Let the abstract fractional space–time operator \((\partial _t + A)^s\) be given, where \(s \in (0,\infty )\) and \(-A :{\textsf{D}}(A) \subseteq X \rightarrow X\) is a linear operator generating a uniformly bounded strongly measurable semigroup \((S(t))_{t\ge 0}\) on a complex Banach space X. We consider the corresponding Dirichlet problem of finding \(u :{\mathbb {R}} \rightarrow X\) such that \(\begin{aligned} \left\{ \begin{aligned} (\partial _t + A)^s u(t)&= 0,&t&\in (t_0, \infty ), \\ u(t)&= g(t),&t&\in (-\infty , t_0], \end{aligned} \right. \end{aligned}\) for given \(t_0 \in {\mathbb {R}}\) and \(g :(-\infty ,t_0] \rightarrow X\) . We define the concept of \(L^p\) -solutions, to which we associate a mild solution formula which expresses u in terms of g and \((S(t))_{t\ge 0}\) 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 \(u(t_0) = x \in \overline{{\textsf{D}}(A)}\) . Moreover, we include a comparison to analogous solution concepts arising from Riemann–Liouville and Caputo type initial value problems.
