By Hino Y., et al.

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10), then λx = y + w, where w = U (t, 0)(λ − P )−1 U (1, t)y. Conversely, defining x by this equation, it follows that (λ − P (t))x = y so ρ(P (t)) ⊃ ρ(P )\{0} . The second assertion follows by the periodicity. Finally, the above formula involving x proves the third assertion. 2 In view of the above lemma, below, in connection with spectral properties of the monodromy operators, the terminology ”monodromy operators” may be referred to as the operator P if this does not cause any danger of confusion.

19). 4) be uniquely solvable in M. 19) act on M. 19) is uniquely solvable in M . Proof. 7. g. [90] for more details) and apply the results obtained above to study the existence of almost periodic solutions to these equations. It may be noted that a necessary condition for the existence of Floquet representation is that the process under consideration is invertible. g. [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.

16) s Then x(t) := e−iµt y(t) must be the unique solution to the following equation t x(t) = e−iµ(t−s) U (t, s)x(s) + e−iµ(t−ξ) U (t, ξ)f (ξ)dξ), ∀t ≥ s. 17) s And vice versa. We show that x(t) should be periodic. 16). 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).

### Almost periodic solutions of differential equations in Banach spaces by Hino Y., et al.

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