Semi-dynamical systems generated by autonomous Caputo fractional differential equations

An autonomous Caputo fractional differential equation of order $alphain(0,1)$ 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 $f:R^+rightarrow R^d$ 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 $mathbb{R}^d$.