(t)f dt ∈ M ∩ Λ(X). R(λ, DΛ(X) )f = 0 Since R(µ, DΛ(X) )f is continuous in µ ∈ ρ(DΛ(X) ) = C\iΛ, we get R(λ, DΛ(X) )f ∈ M ∩ Λ(X) for all λ ∈ ρ(DΛ(X) ). , ρ(DΛ(X) ) ⊂ ρ(DM∩Λ(X) ). 6, σ(DM∩Λ(X) ) ⊂ σ(DΛ(X) ) = iΛ. 30) 54 CHAPTER 2. SPECTRAL CRITERIA σ(DM∩Λ(X) ) ∩ σ(AM∩Λ(X) ) = . 10 to the pair of operators DM∩Λ(X) and AM∩Λ(X) we get the assertion of the theorem.
If (U (t, s))t≥s is strongly continuous, then L is a single-valued operator from D(L) ⊂ BC(R, X) to BC(R, X). 4) be uniquely solvable in M. 19) is also uniquely solvable in this space. Proof. 4) holds. 9 in the next section). This means L is single-valued. Moreover, one can see that L is closed. e. |v| = v + Lv . By assumption it is seen that L is an isomorphism from [D(L)] onto M. g. 11) for sufficiently small k the operator L − F is invertible. Hence there is a unique u ∈ M such that Lu − F u = 0.