  Let the abstract fractional space--time operator $(\partial_t + A)^s$ be
given, where $s \in (0,\infty)$ and $-A \colon \mathsf{D}(A) \subseteq X \to X$
is a linear operator generating a uniformly bounded strongly measurable
semigroup $(S(t))_{t\ge0}$ on a complex Banach space $X$. We consider the
corresponding Dirichlet problem of finding a function $u \colon \mathbb{R} \to
X$ such that $(\partial_t + A)^s u(t) = 0$ on $(t_0, \infty)$ and $u(t) = g(t)$
on $(-\infty, t_0]$, for given $t_0 \in \mathbb{R}$ and $g \colon (-\infty,t_0]
\to X$. We derive a solution formula which expresses $u$ in terms of $g$ and
$(S(t))_{t\ge0}$ and generalizes the well-known variation of constants formula
for the mild solution to the abstract Cauchy problem $u' + Au = 0$ on $(t_0,
\infty)$ with $u(t_0) = x \in \overline{\mathsf{D}(A)}$. Moreover, we include a
comparison to analogous solution concepts arising from Riemann-Liouville and
Caputo type initial value problems.
