By Hino Y., et al.

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

**Example text**

Then for all invariant subspaces M satisfying condition H, σ(PˆM )\{0} = σ(P )\{0}. Proof. For u, v ∈ M , consider the equation (λ − PˆM )u = v . It is equivalent to the equation (λ − P (t))u(t) = v(t), t ∈ R. If λ ∈ ρ(PˆM )\{0} , for every v the first equation has a unique solution u , and u ≤ R(λ, PˆM ) v . Take a function v ∈ M of the form v(t) = yeiµt , for some µ ∈ R ; the existence of such a µ is guaranteed by the axioms of condition H. Then the solution u satisfies u ≤ R(λ, PˆM ) y . Hence, for every y ∈ X the solution of the equation (λ − P (0))u(0) = y has a unique solution u(0) such that u(0) ≤ sup u(t) ≤ R(λ, PˆM ) sup v(t) ≤ R(λ, PˆM ) y .

E. Q(t) = e−iµ V (t, t− 1) and (Tµh )h≥0 denote the evolution semigroup associated with the evolutionary process (V (t, s))t≥s . Then by the same argument as above we can show that since σ(Tµh ) = e−iµ σ(T h ), σ(Tµh ) ∩ S 1 = . 2, the following equation t V (t, ξ)f (ξ)dξ, ∀t ≥ s y(t) = V (t, s)y(s) + s has a unique almost periodic solution y(·) . Let x(t) := eiµt y(t) . Then 40 CHAPTER 2. SPECTRAL CRITERIA t x(t) = eiµt y(t) = U (t, s)eiµs y(s) + U (t, ξ)eiµξ f (ξ)dξ s t iµξ = U (t, s)x(s) + U (t, ξ)e f (ξ)dξ ∀t ≥ s.

55, Chap. 2]) that if the spectrum of the monodromy operator does not circle the origin (of course, it should not contain the origin), then the evolution operators admit Floquet representation. In the example below, in general, Floquet representation does not exist. For instance, if the sectorial operator A has compact 44 CHAPTER 2. SPECTRAL CRITERIA resolvent, then monodromy operator is compact (see [90] for more details). Thus, if dimX = ∞, then monodromy operators cannot be invertible. However, the above results can apply.