In this paper, we derive a new fractional Halanay-like inequality, which is used to characterize the long-term behavior of time fractional neutral functional differential equations (F-NFDEs) of Hale type with order $\alpha \in (0,1)$. The contractivity and dissipativity of F-NFDEs are established under almost the same assumptions as those for classical integer-order NFDEs. In contrast to the exponential decay rate for NFDEs, the F-NFDEs are proved to have a polynomial decay rate. The numerical scheme based on the $\mathcal{L}$1 method together with linear interpolation is constructed and applied in several examples to illustrate the theoretical results and to reveal the quite different long-term decay rate in the solutions between F-NFDEs and NFDEs.

在本文中，我们导出了一个新的分数阶Halanay不等式，用于刻画Hale型时间分数阶中立泛函微分方程（F-NFDE）的长期行为 $\alpha \in （0，\mathrm{1个}）$。F-NFDE的收缩性和耗散性是在与经典整数阶NFDE几乎相同的假设下建立的。与NFDE的指数衰减率相反，事实证明F-NFDE具有多项式衰减率。基于$\mathcal{大号}$构造并结合了线性插值的1种方法在几个示例中用于说明理论结果并揭示F-NFDE和NFDE之间溶液的长期衰减率完全不同。