Analysis of the dynamics of Caputo fractional differential equations

It is known that a finite-dimensional Caputo fractional differential equation, though itself need not generate a semiflow, can be represented as a Volterra integral equation which generates an infinite-dimensional semiflow on the space $\mathfrak{C}=C([0,\mathrm{\infty });{\mathbb{R}}^d)$ under the standard compact-open topology. In this paper we construct a compact absorbing set and an attractor for this semiflow on $\mathfrak{C}$, and then prove that the attractor consists of equi globally Hölder continuous functions. This strengthens the previous work of Doan & Kloeden \cite{DK21} where a bounded (with respect to a weighted norm) attractor was constructed.