Attractors of Caputo semi-dynamical systems

The Volterra integral equation associated with autonomous Caputo fractional differential equation (FDE) of order \(\alpha \in (0,1)\) in \({\mathbb {R}}^d\) was shown by the authors [4] to generate a semi-group on the space \({\mathfrak {C}}\) of continuous functions \(f:{\mathbb {R}}^+\rightarrow {\mathbb {R}}^d\) with the topology uniform convergence on compact subsets. It serves as a semi-dynamical system for the Caputo FDE when restricted to initial functions f(t) \(\equiv \) \(id_{x_0}\) for \(x_0\) \(\in \) \({\mathbb {R}}^d\). Here it is shown that this semi-dynamical system has a global Caputo attractor in \({\mathfrak {C}}\), which is closed, bounded, invariant and attracts constant initial functions, when the vector field function in the Caputo FDE satisfies a dissipativity condition as well as a local Lipschitz condition.