Skip to main content
SpringerLink
Log in

Compact Almost Automorphic Function on Time Scales and Its Application

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, we first propose a concept of compact almost automorphic functions on time scales, and study some basic properties of compact almost automorphic functions on time scales, including an equivalent characterization of compact almost automorphic functions on time scales, composition theorems of compact almost automorphic functions and the completeness of the space of compact almost automorphic functions. Then, as an application of our results, we prove the existence and global exponential stability of a class of Clifford-valued recurrent neural networks with time-varying delays on time scales, and demonstrate the feasibility of our results by an example.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Bochner, S.: A new approach to almost periodicity. Proc. Natl. Acad. Sci. USA 48, 2039–2043 (1962)

    Article  MathSciNet  Google Scholar 

  2. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)

    MATH  Google Scholar 

  3. Li, Y., Kuang, Y.: Periodic solutions of periodic delay Lotka–Volterra equations and systems. J. Math. Anal. Appl. 255(1), 260–280 (2001)

    Article  MathSciNet  Google Scholar 

  4. Li, Y., Kuang, Y.: Periodic solutions in periodic state-dependent delay equations and population models. Proc. Am Math. Soc. 130(5), 1345–1353 (2002)

    Article  MathSciNet  Google Scholar 

  5. Liz, E., Ruiz-Herrera, A.: Delayed population models with Allee effects and exploitation. Math. Biosci. Eng. 12(1), 83–97 (2015)

    Article  MathSciNet  Google Scholar 

  6. Liz, E.: Complex dynamics of survival and extinction in simple population models with harvesting. Theor. Ecol. 3, 209–221 (2010)

    Article  Google Scholar 

  7. Liz, E., Ruiz-Herrera, A.: Attractivity, multistability, and bifurcation in delayed Hopfield’s model with non-monotonic feedback. J. Differ. Equ. 255, 4244–4266 (2013)

  8. Veech, W.A.: Almost automorphic functions on groups. Am. J. Math. 87, 719–751 (1965)

    Article  MathSciNet  Google Scholar 

  9. Hino, Y., Murakami, S.: Almost automorphic solutions for abstract functional differential equations. J. Math. Anal. Appl. 286(2), 741–752 (2003)

    Article  MathSciNet  Google Scholar 

  10. Henríquez, H.R., Lizama, C.: Compact almost automorphic solutions to integral equations with infinite delay. Nonlinear Anal. 71(12), 6029–6037 (2009)

    Article  MathSciNet  Google Scholar 

  11. Andrade, B.D., Cuevas, C.: Compact almost automorphic solutions to semilinear Cauchy problems with non-dense domain. Appl. Math. Comput. 215(8), 2843–2849 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Es-Sebbar, B.: Almost automorphic evolution equations with compact almost automorphic solutions. C. R. Math. 354(11), 1071–1077 (2016)

    Article  MathSciNet  Google Scholar 

  13. Drisi, N., Es-sebbar, B., Ezzinbi, K.: Compact almost automorphic solutions for some nonlinear dissipative differential equations in Banach spaces. Numer. Funct. Anal. Optim. 39(7), 825–841 (2018)

    Article  MathSciNet  Google Scholar 

  14. Hernández, E., Wu, J.H.: Existence, Uniqueness and qualitative properties of global solutions of abstract differential equations with state-dependent delay. Proc. Edinb. Math. Soc. 62(3), 771–788 (2019)

    Article  MathSciNet  Google Scholar 

  15. Es-sebbar, B., Ezzinbi, K., Fatajou, S., Ziat, M.: Compact almost automorphic weak solutions for some monotone differential inclusions: Applications to parabolic and hyperbolic equations. J. Math. Anal. Appl. 486(1), Paper number 123805 (2020)

  16. Hilger, S.: Analysis on measure chains-a unified approach to continuous and discrete calculus. Result Math. 18(1–2), 18–56 (1990)

    Article  MathSciNet  Google Scholar 

  17. Li, Y., Wang, C.: Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales. Abstr Appl Anal. 2011, Article ID 341520 (2011)

  18. Li, Y., Wang, C.: Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales. Adv. Differ. Equ. 2012, Paper number 77 (2012)

  19. Wang, C., Li, Y.: Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales. Ann. Pol. Math. 108, 225–240 (2013)

    Article  MathSciNet  Google Scholar 

  20. Lizama, C., Mesquita, J.G.: Almost automorphic solutions of dynamic equations on time scales. J. Funct. Anal. 265, 2267–2311 (2013)

    Article  MathSciNet  Google Scholar 

  21. Li, Y., Wang, P.: Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete Contin. Dyn. Syst. Ser. S 10(3), 463–473 (2017)

    MathSciNet  Google Scholar 

  22. Clifford, W.K.: Applications of Grassmann’s extensive algebra. Amer. J. Math. 1(4), 350–358 (1878)

  23. Buchholz, S.: A theory of neural computation with Clifford algebras. Ph.D. thesis, University of Kiel, Kiel (2005)

  24. Buchholz, S., Sommer, G.: On Clifford neurons and Clifford multi-layer perceptrons. Neural Netw. 21(7), 925–935 (2008)

    Article  Google Scholar 

  25. Hitzer, E., Nitta, T., Kuroe, Y.: Applications of Cliffords geometric algebra. Adv. Appl. Clifford Algebras 23(2), 377–404 (2013)

    Article  MathSciNet  Google Scholar 

  26. Shen, S., Li, Y.: \(S^p\)-Almost periodic solutions of Clifford-valued fuzzy cellular neural networks with time-varying delays. Neural Process. Lett. 51, 1749–1769 (2020)

    Article  Google Scholar 

  27. Li, Y., Xiang, J.: Existence and global exponential stability of anti-periodic solution for Clifford-valued inertial Cohen–Grossberg neural networks with delays. Neurocomputing 332(7), 259–269 (2019)

    Article  Google Scholar 

  28. Li, Y., Huo, N., Li, B.: On \(\mu \)-pseudo almost periodic solutions for Clifford-valued neutral type neural networks with delays in the leakage term. IEEE Trans. Neural Netw. Learn. Syst. 32(3), 1365–1374 (2021)

    Article  MathSciNet  Google Scholar 

  29. Li, B., Li, Y.: Existence and global exponential stability of almost automorphic solution for Clifford-valued high-order Hopfield neural networks with leakage delays. Complexity 2019, Article ID 6751806, 13 (2019)

  30. Li, Y., Wang, Y., Li, B.: The existence and global exponential stability of \(\mu \)-pseudo almost periodic solutions of Clifford-valued semi-linear delay differential equations and an application. Adv. Appl. Clifford Algebras 29, Paper number 105 (2019)

  31. Huo, N., Li, B., Li, Y.: Anti-periodic solutions for Clifford-valued high-order Hopfield neural networks with state-dependent and leakage delays. Int. J. Appl. Math. Comput. Sci. 30(1), 83–98 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Li, B., Li, Y.: Existence and global exponential stability of pseudo almost periodic solution for Clifford-valued neutral high-order Hopfield neural networks with leakage delays. IEEE Access 7, 150213–150225 (2019)

    Article  Google Scholar 

  33. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman Books Limited, London (1982)

    MATH  Google Scholar 

  34. Diagana, T.: Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces. Springer, New York (2013)

    Book  Google Scholar 

  35. N’Guérékata, G.M.: Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces. Kluwer, New York (2001)

  36. Li, Y., Yang, L., Wu, W.: Square-mean almost periodic solution for stochastic Hopfield neural networks with time-varying delays on timescales. Neural Comput Appl. 26, 1073–1084 (2015)

    Article  Google Scholar 

  37. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. An Introuduction with Applications. Birkhäuser, Boston (2001)

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Editor and the anonymous referees for their helpful comments and valuable suggestions regarding this article.

Funding

This work is supported by the National Natural Science Foundation of China under Grant 11861072.

Author information

Authors and Affiliations

  1. Department of Mathematics, Yunnan University, Kunming, 650091, Yunnan, China

    Yongkun Li & Shiping Shen

Authors
  1. Yongkun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shiping Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongkun Li.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

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

Li, Y., Shen, S. Compact Almost Automorphic Function on Time Scales and Its Application. Qual. Theory Dyn. Syst. 20, 86 (2021). https://doi.org/10.1007/s12346-021-00522-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-021-00522-5

Keywords

Mathematics Subject Classification

Search

Navigation