Skip to main content
SpringerLink
Log in

Hyers–Ulam Stability for Nonautonomous Semilinear Dynamics on Bounded Intervals

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove that any nonautonomous semilinear dynamics on a bounded interval with the property that the nonlinear part is Lipschitz with a sufficiently small Lipschitz constant exhibits Hyers–Ulam stability.

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. Backes, L., Dragičević, D.: Shadowing for nonautonomous dynamics. Adv. Nonlinear Stud. 19, 425–436 (2019)

    Article  MathSciNet  Google Scholar 

  2. Backes, L., Dragičević, D.: Shadowing for infinite dimensional dynamics and exponential trichotomies. Proc. Roy. Soc. Edinb. Sect. A (2019). https://doi.org/10.1017/prm.2020.42

    Article  MATH  Google Scholar 

  3. Backes, L., Dragičević, D.: Quasi-shadowing for partially hyperbolic dynamics on Banach spaces. J. Math. Anal. Appl. 492, 124445 (2020)

    Article  MathSciNet  Google Scholar 

  4. Barbu, D., Buşe, C., Tabassum, A.: Hyers-Ulam stability and discrete dichotomy. J. Math. Anal. Appl. 423, 1738–1752 (2015)

    Article  MathSciNet  Google Scholar 

  5. Brzdȩk, J., Popa, D., Xu, B.: The Hyers-Ulam stability of nonlinear recurrences. J. Math. Anal. Appl. 335, 443–449 (2007)

    Article  MathSciNet  Google Scholar 

  6. Buşe, C., Barbu, D., Tabassum, A.: Hyers-Ulam stability and exponential dichotomy of linear differential periodic systems are equivalent. Electron. J. Qual. Theory Differ. Equ. 58, 1–12 (2015)

    Article  MathSciNet  Google Scholar 

  7. Dragičević, D.: On the Hyers-Ulam stability of certain nonautonomous and nonlinear difference equations. Aequat. Math (2021). https://doi.org/10.1007/s00010-020-00774-7

  8. Fukutaka, R., Onitsuka, M.: A necessary and sufficient condition for Hyers-Ulam stability of first-order periodic linear differential equations. Appl. Math. Lett. 100, 106040 (2020)

  9. Jung, S.-M.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17, 1135–1140 (2004)

    Article  MathSciNet  Google Scholar 

  10. Jung, S.-M.: Hyers-Ulam stability of linear differential equations of first order II. Appl. Math. Lett. 19, 854–858 (2006)

    Article  MathSciNet  Google Scholar 

  11. Jung, S.-M.: Hyers-Ulam stability of linear differential equations of first order III. J. Math. Anal. Appl. 311, 139–146 (2005)

    Article  MathSciNet  Google Scholar 

  12. Kang, D., Kim, H.B.: Generalized Hyers-Ulam stability of diffusion equation in the n-dimensional Euclidean space, \({\mathbb{R}^n}\). Appl. Math. Lett. 103, 106169 (2020)

    Article  MathSciNet  Google Scholar 

  13. Li, T., Zada, A.: Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces. Adv. Differ. Equa. 153, 1–8 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Lungu, N., Popa, D.: Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 385, 86–91 (2012)

    Article  MathSciNet  Google Scholar 

  15. Onitsuka, M.: Hyers-Ulam stability of first order linear differential equations of Carathéodory type and its application. Appl. Math. Lett. 90, 61–68 (2019)

    Article  MathSciNet  Google Scholar 

  16. Onitsuka, M., Shoji, T.: Hyers-Ulam stability of first-order homogeneous linear differential equations with a real-valued coefficient. Appl. Math. Lett. 63, 102–108 (2017)

    Article  MathSciNet  Google Scholar 

  17. Popa, D.: Hyers-Ulam-Rassias stability of a linear recurrence. J. Math. Anal. Appl. 369, 591–597 (2005)

    Article  MathSciNet  Google Scholar 

  18. Popa, D., Raşa, I.: On the Hyers-Ulam stability of the linear differential equations. J. Math. Anal. Appl. 381, 530–537 (2011)

    Article  MathSciNet  Google Scholar 

  19. Popa, D., Raşa, I.: Hyers-Ulam stability of the linear differential operator with nonconstant coefficients. Appl. Math. Comput. 219, 1562–1568 (2012)

    Article  MathSciNet  Google Scholar 

  20. Takahashi, S.-E., Takagi, H., Miura, T., Miyajima, S.: The Hyers-Ulam stability constants of first order linear differential operators. J. Math. Anal. Appl. 286, 136–146 (2003)

    Article  MathSciNet  Google Scholar 

  21. Wang, G., Zhou, M., Sun, L.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024–1028 (2008)

    Article  MathSciNet  Google Scholar 

  22. Wang, J., Fečkan, M., Tian, Y.: Stability analysis for a general class of non-instantaneous impulsive differential equations. Medit. J. Math. 14, 21–46 (2017)

    Article  MathSciNet  Google Scholar 

  23. Wang, J., Fečkan, M., Zhou, Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)

    Article  MathSciNet  Google Scholar 

  24. Zada, A., Shah, O., Shah, R.: Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems. Appl. Math. Comput. 271, 512–518 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

D. D. was supported in part by Croatian Science Foundation under the project IP-2019-04-1239 and by the University of Rijeka under the projects uniri-prirod-18-9 and uniri-prprirod-19-16. The author is grateful to Lucas Backes for a joint collaboration that inspired ideas of the present paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000, Rijeka, Croatia

    Davor Dragičević

Authors
  1. Davor Dragičević
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Davor Dragičević.

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

Dragičević, D. Hyers–Ulam Stability for Nonautonomous Semilinear Dynamics on Bounded Intervals. Mediterr. J. Math. 18, 71 (2021). https://doi.org/10.1007/s00009-021-01729-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-021-01729-1

Keywords

Mathematics Subject Classification

Search

Navigation