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Asymptotic stability of fractional resolvent families
Journal of Evolution Equations ( IF 1.1 ) Pub Date : 2021-05-12 , DOI: 10.1007/s00028-021-00694-2
Chen-Yu Li , Miao Li

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.



中文翻译:

分数分辨子族的渐近稳定性

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\(\ alpha \)-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\(S_ \ alpha(t)\)\(\ Vert S_ \ alpha(t)\ Vert = O(t ^ {-\ alpha})\,(t \ rightarrow \ infty)\)。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。

更新日期:2021-05-12
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