Well-posedness of fractional differential equations with finite delay in complex Banach spaces

Abstract

We study the well-posedness of the fractional differential equations with finite delay: D^∝u(t) + BD^βu(t) = Au(t) + Fu_t + f(t); (0 ≤ t ≤ 2π ) on Lebesgue-Bochner spaces L^p(T;X) and periodic Besov spaces B^s_p;_q(T;X), where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊏  D(B), 0 ≤  β < ∝ , ∝ ≥ 1 and F is a bounded linear operator from L^p([-2π; 0];X) (resp. B^s_p;_q([- 2π; 0];X)) into X. We give necessary and sufficient conditions for the L^p-well- posedness and the B^s_p;_q-well-posedness of above equations by using known operator- valued Fourier multiplier theorems.