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.

作者展示了与\({\mathbb {R}}^d\)中\(\alpha \in (0,1)\)阶的自主 Caputo 分数阶微分方程 (FDE) 相关的 Volterra 积分方程 [4 ] 在连续函数\(f:{\mathbb {R}}^+\rightarrow {\mathbb {R}}^d\)的空间\({\mathfrak {C}}\)上生成半群拓扑一致收敛于紧子集。当限制于初始函数f ( t ) \(\equiv \) \(id_{x_0}\) for \(x_0\) \(\in \) \({ \mathbb {R}}^d\) 。此处表明，该半动力系统在\({\mathfrak {C}}\)中具有全局 Caputo 吸引子，当 Caputo FDE 中的向量场函数为闭、有界、不变且吸引常数初始函数时，该系统具有全局 Caputo 吸引子满足耗散条件以及局部 Lipschitz 条件。