By Hino Y., et al.

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

**Example text**

Then the following inhomogeneous equation du = (−A + B(t))u + f (t) dt has a unique almost periodic solution u such that sp(u) ⊂ {λ + 2πk, k ∈ Z, λ ∈ sp(f )}. We now show Claim 2 Let the conditions of Claim 1 be satisfied except for the compactness of the resolvent of A . 21) dt has an exponential dichotomy if and only if the spectrum of the monodromy operator does not intersect the unit circle. Moreover, if A has compact resolvent, it has an exponential dichotomy if and only if all multipliers have modulus different from one.

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. 15). The uniqueness of x(·) follows from that of the solution y(·) . We now prove the converse. Let y(t) be the unique almost periodic solution to the equation t U (t, ξ)eiµξ f (ξ)dξ, ∀t ≥ s. 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.

Moreover, let f ∈ AP (X) such that σ(P ) ∩ {eiλ , λ ∈ sp(f )} = . 4) has an almost periodic solution which is unique in M(f ). Proof. 2 it follows that the function space M(f ) satisfies condition H. 8) we have σ(S(1)|M(f ) ) = eσ(D|M(f ) ) , where D|M(f ) is the generator of (S(t)|M(f ) )t≥0 . 6 we have σ(D|M(f ) ) = iΛ, where Λ = {λ + 2πk, λ ∈ sp(f ), k ∈ Z}. Hence, since eσ(D|M(f ) ) = eiΛ ⊂ eisp(f ) ⊂ eiΛ , we have σ(S(1)|M(f ) ) = eσ(D|M(f ) ) = eisp(f ) . σ(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 ) ).

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

by William

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