Hyers–Ulam stability for quantum equations

Abstract

We introduce and study the Hyers–Ulam stability (HUS) of a Cayley quantum (q-difference) equation of first order, where the constant coefficient is allowed to range over the complex numbers. In particular, if this coefficient is non-zero, then the quantum equation has Hyers–Ulam stability for certain values of the Cayley parameter, and we establish the best (minimal) HUS constant in terms of the coefficient only, independent of q and the Cayley parameter. If the Cayley parameter equals one half, then there is no Hyers–Ulam stability for any coefficient value in the complex plane.

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

References

  1. 1.

    Anderson, D.R.: The discrete diamond-alpha imaginary ellipse and Hyers–Ulam stability. Int. J. Diff. Equ. 14(1), 25–38 (2019)

    MathSciNet  Google Scholar 

  2. 2.

    Anderson, D.R., Onitsuka, M.: Hyers–Ulam stability for a discrete time scale with two step sizes. Appl. Math. Comput. 344–345, 128–140 (2019)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Anderson, D.R., Onitsuka, M.: Best constant for Hyers–Ulam stability of second-order \(h\)-difference equations with constant coefficients. Results Math. 74, 151 (2019). https://doi.org/10.1007/s00025-019-1077-9

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Anderson, D.R., Onitsuka, M.: Hyers-Ulam stability for quantum equations of Euler type. Discrete Dyn. Nat. Soc. 2020, 10 (2020). https://doi.org/10.1155/2020/5626481

    MathSciNet  Article  Google Scholar 

  5. 5.

    Baias, A.R., Popa, D.: On Ulam stability of a linear difference equation in Banach spaces. Bull. Malays. Math. Sci. Soc. (2019). https://doi.org/10.1007/s40840-019-00744-6

    MATH  Article  Google Scholar 

  6. 6.

    Brzdȩk, J., Wójcik, P.: On approximate solutions of some difference equations. Bull. Aust. Math. Soc. 95(3), 76–481 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Buşe, C., O’Regan, D., Saierli, O.: Hyers–Ulam stability for linear differences with time dependent and periodic coefficients. Symmetry 11, 512 (2019). https://doi.org/10.3390/sym11040512

    MATH  Article  Google Scholar 

  8. 8.

    Cieśliński, J.L.: Improved \(q\)-exponential and \(q\)-trigonometric functions. Appl. Math. Lett. 24, 2110–2114 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Jung, S.-M., Nam, Y.W.: Hyers–Ulam stability of Pielou logistic difference equation. J. Nonlinear Sci. Appl. 10, 3115–3122 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2001)

    Google Scholar 

  11. 11.

    Nam, Y.W.: Hyers–Ulam stability of hyperbolic Möbius difference equation. Filomat 32(13), 4555–4575 (2018). https://doi.org/10.2298/FIL1813555N

    MathSciNet  Article  Google Scholar 

  12. 12.

    Nam, Y.W.: Hyers–Ulam stability of elliptic Möbius difference equation. Cogent Math. Stat. 5(1), 1–9 (2018)

    MATH  Article  Google Scholar 

  13. 13.

    Nam, Y.W.: Hyers–Ulam stability of loxodromic Möbius difference equation. Appl. Math. Comput. 356, 119–136 (2019)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Onitsuka, M.: Influence of the step size on Hyers–Ulam stability of first-order homogeneous linear difference equations. Int. J. Differ. Equ. 12(2), 281–302 (2017)

    MathSciNet  Google Scholar 

  15. 15.

    Onitsuka, M.: Hyers–Ulam stability of second-order nonhomogeneous linear difference equations with a constant step size. J. Comput. Anal. Appl. 28(1), 152–165 (2020)

    Google Scholar 

  16. 16.

    Rasouli, H., Abbaszadeh, S., Eshaghi, M.: Approximately linear recurrences. J. Appl. Anal. 24(1), 81–85 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Satco, B.-R.: Ulam-type stability for differential equations driven by measures. Math. Nachr. (2019). https://doi.org/10.1002/mana.201800481

    MATH  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for reading carefully and giving valuable comments to improve the quality of the paper. The second author was supported by JSPS KAKENHI Grant Number JP20K03668.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Douglas R. Anderson.

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

Verify currency and authenticity via CrossMark

Cite this article

Anderson, D.R., Onitsuka, M. Hyers–Ulam stability for quantum equations. Aequat. Math. (2020). https://doi.org/10.1007/s00010-020-00734-1

Download citation

Keywords

  • Stability
  • Hyers–Ulam
  • Quantum calculus
  • Best constant

Mathematics Subject Classification

  • 39A06
  • 39A13
  • 39A30
  • 34N05