Skip to main content
SpringerLink
Log in

On the compactness of fractional resolvent operator functions

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

Abstract

We study and characterize the compactness of resolvent families of operators associated to fractional differential equations. We show an application in the study of existence of mild solutions for a class of semilinear fractional differential equations with non-local initial conditions.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69, 3692–3705 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monogr. Math. Birkhäuser, Basel (2001)

    Book  MATH  Google Scholar 

  3. Bazhlekova, E.: Fractional evolution equations in Banach spaces. Ph.D. Thesis, Eindhoven University of Technology (2001)

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Goldstein, J.A.: Semigroups of linear operators and applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1985)

    Google Scholar 

  6. Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics. Springer, New York (2000)

    MATH  Google Scholar 

  7. Fan, Z.: Characterization of compactness for resolvents and its applications. Appl. Math. Comput. 232, 60–67 (2014)

    MathSciNet  Google Scholar 

  8. Haase, M.: The complex inversion formula revisited. J. Aust. Math. Soc. 84(1), 73–83 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing Co., Inc, River Edge (2000)

    Book  MATH  Google Scholar 

  10. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics studies 204, Elsevier Science B.V., Amsterdam (2006)

    MATH  Google Scholar 

  11. Li, M., Chen, C., Li, F.: On fractional powers of generators of fractional resolvent families. J. Funct. Anal. 259(10), 2702–2726 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, K., Peng, J., Jia, J.: Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives. J. Funct. Anal. 263(2), 476–510 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lizama, C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lizama, C., N’Guérékata, G.M.: Mild solutions for abstract fractional differential equations. Appl. Anal. 92(8), 1731–1754 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lizama, C., Pozo, J.C.: Existence of mild solutions for semilinear integrodifferential equations with nonlocal conditions. Abstr. Appl. Anal. 2012, p. 15 (2012), Article ID 647103. doi:10.1155/2012/647103

  16. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  17. Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44. Springer, New York (1983)

    Book  MATH  Google Scholar 

  18. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  19. Vasilév, V., Piskarev, S.: Differential equations in Banach spaces. II. Theory of cosine operator functions. Functional analysis. J. Math. Sci. (N. Y.) 122(2), 3055–3174 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Prüss, J.: Positivity and regularity of hyperbolic Volterra equations in Banach spaces. Math. Ann. 279, 317–344 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Prüss, J.: Evolutionary Integral Equations and Applications. Monographs Math., vol. 87. Birkhäuser, Basel (1993)

    Book  MATH  Google Scholar 

  22. Travis, C.C., Webb, G.F.: Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houst. J. Math. 3(4), 555–567 (1977)

    MathSciNet  MATH  Google Scholar 

  23. Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252, 202–235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Weis, L.W.: A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory. J. Math. Anal. Appl. 129(1), 6–23 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

C. Lizama is partially supported by FONDECYT Grant #1140258 and R. Ponce is partially supported by FONDECYT Grant #11130619.

Author information

Authors and Affiliations

  1. Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencia, Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago, Chile

    Carlos Lizama

  2. Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

    Aldo Pereira & Rodrigo Ponce

Authors
  1. Carlos Lizama
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aldo Pereira
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rodrigo Ponce
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Lizama.

Additional information

Communicated by Jerome A. Goldstein.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lizama, C., Pereira, A. & Ponce, R. On the compactness of fractional resolvent operator functions. Semigroup Forum 93, 363–374 (2016). https://doi.org/10.1007/s00233-016-9788-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-016-9788-7

Keywords

Search

Navigation