Skip to main content
SpringerLink
Log in

Stability of a generalization of Cauchy’s and the quadratic functional equations

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

Using the fixed point method, we investigate the stability of the following generalization of Cauchy’s and the quadratic functional equations

$$\begin{aligned} \sum _{k=0}^{n-1} f(x+ b_{k}y)=nf(x)+nf(y), \end{aligned}$$

where \(n \in \mathbb {N}_{2}\), \(b_{k}=\exp (\frac{2i\pi k}{n})\) for \(0\le k \le n-1\), in Banach spaces. Also, we prove the hyperstability results of this equation by the fixed point method.

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

References

  1. Almahalebi, M.: On the hyperstability of \(\sigma \)-Drygas functional equation on semigroups. Aequ. Math. 90, 849–857 (2016)

    Article  MathSciNet  Google Scholar 

  2. Almahalebi, M., Chahbi, A.: Hyperstability of the Jensen functional equation in ultrametric spaces. Aequ. Math. 91, 647–661 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bahyrycz, A., Piszczek, M.: Hyperstability of the Jensen functional equation. Acta Math. Hung. 142, 353–365 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baker, J.A.: A general functional equation and its stability. Proc. Am. Math. Soc. 133, 1657–1664 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourgin, D.G.: Approximately isometric and multiplicative transformations on continuous function rings. Duke Math. J. 16, 385–397 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brzdȩk, J., Chudziak, J., Páles, Z.: A fixed point approach to stability of functional equations. Nonlinear Anal. 74, 6728–6732 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brzdȩk, J.: Remarks on hyperstability of the Cauchy functional equation. Aequ. Math. 86, 255–267 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brzdȩk, J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hung. 141, 58–67 (2013)

    Article  MathSciNet  Google Scholar 

  10. Brzdȩk, J., Ciepliński, K.: A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal. 74, 6861–6867 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brzdȩk, J., Ciepliński, K.: Hyperstability and superstability. Abstr. Appl. Anal. 2013, 13 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Brzdȩk, J.: A hyperstability result for the Cauchy equation. Bull. Aust. Math. Soc. 89, 33–40 (2014)

    Article  MathSciNet  Google Scholar 

  13. Brzdȩk, J.: On Ulam’s type stability of the Cauchy additive equation. Sci. World J. 2014(540164), 7 (2014)

    Google Scholar 

  14. Chahbi, A., Almahalebi, M., Charifi, A., Kabbaj, S.: Generalized Jensen functional equation on restricted domain. Ann. West Univ. Timisoara-Math. 52, 29–39 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Chahbi, A., Charifi, A., Bouikhalene, B., Kabbaj, S.: Nonarchimedean stability of a Pexider \(K\)-quadratic functional equation. Arab J. Math. Sci. 21, 67–83 (2015)

    MathSciNet  Google Scholar 

  16. Găvruţa, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gselmann, E.: Hyperstability of a functional equation. Acta Math. Hung. 124, 179–188 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Łukasik, R.: Some generalization of Cauchy’s and the quadratic functional equations. Aequ. Math. 83, 75–86 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Maksa, G., Páles, Z.: Hyperstability of a class of linear functional equations. Acta Math. 17, 107–112 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Piszczek, M.: Remark on hyperstability of the general linear equation. Aequ. Math. 88, 163–168 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Stetkær, H.: Functional equations involving means of functions on the complex plane. Aequ. Math. 55, 47–62 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ulam, S.M.: Problems in Modern Mathematics. Wiley, New York (1960)

    MATH  Google Scholar 

Download references

Acknowledgements

The author is very thankful to the referee for valuable suggestions. Also my sincere regards and gratitude go to Professor Janusz Brzdȩk for his important contributions on the hyperstability of functional equations.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, BP 133, Kenitra, Morocco

    Muaadh Almahalebi

Authors
  1. Muaadh Almahalebi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muaadh Almahalebi.

Ethics declarations

Conflict of interest

The author declares that there is no conflict of interest regarding the publication of this article.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Almahalebi, M. Stability of a generalization of Cauchy’s and the quadratic functional equations. J. Fixed Point Theory Appl. 20, 12 (2018). https://doi.org/10.1007/s11784-018-0503-z

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11784-018-0503-z

Keywords

Mathematics Subject Classification

Search

Navigation