Skip to main content

A novel algebraic characteristic of fractional resolvent families

Semigroup Forum volume 99, pages 293–302 (2019)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001)

    Book  Google Scholar 

  2. Bajlekova, E.: Fractional Evolution Equations in Banach Spaces. Ph.D. Thesis, Eindhoven University of Technology (2001)

  3. Chen, C., Li, M.: On fractional resolvent operator functions. Semigroup Forum 80, 121–142 (2010)

    MathSciNet  Article  Google Scholar 

  4. Li, F.B., Li, M.: On maximal regularity and semivariation of \(\alpha \)-times resolvent families. Adv. Pure Math. 3, 680–684 (2013)

    Article  Google Scholar 

  5. Li, K.X., Peng, J.G., Jia, J.X.: Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives. J. Funct. Anal. 263, 476–510 (2012)

    MathSciNet  Article  Google Scholar 

  6. Li, M., Chen, C., Li, F.B.: On fractional powers of generators of fractional resolvent families. J. Funct. Anal. 259, 2702–2726 (2010)

    MathSciNet  Article  Google Scholar 

  7. Li, M., Zheng, Q.: On spectral inclusions and approximations of \(\alpha \)-times resolvent families. Semigroup Forum 69, 356–368 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Mei, Z.D., Peng, J.G.: On general fractional abstract Cauchy problem. Commun. Pure Appl. Anal. 12, 2753–2772 (2013)

    MathSciNet  Article  Google Scholar 

  9. Mei, Z.D., Peng, J.G., Jia, J.X.: A new characteristic property of Mittag-Leffler functions and fractional cosine functions. Stud. Math. 220, 119–140 (2014)

    MathSciNet  Article  Google Scholar 

  10. Peng, J., Li, K.: A novel characteristic of solution operator for the fractional abstract Cauchy problem. J. Math. Anal. Appl. 385, 786–796 (2012)

    MathSciNet  Article  Google Scholar 

  11. Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic, San Diego (1999)

    MATH  Google Scholar 

  12. Prüss, J.: Evolutionary Integral Equations and Applications. Birkhäuser, Basel (1993)

    Book  Google Scholar 

  13. Sato, R., Shaw, S.: Strong and uniform mean stability of cosine and sine operator functions. J. Math. Anal. Appl. 330, 1293–1306 (2007)

    MathSciNet  Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan University Jinjiang College, Meishan, 620860, People’s Republic of China

    Jie Mei

  2. Department of Mathematics, Sichuan University, Chengdu, 610064, People’s Republic of China

    Chuang Chen & Miao Li

Authors
  1. Jie Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chuang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chuang Chen.

Additional information

Communicated by Abdelaziz Rhandi.

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

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