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Asymptotic Stability of Riemann–Liouville Fractional Resolvent Families

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Abstract

In this paper, we investigate the asymptotic stability of Riemann–Liouville fractional resolvent families (R–L resolvent families) on Banach spaces and ordered Banach spaces. If the generator A satisfies some natural requirement, we show that an \(\alpha \)-times R–L resolvent family \(\{R_\alpha (t)\}\) with generator A is uniformly stable iff \(0\in \rho (A)\); Next, we prove the subordination principle of R–L resolvent family. For a positive \(t_{0}\)-bounded \(\alpha \)-times R–L resolvent family on an ordered Banach space, we show that it can not be uniformly stable if \(\alpha \in (1,2)\); in the case of \(\alpha \in (0,1)\), we show that A generates a positive R–L resolvent family iff \(-A\) is a sectorial operator with \(\omega (-A)\le \frac{\pi }{2}\). Several results on orbit stability are also given by using contour integrals, Tauberian theorems and subordination principles.

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  1. College of Computer Science, Chengdu University, Chengdu, China

    Li Chen-Yu

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Chen-Yu, L. Asymptotic Stability of Riemann–Liouville Fractional Resolvent Families. Results Math 78, 242 (2023). https://doi.org/10.1007/s00025-023-02021-2

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