Skip to main content
Log in

Abstract differential equations and Caputo fractional derivative

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript


In this work we consider the abstract Cauchy problem with Caputo fractional time derivative of order \(\alpha \in (0,1]\), and discuss the continuity of the respective solutions regarding the parameter \(\alpha \). We also present a study about the continuity of the Mittag-Leffler families of operators (for \(\alpha \in (0,1]\)), when they are induced by sectorial operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others


  1. Almeida, M.F., Ferreira, L.C.F.: Self-similarity, symmetries and asymptotic behavior in Morrey spaces for a fractional wave equation. Differ. Integral Equ. 25, 957–976 (2012)

    MathSciNet  MATH  Google Scholar 

  2. de Andrade, B., Carvalho, A.N., Carvalho-Neto, P.M., Marín-Rubio, P.: Semilinear fractional differential equations: global solutions, critical non-linearities and comparison results. Topol. Methods Nonlinear Anal. 45, 439–467 (2015)

    Article  MathSciNet  Google Scholar 

  3. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems, 2nd edn. Birkhäuser Basel (2011)

  4. Carvalho-Neto, P. M.: Fractional Differential Equations: A Novel Study of Local and Global Solutions in Banach Spaces. Ph.D. Thesis, Universidade de São Paulo, São Carlos (2013)

  5. Duan, J.-S., Chaolu, T., Rach, R.: Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach-Adomian-Meyers modified decomposition method. Appl. Math. Comput. 218, 8370–8392 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Dunford, N., Schwartz, J.T.: Linear Operators. General Theory. Interscience Publishers, New York, Part I (1958)

    MATH  Google Scholar 

  7. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions. Related Topics and Applications. Springer, Berlin, Heidelberg (2014)

    MATH  Google Scholar 

  8. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)

    MATH  Google Scholar 

  9. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)

  10. Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. Amer. Mathematical Soc, Providence, RI (1996)

    Book  Google Scholar 

  11. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics, vol. 132. Springer, Berlin (1995)

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

    MathSciNet  MATH  Google Scholar 

  13. Mikusiński, J.: The Bochner Integral. Birkhäuser, Basel (1978)

    Book  Google Scholar 

  14. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

  15. Wazwaz, A.M., Khuri, S.A.: A reliable technique for solving the weakly singular second-kind Volterra-type integral equations. Appl. Math. Comput. 80, 287–299 (1996)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to P. M. Carvalho-Neto.

Additional information

Communicated by Abdelaziz Rhandi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carvalho-Neto, P.M. Abstract differential equations and Caputo fractional derivative. Semigroup Forum 104, 561–583 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: