In this paper, we investigate the asymptotic stability of fractional resolvent families on Banach spaces and ordered Banach spaces. We show that an \(\alpha \)-times resolvent family \(S_\alpha (t)\) with generator A is uniformly Abel-stable if and only if \(0\in \rho (A)\), and if in addition \(S_\alpha (t)\) is analytic and bounded, then \(S_\alpha (t)\) is uniformly stable with \(\Vert S_\alpha (t)\Vert = O(t^{-\alpha })\, (t \rightarrow \infty )\). For a bounded positive \(\alpha \)-times resolvent family on an ordered Banach space, we show that it cannot be uniformly stable if \(\alpha \in (1,2)\); in the case of \(\alpha \in (0,1)\), \(0 \in \rho (A)\) implies the same decay rate \(t^{-\alpha }\). Several results on strong stability are also given by using contour integrals, Tauberian theorems and subordination principles.

在本文中，我们研究了分数分解家庭在Banach空间和有序Banach空间上的渐近稳定性。我们证明，当且仅当\（0 \ in \ rho（A）\）时，带有生成器A的\（\ alpha \）-时间分辨物族\（S_ \ alpha（t）\）一致是Abel稳定的。如果另外\（S_ \ alpha（t）\）是有界的，则\（S_ \ alpha（t）\）在\（\ Vert S_ \ alpha（t）\ Vert = O（t ^ {-\ alpha}）\，（t \ rightarrow \ infty）\）。对于有序Banach空间上的有界正\（\ alpha \）-时间分辨子族，我们证明如果\（\ alpha \ in（1,2）\）时，它不可能是一致稳定的; 在\（\ alpha \ in（0,1）\）的情况下，\（0 \ in \ rho（A）\）表示相同的衰减率\（t ^ {-\ alpha} \）。通过使用轮廓积分，Tauberian定理和从属原理，还给出了一些有关强稳定性的结果。