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 Dtαu(t)=Au(t)+f(t),  t>0, where 0<α<1. When A is the generator of a C0-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∈C([0,1];X) if and only if the operator I-Sα(1) is invertible. Here, we use the representation Sα(t)x=∫0∞Φα(s)T(stα)x ds,  t>0 in which Φα is a Wright type function. For the first order case, that is, α=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 Dtαu(t)=Au(t)+f(t,u(t)), t>0, 0<α<1