Skip to main content
Log in


We prove a non-stability result for linear recurrences with constant coefficients in Banach spaces. As a consequence we obtain a complete solution of the problem of the Hyers-Ulam stability for those congruences in the complex Banach space.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. R. P. Agarwal, B. Xu, andW. Zhang, Stability of functional equations in single variable.J. Math. Anal. Appl. 288 (2003), 852–869.

    Article  MATH  MathSciNet  Google Scholar 

  2. C. Borelli andG. L. Forti, On a general Hyers-Ulam stability result.Internat. J. Math. Math. Sci. 18 (1995), 229–236.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. G. Bourgin, Classes of transformations and bordering transformations.Bull. Am. Math. Soc. 57 (1951), 223–237.

    Article  MATH  MathSciNet  Google Scholar 

  4. G. L. Forti, An existence and stability theorem for a class of functional equations.Stochastica 4 (1980), 23–30

    MATH  MathSciNet  Google Scholar 

  5. —, Hyers-Ulam stability of functional equations in several variables.Aequationes Math.50 (1995), 143–190.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Ger, A survey of recent results on stability of functional equations. In:Proc. of the 4th International Conference on Functional Equations and Inequalities (Cracow), Pedagogical University of Cracow, Poland, 1994, pp. 5–36.

    Google Scholar 

  7. P. M. Gruber, Stability of isometries.Trans. Amer. Math. Soc. 245 (1978), 263–277.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. H. Hyers, On the stability of the linear functional equation.Proc. Nat. Acad. Sci. USA 27 (1941), 222–224.

    Article  MathSciNet  Google Scholar 

  9. D. H. Hyers, G. Isac, and T. M. Rassias,Stability of Functional Equations in Several Variables. Birkhäuser, 1998.

  10. Z. Moszner, Sur la stabilité de l’équation d’homomorphisme.Aequationes Math. 29 (1985), 290–306.

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Popa, Hyers-Ulam stability of the linear recurrence with constant coefficients.Adv. Difference Equ. 2005:2 (2005), 101–107.

    Article  MATH  Google Scholar 

  12. J. Rätz, On approximately additive mappings. In:General inequalities 2 (Proc. Second Internat. Conf., Oberwolfach, 1978), Birkhäuser, Basel-Boston, Mass., 1980, pp. 233–251.

    Google Scholar 

  13. S. M. Ulam,Problems in Modern Mathematics, Science Editions. John-Wiley & Sons Inc., New York, 1964.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding authors

Correspondence to J. Brzdk, D. Popa or B. Xu.

Additional information

Communicated by: A. Kreuzer

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brzdk, J., Popa, D. & Xu, B. Note on nonstability of the linear recurrence. Abh.Math.Semin.Univ.Hambg. 76, 183–189 (2006).

Download citation

  • Received:

  • Issue Date:

  • DOI:

2000 Mathematics Subject Classification

Key words and phrases