Best Ulam constant for p-order quaternion linear difference equation

• Yuqun Zou ^[1] ; JinRong Wang ^[1]
  1. [1] Guizhou University

     Guizhou University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 5, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we determine the best Ulam constant for p-order quaternion linear difference equations. Firstly, we use the constant variation formula to derive the general solution of this equation with p distinct characteristic roots. Secondly, we present the Hyers-Ulam stability of such an equation when the modulus of the characteristic roots is not equal to 1. Finally, the best Ulam constant of this equation is determined when the modulus of the characteristic roots is greater than 1. Additionally, as corollaries of our main results, we obtain the best Ulam constants for the first-order and second-order cases, as well as for the complex linear difference equations. Examples are given to illustrate the obtained results.

• Referencias bibliográficas
  • 1. Tayebi, A.: Unit quaternion-based output feedback for the attitude tracking problem. IEEE Trans. Autom. Control 53, 1516–1520 (2008)
  • 2. Seelen, L.J.H., Padding, J.T., Kuipers, J.A.M.: Improved quaternion-based integration scheme for rigid body motion. Acta Mech. 227, 3381–3389...
  • 3. Adler, S.L.: Quaternionic quantum field theory. Commun. Math. Phys. 104, 611–656 (1986)
  • 4. Adler, S.L.: Quaternionic quantum mechanics and quantum fields. Oxford University Press, New York (1995)
  • 5. Kou, K.I., Xia, Y.: Linear quaternion differential equations: basic theory and fundamental results. Stud. Appl. Math. 141, 3–45 (2018)
  • 6. Cai, Z., Kou, I.K.: Laplace transform: a new approach in solving linear quaternion differential equations. Mathematical Methods in the...
  • 7. Kou, K.I., Liu, W., Xia, Y.: Solve the linear quaternion-valued differential equations having multiple eigenvalues. J. Math. Phys. 60,...
  • 8. De Leo, S., Ducati, G.C.: Solving simple quaternionic differential equations. J. Math. Phys. 44, 2224– 2233 (2003)
  • 9. Gasull, A., Llibre, J., Zhang, X.: One-dimensional quaternion homogeneous polynomial differential equations. J. Math. Phys. 50, 082705...
  • 10. Zhang, X.: Global structure of quaternion polynomial differential equations. Commun. Math. Phys. 303, 301–316 (2011)
  • 11. Wang, C., Qin, G., Agarwal, R.: Quaternionic exponentially dichotomous operators through S-spectral splitting and applications to Cauchy...
  • 12. Ulam, S.M.: A collection of mathematical problems. Interscience Tracts in Pure and Applied Mathematics, New York (1960)
  • 13. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. U.S.A. 27, 222–224 (1941)
  • 14. Ciepli ´nski, K.: On Ulam stability of a functional equation. RM 75, 151 (2020)
  • 15. Brzd¸ek, J., Ciepli ´nski, K., Le´sniak, Z.: On Ulam’s type stability of the linear equation and related issues. Discret. Dyn. Nat. Soc....
  • 16. Wang, J., Li, X.: A uniform method to Ulam-Hyers stability for some linear fractional equations. Mediterr. J. Math. 13, 625–635 (2016)
  • 17. Ciepli ´nski, K.: On perturbations of two general equations in several variables. Math. Ann. 385, 921–937 (2023)
  • 18. Brzd¸ek, J., Popa, D., Xu, B.: The Hyers-Ulam stability of nonlinear recurrences. J. Math. Anal. Appl. 335, 443–449 (2007)
  • 19. Brzd¸ek, J., Popa, D., Xu, B.: Remarks on stability of linear recurrence of higher order. Appl. Math. Lett. 23, 1459–1463 (2010)
  • 20. Brzd¸ek, J.: Remarks on approximate solutions to difference equations in various spaces. Symmetry 15, 1829 (2023)
  • 21. Brzd¸ek, J., Popa, D., Ra¸sa, I., et al.: Ulam stability of operators. Academic Press, Boston (2018)
  • 22. Hatori, O., Kobayashi, K., Miura, T., et al.: On the best constant of Hyers-Ulam stability. Journal of Nonlinear and Convex Analysis 5,...
  • 23. Takahasi, S.E., Takagi, H., Miura, T., et al.: The Hyers-Ulam stability constants of first order linear differential operators. J. Math....
  • 24. Hirasawa, G., Miura, T.: Hyers-Ulam stability of a closed operator in a Hilbert space. Bull. Korean Math. Soc. 43, 107–117 (2006)
  • 25. Onitsuka, M.: Hyers-Ulam stability of first-order nonhomogeneous linear difference equations with a constant stepsize. Appl. Math. Comput....
  • 26. Anderson, D.R., Onitsuka, M.: Best constant for Hyers-Ulam stability of second-order h-difference equations with constant coefficients....
  • 27. Brzd¸ek, J.: Some remarks on the best Ulam constant. Symmetry 16, 1644 (2024)
  • 28. Cheng, D., Kou, K.I., Xia, Y.: A unified analysis of linear quaternion dynami equations on time scales. Journal of Applied Analysis and...
  • 29. Li, Z., Wang, C., Agarwal, R.P., et al.: Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on...
  • 30. Wang, C., Chen, D., Li, Z.: General theory of the higher-order quaternion linear difference equations via the complex adjoint matrix and...
  • 31. Chen, D., Feˇckan, M., Wang, J.: Linear quaternion-valued difference equations: representation of solutions, controllability and observability....
  • 32. Wang, J., Wang, J., Liu, R.: Hyers-Ulam stability to linear nonhomogeneous quaternion-valued matrix difference equations via complex representation,...
  • 33. Zou, Y., Feˇckan, M., Wang, J.: Hyers-Ulam stability of linear recurrence with constant coefficients over the quaternion skew yield, Qualitative...
  • 34. Baias, A.R., Blaga, F., Popa, D.: Best Ulam constant for a linear difference equation. Carpathian Journal of Mathematics 35, 13–22 (2019)
  • 35. Baias, A.R., Popa, D.: On Ulam stability of a linear difference equation in Banach spaces. Bulletin of the Malaysian Mathematical Sciences...
  • 36. Baias, A.R., Popa, D.: On the best Ulam constant of a higher order linear difference equation. Bulletin des Sciences Mathématiques 166,...
  • 37. Chen, L.: Definition of determinant and Cramer solutions over quaternion field. Acta Math. Sinica 7, 171–180 (1991)
  • 38. Kyrchei, I.I.: Cramer’s rule for quaternionic systems of linear equations. J. Math. Sci. 155, 839–858 (2008)
  • 39. Chen, L.: Inverse matrix and properties of double determinant over quaternion field. Science in China. Series A 34, 528–540 (1991)
  • 40. Kou, K., Liu, M., Tao, S.: Herglotz’s theorem and quaternion series of positive term. Mathematical Methods in the Applied Sciences 39,...
  • 41. Kelly, W.G., Peterson, A.C.: Difference Equations: An introduction with applications. Academic Press, New York (1991)
  • 42. Huang, L., So, W.: Quadratic formulas for quaternions. Appl. Math. Lett. 15, 533–540 (2002)
  • 43. Janovská, D., Opfer, G.: A note on the computation of all zeros of simple quaternionic polynomials. SIAM J. Numer. Anal. 48, 244–256 (2010)