Well-posedness of equations with fractional derivative via the method of sum

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, 0 < α < 1 and Dα is the fractional derivative in the sense of Weyl. Although this problem is not always well-posed in Lp(0, 2π;X) or periodic continuous function spaces Cper([0, 2π];X), we show by using the method of sum that it is well-posed in some subspaces of Lp(0, 2π;X) or Cper([0, 2π];X).