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A novel algebraic characteristic of fractional resolvent families

Abstract

In this paper, we give a novel one-parameter algebraic functional equation for fractional resolvent families. With the help of this functional equation, we are able to show that all fractional resolvent families, except \(C_0\)-semigroups, are never exponentially stable.

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Acknowledgements

We are grateful to Dr. J. Pastor for some helpful discussions, and for suggesting Remark 3.6(2) and Proposition 4.1. We are also grateful to the referees for a painstaking reading of the paper and for pointing out several inaccuracies.

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Correspondence to Chuang Chen.

Additional information

Communicated by Abdelaziz Rhandi.

This project was supported by the NSFC-RFBR Programme of China (No. 11611530677).

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Mei, J., Chen, C. & Li, M. A novel algebraic characteristic of fractional resolvent families. Semigroup Forum 99, 293–302 (2019). https://doi.org/10.1007/s00233-018-9964-z

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  • DOI: https://doi.org/10.1007/s00233-018-9964-z

Keywords

  • Fractional resolvent families
  • Fractional Cauchy problems
  • Algebraic functional equation
  • Laplace transform
  • Exponential stability