Dirichlet problems associated to abstract nonlocal space-time differential operators

Let the abstract fractional space--time operator (∂ₜ + A)ˢ be given, where s ∈ (0,∞) and -A D(A) ⊆ X → X is a linear operator generating a uniformly bounded strongly measurable semigroup (S(t))t≥0 on a complex Banach space X. We consider the corresponding Dirichlet problem of finding a function u R → X such that (∂ₜ + A)ˢ u(t) = 0 on (t₀, ∞) and $u(t) = g(t)$ on (-∞, t₀], for given t₀ ∈ R and g (-∞,t₀] → X. We derive a solution formula which expresses u in terms of g and (S(t))t≥0 and generalizes the well-known variation of constants formula for the mild solution to the abstract Cauchy problem $u' + Au = 0$ on (t₀, ∞) with u(t₀) = x ∈ D(A)̄. Moreover, we include a comparison to analogous solution concepts arising from Riemann-Liouville and Caputo type initial value problems.