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

Semi-dynamical systems generated by autonomous Caputo fractional differential equations