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) ) ₓ₀ 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) ) ₓ₀ 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.