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

We study the well-posedness of the equations with fractional derivative D . u(t) = Au(t)+ f(t) (0 . t . 2.), where A is a closed operator in a Banach space X, 0 < . < 1 and D . is the fractional derivative in the sense of Weyl. Although this problem is not always well-posed in L p(0, 2.;X) or periodic continuous function spaces C per([0, 2.];X), we show by using the method of sum that it is well-posed in some subspaces of L p(0, 2.;X) or C per([0, 2.];X).