Semi-Dynamical Systems Generated by Autonomous Caputo Fractional Differential Equations

An autonomous Caputo fractional differential equation of order α ∈ (0,1) in a finite dimensional space whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space C of continuous functions with the topology uniform convergence on compact subsets. This contrasts with a recent result of Cong and Tuan (J. Integral Equ. Appl.: 29, 585–608, 2017 ), which showed that such equations do not, in general, generate a dynamical system on the state space.