Skip to main content
SpringerLink
Log in

Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic PC-mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic PC-mild solution of a two-dimensional impulsive fractional predator-prey system with diffusion are investigated.

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. K. Adolfsson, J. Enelund, P. Olsson: On the fractional order model of viscoelasticity. Mech. Time-Depend. Mat. 9 (2005), 15–34. doi

    Article  Google Scholar 

  2. E. Ait Dads, O. Arino: Exponential dichotomy and existence of pseudo almost-periodic solutions of some differential equations. Nonlinear Anal., Theory Methods Appl. 27 (1996), 369–386.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. U. Akhmet, M. Beklioglu, T. Ergenc, V. I. Tkachenko: An impulsive ratio-dependent predator-prey system with diffusion. Nonlinear Anal., Real World Appl. 7 (2006), 1255–1267.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Cao, Q. Yang, Z. Huang: Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 224–234.

    Article  MathSciNet  MATH  Google Scholar 

  5. Y. K. Chang, R. Zhang, G. M. N’Guérékata: Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations. Comput. Math. Appl. 64 (2012), 3160–3170.

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Chérif: Pseudo almost periodic solutions of impulsive differential equations with delay. Differ. Equ. Dyn. Syst. 22 (2014), 73–91.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Debbouche, M. M. El-borai: Weak almost periodic and optimal mild solutions of fractional evolution equations. Electron. J. Differ. Equ. (electronic only) 2009 (2009), No. 46, 8 pages.

    Google Scholar 

  8. T. Diagana, G. M. N’Guérékata: Pseudo almost periodic mild solutions to hyperbolic evolution equations in intermediate Banach spaces. Appl. Anal. 85 (2006), 769–780.

    Article  MathSciNet  MATH  Google Scholar 

  9. H.-S. Ding, J. Liang, G. M. N’Guérékata, T. J. Xiao: Mild pseudo-almost periodic solutions of nonautonomous semilinear evolution equations. Math. Comput. Modelling 45 (2007), 579–584.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Enelund, P. Olsson: Damping described by fading memory-analysis and application to fractional derivative models. Int. J. Solids Struct. 36 (1999), 939–970.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. M. Fink: Almost Periodic Differential Equations. Lecture Notes in Mathematics 377, Springer, New York, 1974.

    Google Scholar 

  12. H. R. Henríquez, B. de Andrade, M. Rabelo: Existence of almost periodic solutions for a class of abstract impulsive differential equations. ISRN Math. Anal. 2011 (2011), Article ID 632687, 21 pages.

    Google Scholar 

  13. J. Hong, R. Obaya, A. Sanz: Almost-periodic-type solutions of some differential equations with piecewise constant argument. Nonlinear Anal., TheoryMethods Appl., Ser. A, Theory Methods 45 (2001), 661–688.

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. Li, C. Wang: Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales. Adv. Difference Equ. 2012 (2012), Article ID 77, 24 pages.

    Google Scholar 

  15. J. Liu, C. Zhang: Existence and stability of almost periodic solutions for impulsive differential equations. Adv. Difference Equ. 2012 (2012), Article ID 34, 14 pages.

    Google Scholar 

  16. J. Liu, C. Zhang: Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations. Adv. Difference Equ. 2013 (2013), 2013:11, 21 pages.

    MathSciNet  Google Scholar 

  17. J. Liu, C. Zhang: Existence and stability of almost periodic solutions to impulsive stochastic differential equations. Cubo 15 (2013), 77–96.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Liu, C. Zhang: Existence of almost periodic solutions for impulsive neutral functional differential equations. Abstr. Appl. Anal. 2014 (2014), Article ID 782018, 11 pages.

    Google Scholar 

  19. A. Pazy: Semigroup of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44, Springer, New York, 1983.

    Google Scholar 

  20. I. Podlubny: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering 198, Academic Press, San Diego, 1999.

    Google Scholar 

  21. A. M. Samoilenko, N. A. Perestyuk: Impulsive Differential Equations. World Scientific Series on Nonlinear Science. Series A. 14, World Scientific, Singapore, 1995.

    Google Scholar 

  22. G. T. Stamov: Almost Periodic Solutions of Impulsive Differential Equations. Lecture Notes in Mathematics 2047, Springer, Berlin, 2012.

    Google Scholar 

  23. G. T. Stamov, J. O. Alzabut: Almost periodic solutions for abstract impulsive differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 2457–2464.

    Article  MathSciNet  MATH  Google Scholar 

  24. G. T. Stamov, I. M. Stamova: Almost periodic solutions for impulsive fractional differential equations. Dyn. Syst. 29 (2014), 119–132.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. R. Wang, M. Fečkan, Y. Zhou: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8 (2011), 345–361.

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Zhang: Pseudo almost perioidc solutions of some differential equations. J. Math. Anal. Appl. 181 (1994), 62–76.

    Article  MathSciNet  Google Scholar 

  27. C. Zhang: Pseudo almost perioidc solutions of some differential equations. II. J. Math. Anal. Appl. 192 (1995), 543–561.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Zhejiang University of Technology, Liuhe Road 288, Hangzhou, 310023, Zhejiang, China

    Zhinan Xia

Authors
  1. Zhinan Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhinan Xia.

Additional information

This research is supported by the National Natural Science Foundation of China (Grant Nos. 11501507, 11426201).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xia, Z. Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces. Czech Math J 67, 123–141 (2017). https://doi.org/10.21136/CMJ.2017.0398-15

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2017.0398-15

Keywords

MSC 2010

Search

Navigation