Skip to main content
SpringerLink
Log in

Hyers–Ulam Stability of Linear Recurrence with Constant Coefficients Over the Quaternion Skew Yield

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

Abstract

In this paper, we discuss the Hyers–Ulam stability of the linear recurrence over the quaternion skew yield by considering the associated first-order matrix difference equations in complex Banach space. Firstly, we show the Hyers–Ulam stability result for the first-order quaternion-valued linear recurrence. Secondly, we give the sufficient conditions to present the Hyers–Ulam stability of the second-order quaternion-valued linear recurrence. Further, we extend to derive the general result for the higher-order quaternion-valued linear recurrence. Finally, examples are presented to illustrate the theoretical results.

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

Data Availability

No data.

References

  1. Betsch, P., Siebert, R.: Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration. Int. J. Num. Methods Eng. 79, 444–473 (2009)

    Article  MATH  Google Scholar 

  2. Conway, J.H., Smith, D.A.: On quaternions and octonions: their geometry, arithmetic, and symmetry. A K Peters, Natick (2003)

    Book  MATH  Google Scholar 

  3. Pereira, R., Rocha, P., Vettori, P.: Algebraic tools for the study of quaternionic behavioral systems. Linear Algebra Appl. 400, 121–140 (2005)

    Article  MATH  Google Scholar 

  4. Adler, S.L., Finkelstein, D.R.: Quaternionic quantum mechanics and quantum fields. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  5. Kou, K.I., Xia, Y.H.: Linear quaternion differential equations: basic theory and fundamental results. Stud. Appl. Math. 141, 3–45 (2018)

    Article  MATH  Google Scholar 

  6. Kou, K.I., Liu, W.K., Xia, Y.H.: Solve the linear quaternion-valued differential equations having multiple eigenvalues. J. Math. Phys. 60, 023510 (2019)

    MATH  Google Scholar 

  7. Xia, Y.H., Huang, H., Kou, K.I.: An algorithm for solving linear nonhomogeneous quaternion-valued differential equations and some open problems. Discrete Contin. Dyn. Syst. S 15, 1685 (2022)

    Article  MATH  Google Scholar 

  8. Kyrchei, I.: Linear differential systems over the quaternion skew field, 2018, arXiv:1812.03397v1

  9. Cai, Z.F., Kou, K.I.: Laplace transform: a new approach in solving linear quaternion differential equations. Math. Methods Appl. Sci. 41, 4033–4048 (2018)

    Article  MATH  Google Scholar 

  10. Xia, Y.H., Kou, K.I., Liu, Y.: Theory and applications of quaternion-valued differential equations. Science Press, Beijing (2021)

    Google Scholar 

  11. Suo, L., Fečkan, M., Wang, J.: Quaternion-valued linear impulsive differential equations. Qual. Theory Dyn. Syst. 20, 1–78 (2021)

    Article  MATH  Google Scholar 

  12. Cheng, D., Kou, K.I., Xia, Y.H.: A unified analysis of linear quaternion dynamic equations on time scales. J. Appl. Anal. Comput. 8, 172–201 (2018)

    MATH  Google Scholar 

  13. Cheng, D., Kou, K.I., Xia, Y.H., Xu, J.: Sampling expansions associated with quaternion difference equations, 2019, arXiv:1903.05540

  14. Ulam, S.M.: A collection of mathematical problems. Interscience Tracts in Pure and Applied Mathematics, New York (1960)

  15. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27, 222–224 (1941)

    Article  MATH  Google Scholar 

  16. Hyers, D.H., Isac, G., Rassias, T.M.: Stability of functional equations in several variables. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser (1998)

    Book  MATH  Google Scholar 

  17. Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MATH  Google Scholar 

  18. Brillouët-Belluot, N., Brzdȩk, J., Ciepliński, K.: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012, 716936 (2012)

    Article  MATH  Google Scholar 

  19. Jung, S.M., Popa, D., Rassias, M.T.: On the stability of the linear functional equation in a single variable on complete metric groups. J. Glob. Optim. 59, 165–171 (2014)

    Article  MATH  Google Scholar 

  20. Ravi, K., Murali, R., Selvan, A.P.: Ulam stability of a general nth order linear differential equation with constant coefficients. Asian J. Math. Comput. Res. 11, 61–68 (2016)

    Google Scholar 

  21. Wang, J., Li, X.: A uniform method to Ulam-Hyers stability for some linear fractional equations. Mediterr. J. Math. 13, 625–635 (2016)

    Article  MATH  Google Scholar 

  22. Brzdȩk, J., Popa, D., Raşa, I., Xu, B.: Ulam stability of operators. Academic Press, Boston (2018)

    MATH  Google Scholar 

  23. Jung, S.M.: Hyers-Ulam stability of isometries on bounded domains. Open Math. 19, 675–689 (2021)

    Article  MATH  Google Scholar 

  24. Chen, D., Fečkan, M., Wang, J.: On the stability of linear quaternion-valued differential equations. Qual. Theory Dyn. Syst. 21, 9 (2022)

    Article  MATH  Google Scholar 

  25. Chen, D., Fečkan, M., Wang, J.: Hyers-Ulam stability for linear quaternion-valued differential equations with constant coefficient. R. Mt. J. Math. 52, 1237–1250 (2022)

    MATH  Google Scholar 

  26. Popa, D.: Hyers–Ulam–Rassias stability of a linear recurrence. J. Math. Anal. Appl. 309, 591–597 (2005)

    Article  MATH  Google Scholar 

  27. Popa, D.: Hyers–Ulam stability of the linear recurrence with constant coefficients. Adv. Differ. Equ. 2005, 1–7 (2005)

    Article  MATH  Google Scholar 

  28. Brzdȩk, J., Popa, D., Xu, B.: Remarks on stability of linear recurrence of higher order. Appl. Math. Lett. 23, 1459–1463 (2010)

    Article  MATH  Google Scholar 

  29. Brzdȩk, J., Popa, D., Xu, B.: Note on nonstability of the linear recurrence. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 76, 183–189 (2006)

    Article  MATH  Google Scholar 

  30. Brzdȩk, J., Popa, D., Xu, B.: On nonstability of the linear recurrence of order one. J. Math. Anal. Appl. 367, 146–153 (2010)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  32. Brzdȩk, J., Wójcik, P.: On approximate solutions of some difference equations. Bull. Aust. Math. Soc. 95, 476–481 (2017)

    Article  MATH  Google Scholar 

  33. Xu, B., Brzdȩk, J.: Hyers–Ulam stability of a system of first order linear recurrences with constant coefficients. Discr. Dyn. Nat. Soc. 2015, 269356 (2015)

    MATH  Google Scholar 

  34. Baias, A.R., Blaga, F., Popa, D.: On the best Ulam constant of a first order linear difference equation in Banach spaces. Acta Math. Hungarica 163, 563–575 (2021)

    Article  MATH  Google Scholar 

  35. Baias, A.R., Popa, D.: On the best Ulam constant of a higher order linear difference equation. Bull. des Sci. Mathématiques 166, 102928 (2021)

    Article  MATH  Google Scholar 

  36. Wang, J., Fečkan, M.: Ulam–Hyers–Rassias stability for semilinear equations. Discontin. Nonlinearity Complex. 3, 379–388 (2014)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from the editor too.

Author information

Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, China

    Yuqun Zou & JinRong Wang

  2. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48, Bratislava, Slovakia

    Michal Fečkan

  3. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Michal Fečkan

Authors
  1. Yuqun Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michal Fečkan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. JinRong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JinRong Wang.

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.

This work is partially supported by the National Natural Science Foundation of China (12161015), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), Major Project of Guizhou Postgraduate Education and Teaching Reform (YJSJGKT[2021]041), Super Computing Algorithm and Application Laboratory of Guizhou University and Guian Scientific Innovation Company (K22-0116-003), the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No.1/0358/20 and No.2/0127/20.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zou, Y., Fečkan, M. & Wang, J. Hyers–Ulam Stability of Linear Recurrence with Constant Coefficients Over the Quaternion Skew Yield. Qual. Theory Dyn. Syst. 22, 3 (2023). https://doi.org/10.1007/s12346-022-00695-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-022-00695-7

Keywords

Search

Navigation