Existence and Ulam’s Stability Results for the First-Order Hahn Difference Systems

• Xinya Zhai ^[1] ; JinRong Wang ^[1]
  1. [1] Guizhou University

     Guizhou University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 1, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we first derive the expression of the solutions of the first-order linear Hahn difference systems to obtain the Ulam’s stability results. Then, we utilize the Banach fixed point theorem to demonstrate the existence and uniqueness of the solution and further study the Ulam’s stability of the first-order nolinear Hahn difference system.

    Eventually, some examples are offered to illusrate these theoretical results.

• Referencias bibliográficas
  • 1. Hahn, W.: Über Orthogonalpolynome, die q-differenzenlgleichungen genügen. Math. Nachr. 2, 4–34 (1949)
  • 2. Jackson, F.H.: On q-functions and a certain difference operator. Earth Environ. Sci. Trans. R. Soc. Edinb. 46, 253–281 (1909)
  • 3. Risha, M.A., Annaby, M.H., Mansour, Z.S., et al.: Linear q-difference equations. Zeitschrift für Analysis und ihre Anwendungen 26, 481–494...
  • 4. Annaby, M.H., Mansour, Z.S.: q-Taylor and interpolation series for Jackson q-difference operators. J. Math. Anal. Appl. 344, 472–483 (2008)
  • 5. Birkhoff, G.D.: General theory of linear difference equations. Trans. Am. Math. Soc. 12, 243–284 (1911)
  • 6. Bird, M.T.: On generalizations of sum formulas of the Euler-MacLaurin type. Am. J. Math. 58, 487–503 (1936)
  • 7. Jordan, C.: Calculus of Finite Differences. AMS Chelsea Publishing, New York (1965)
  • 8. Kwon, K.H., Lee, D.W., Park, S.B., et al.: Hahn class orthogonal polynomials. Kyungpook Math. J. 38, 259–281 (1998)
  • 9. Foupouagnigni, M.: Laguerre-Hahn Orthogonal Polynomials with Respect to the Hahn Operator: Fourth-Order Difference Equation for the rth...
  • 10. Petronilho, J.: Generic formulas for the values at the singular points of some special monic classical Hq,ω-orthogonal polynomials. J....
  • 11. Lesky, P.A.: Charakterisierung der q-Orthogonalpolynome in x. Monatshefte für Mathematik 144, 297–316 (2005)
  • 12. Tariboon, J., Ntouyas, S.K., Sudsutad, W.: New concepts of Hahn calculus and impulsive Hahn difference equations. Adv. Differ. Equ. 2016,...
  • 13. Tariboon, J., Ntouyas, S.K., Sutthasin, B.: Impulsive fractional quantum Hahn difference boundary value problems. Adv. Difference Equ....
  • 14. Annaby, M.H., Hamza, A.E., Aldwoah, K.A.: Hahn difference operator and associated JacksonNörlund integrals. J. Optim. Theory Appl. 154,...
  • 15. Hamza, A.E., Ahmed, S.M.: Theory of linear Hahn difference equations. J. Adv. Math. 4, 441–461 (2013)
  • 16. Hamza, A.E., Ahmed, S.M.: Existence and uniqueness of solutions of Hahn difference equations. Adv. Differ. Equ. 2013, 316 (2013)
  • 17. Hamza, A.E., Abdelkhaliq, M.M.: Hahn difference equations in Banach algebras. Adv. Differ. Equ. 161, 1–25 (2016)
  • 18. Hamza, A.E., Zaghrout, A.S., Ahmed, S.M.: Characterization of stability of first order Hahn difference equations. J. Adv. Math. 5, 678–687...
  • 19. Li, T., Zada, A.: Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded...
  • 20. Zada, A., Zada, B.: Hyers-Ulam stability and exponential dichotomy of discrete semigroup. Appl. Math. E-Notes 19, 527–534 (2019)
  • 21. Li, T., Frassu, S., Viglialoro, G.: Combining effects ensuring boundedness in an attraction-repulsion chemotaxis model with production...
  • 22. Columbu, A., Fuentes, R.D., Frassu, S.: Uniform-in-time boundedness in a class of local and nonlocal nonlinear attraction-repulsion chemotaxis...
  • 23. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)
  • 24. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27, 222–224 (1941)
  • 25. Rassias, T.M.: On the stability of the linear maping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
  • 26. Rus, I.A.: Ulam stability of ordinary differential equations, Studia Universitais Babes-Bolyai. Mathematica 54, 125–133 (2009)
  • 27. Wang, G., Zhou, M., Sun, L.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024–1028 (2008)
  • 28. Li, Y., Shen, Y.: Hyers-Ulam stability of linear differential equations of second order. Appl. Math. Lett. 23, 306–309 (2010)
  • 29. Xue, J.: Hyers-Ulam stability of linear differential equations of second order with constant coefficient. Ital. J. Pure Appl. Math. 32,...
  • 30. Buse, C., Barbu, D., Tabassum, A.: Hyers-Ulam stability and exponential dichotomy of linear differential periodic systems are equivalent....
  • 31. Zada, A., Shah, O., Shah, R.: Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems. Appl. Math. Comput....
  • 32. Jung, S.M.: Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients. J. Math. Anal. Appl....
  • 33. Zada, B.: Uniform exponential stability in the sense of Hyers and Ulam for periodic time varying linear systems. Differ. Eq. Appl. 10,...
  • 34. Etemad, S., Tellab, B., Alzabut, J., et al.: Approximate solutions and Hyers-Ulam stability for a system of the coupled fractional thermostat...
  • 35. Jung, S.M.: Hyers-Ulam stability of the first-order matrix difference equations. Adv. Differ. Equ. 2015, 170 (2015)
  • 36. Nam, Y.W.: Hyers-Ulam stability of loxodromic Möbius difference equation. Appl. Math. Comput. 356, 119–136 (2019)
  • 37. Anderson, D.R., Onitsuka, M.: Hyers-Ulam stability for quantum equations of Euler type. Discret. Dyn. Nat. Soc. 2020, 5626481 (2020)
  • 38. Alzabut, J., Abdeljawad, T., Baleanu, D.: Nonlinear delay fractional difference equations wich applications on discrete fractional Lotka-Volterra...
  • 39. Boutiara, A., Etemad, S., Alzabut, J., et al.: On a nonlinear sequential four-point fractional q-difference equation involving q-integral...
  • 40. Alzabut, J., Selvam, A.G.M., Vignesh, D., et al.: Solvability and stability of nonlinear hybrid - difference equations of fractional-order....
  • 41. Hamza, A.E., Alghamdi, M.A., Alasmi, S.A.: Hyers-Ulam and Hyers-Ulam-Rassias stability of firstorder linear quantum difference equations....
  • 42. Hamza, A.E., Shehata, E.M.: Some inequalities based on a general quantum difference operator. J. Inequal. Appl. 2015, 1–12 (2015)