Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations

We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem , where When is the generator of a -semigroup on a Banach space , we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary condition admits a unique mild solution for each if and only if the operator is invertible. Here, we use the representation in which is a Wright type function. For the first order case, that is, , the corresponding result was proved by Prüss in 1984. In case is a Banach lattice and the semigroup is positive, we obtain existence of solutions of the semilinear problem