Well-posedness of equations with fractional derivative

Abstract

We study the well-posedness of the equations with fractional derivative Dαu(t) = Au(t)+ f(t), 0 ≤ t ≤ 2π, where A is a closed operator in a Banach space X, α > 0 and Dα is the fractional derivative in the sense of Weyl. Using known results on Lp-multipliers, we give necessary and/or sufficient conditions for the Lp-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.