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 Dt ? u (t) = A u (t) + f (t), t > 0, where 0 < ? < 1. When A is the generator of a C 0 -semigroup (T (t))t ? 0 on a Banach space X, 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 u (0) = u (1) admits a unique mild solution for each f E C ([ 0,1 ]; X) if and only if the operator I - S ? (1) is invertible. Here, we use the representation S? (t) x = ?0 ? ?? (s) T (s t ?) x d s, t > 0 in which ? ? is a Wright type function. For the first order case, that is, ? = 1, the corresponding result was proved by Prüss in 1984. In case X is a Banach lattice and the semigroup (T (t))t ? 0 is positive, we obtain existence of solutions of the semilinear problem Dt ? u (t) = A u (t) + f (t, u (t)), t > 0, 0 < ? < 1. © 2013 Valentin Keyantuo et al.