Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations

Abstract

We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem Dtα u (t) = A u (t) + f (t), t > 0, where 0 < α < 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 α (1) is invertible. Here, we use the representation Sα (t) x = ∫0∞ Φα (s) T (s t α) x d s, t > 0 in which Φ α 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 Dtα u (t) = A u (t) + f (t, u (t)), t > 0, 0 < α < 1. © 2013 Valentin Keyantuo et al.