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23) uniquely in M. Even in this case one can still find conditions on A so that a given solution uf ∈ BU C(R, X) belongs to M. In this direction, one interesting criterion is the countability of the imaginary spectrum of the operator A which is based on the spectral inclusion ispAP (uf ) ⊂ iR ∩ σ(A), where spAP (uf ) is called (in terminology of [137]) the set of points of non-almost periodicity). 23) has a solution uf ∈ BU C(R, X), then it has a solution in M (which may be different from uf ).

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.

We recall the following notion. 32) we understand a bounded uniformly continuous function u : R → X such that t t1 dt1 0 and tn−1 u(s)ds ∈ D(A) dt2 ... 0 0 CHAPTER 2. tn−1 xn−1 + A t1 dt1 0 t t1 dt1 + 0 57 tn−1 dt2 ... 0 u(s)ds + 0 tn−1 f (s)ds (t ∈ R) dt2 ... , xn−1 ∈ X. 32) holds for all t ∈ R. 7 It may be noted that a classical solution is also a mild solution. In case n = 1 if A generates a strongly continuous semigroup the above definition of mild solution on R coincides with that in Section 1 of Chapter 1.

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


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