Attractors of Caputo semi-dynamical systems

The Volterra integral equation associated with autonomous Caputo fractional differential equation (FDE) of order \(\alpha \in (0,1)\) $\alpha \in (0,1)$ in \({\mathbb {R}}^d\) ${\mathbb{R}}^d$ was shown by the authors [4] to generate a semi-group on the space \({\mathfrak {C}}\) $\mathfrak{C}$ of continuous functions \(f:{\mathbb {R}}^+\rightarrow {\mathbb {R}}^d\) $f:{\mathbb{R}}^{+}\to {\mathbb{R}}^d$ 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) \(\equiv \) $\equiv$ \(id_{x_0}\) $i{d}_{x_0}$ for \(x_0\) $x_0$ \(\in \) $\in$ \({\mathbb {R}}^d\) ${\mathbb{R}}^d$ . Here it is shown that this semi-dynamical system has a global Caputo attractor in \({\mathfrak {C}}\) $\mathfrak{C}$ , 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.