Normal Periodic Solutions For The Fractional Abstract Cauchy Problem

We show that if A is a closed linear operator defined in a Banach space X and there exist t≥ 0 and M> 0 such that {(im)α}|m|>t0⊂ρ(A), the resolvent set of A, and ∥(im)α(A+(im)αI)−1∥≤M for all |m|>t0,m∈Z, then, for each 1p<α≤2p and 1 < p< 2 , the abstract Cauchy problem with periodic boundary conditions {DtαGLu(t)+Au(t)=f(t),t∈(0,2π);u(0)=u(2π), where DαGL denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each f∈L2πp(R,X) that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle ϕ_A∈ (0 , απ/ 2) and ∫02πf(t)dt∈Ran(A).