Skip to main content
SpringerLink
Log in

Asymptotic stability of the Cauchy and Jensen functional equations

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations.

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Badora R.: On some generalized invariant means and their application to the stability of the Hyers–Ulam type. Ann. Polon. Math. 58, 147–159 (1993)

    MathSciNet  MATH  Google Scholar 

  3. Badora R., Ger R., Páles Zs.: Additive selections and the stability of the Cauchy functional equation. ANZIAM J. 44, 323–337 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Badora R., Páles Zs., Székelyhidi L.: Monomial selections of set-valued maps. Aequationes Math. 58, 214–222 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Baron and P. Volkmann, On functions close to homomorphisms between square symmetric structures, Seminar LV (http://www.mathematik.uni-karlsruhe.de/~semlv/) (14) (2002), 12 pp. (electronic).

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Czerwik, Functional Equations and Inequalities in Several Variables. World Scientific Publishing Co., Inc. (River Edge, NJ, 2002).

  10. L. Cădariu and V. Radu, The stability of Jensen’s functional equation: a fixed point approach, Automat. Comput. Appl. Math., 11 (2002), 27–32 (2003).

  11. L. Cădariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, JIPAM. J. Inequal. Pure Appl. Math. 4 (2003).

  12. Forti G.-L.: Hyers–Ulam stability of functional equations in several variables. Aequationes Math. 50, 143–190 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Forti G.-L.: Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations. J. Math. Anal. Appl. 295, 127–133 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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, 1994), pp. 5–36, Pedagogical University of Cracow.

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

    Article  MathSciNet  MATH  Google Scholar 

  16. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables Progress in Nonlinear Differential Equations and their Applications, 34. Birkhäuser Boston Inc. (Boston, MA, 1998).

  17. Jung S.-M.: Hyers–Ulam–Rassias stability of Jensen’s equation and its application. Proc. Amer. Math. Soc. 126, 3137–3143 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Losonczi L.: On the stability of Hosszú’s functional equation. Results Math. 29, 305–310 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Maligranda L.: A result of Tosio Aoki about a generalization of Hyers–Ulam stability of additive functions – a question of priority. Aequationes Math. 75, 289–296 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nikodem K., Páles Zs., Wa̧sowicz Sz.: Abstract separation theorems of Rodé type and their applications. Ann. Polon. Math. 72, 207–217 (1999)

    MathSciNet  MATH  Google Scholar 

  21. Zs. Páles, Generalized stability of the Cauchy functional equation, Aequationes Math., 56 (1998), 222–232.

  22. Zs. Páles, P. Volkmann and R. D. Luce, Hyers–Ulam stability of functional equations with a square-symmetric operation, Proc. Natl. Acad. Sci. USA 95 (1998), 12772–12775 (electronic).

  23. Gy. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis Vol. I. Springer-Verlag (Berlin, 1925).

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

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Skof, On the approximation of locally \({\delta}\) -additive mappings. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 377–389 (1986).

  26. Száz Á.: An instructive treatment and some natural extensions of a set-valued function of Páles. Math. Pannon. 24, 77–108 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Székelyhidi L.: Remarks on Hyers’s theorem. Publ. Math. Debrecen 34, 131–135 (1987)

    MathSciNet  MATH  Google Scholar 

  28. S. M. Ulam, A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience Publishers (New York, 1960).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Pedagogical University of Cracow, Cracow, Poland

    A. Bahyrycz & M. Piszczek

  2. Faculty of Applied Mathematics, AGH University of Science and Technology, Mickiewicza 30, 3D-059, Cracow, Poland

    A. Bahyrycz

  3. Institute of Mathematics, University of Debrecen, Debrecen, Hungary

    Zs. Páles

Authors
  1. A. Bahyrycz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zs. Páles
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Piszczek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zs. Páles.

Additional information

Zs. Páles: This research of the second author was supported by the Hungarian Scientific Research Fund (OTKA) Grant K111651.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bahyrycz, A., Páles, Z. & Piszczek, M. Asymptotic stability of the Cauchy and Jensen functional equations. Acta Math. Hungar. 150, 131–141 (2016). https://doi.org/10.1007/s10474-016-0629-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-016-0629-7

Keywords and phrases

Mathematics Subject Classification

Search

Navigation