Semi-dynamical systems generated by autonomous Caputo fractional differential equations

An autonomous Caputo fractional differential equation of order $\alpha \in (0,1)$ in ${\mathbb{R}}^d$ whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space $\mathfrak{C}$ of continuous functions $f:{\mathbb{R}}^{+}\to {\mathbb{R}}^d$ with the topology uniform convergence on compact subsets. This contrasts with a recent result of Cong & Tuan \cite{cong}, which showed that such equations do not, in general, generate a dynamical system on the space ${\mathbb{R}}^d$.