Skip to main content
SpringerLink
Log in

On the hyperstability of \({\sigma}\)-Drygas functional equation on semigroups

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

In this paper, we obtain hyperstability results for the \({\sigma}\)-Drygas functional equation and the inhomogeneous \({\sigma}\)-Drygas functional equation on semigroups. Namely, we show that a function satisfying the \({\sigma}\)-Drygas equation approximately must be exactly the solution of it.

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. Ait Sibaha, M., Bouikhalene, B., Elqorachi, E.: Hyers-Ulam-Rassias stability of the K-quadratic functional equation. J. Ineq. Pure Appl. Math, 8 (2007)

  2. Almahalebi M., Charifi A., Kabbaj S.: Hyperstability of a monomial functional equation. J. Sci. Res. Rep. 3(20), 2685–2693 (2014)

    Google Scholar 

  3. Almahalebi M., Kabbaj S.: Hyperstability of a Cauchy-Jensen type functional equation. Adv. Res. 2(12), 1017–1025 (2014)

    Article  Google Scholar 

  4. Almahalebi M., Park C.: On the hyperstability of a functional equation in commutative groups. J. Comput. Anal. Appl. 20(1), 826–833 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Almahalebi, M., Charifi, A., Kabbaj, S.: Hyperstability of a Cauchy functional equation. J. Nonlinear Anal. Optim. (preprint)

  6. Arriola L.M., Beyer W.A.: Stability of the Cauchy functional equation over p-adic fields. Real Anal. Exch. 31(1), 125–132 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Bahyrycz A., Piszczek M.: Hyperstability of the Jensen functional equation. Acta Math. Hung. 142(2), 353–365 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Baker J.: A general functional equation and its stability. Proc. Am. Math. Soc. 133(6), 1657–1664 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brzdȩk J., Chudziak J., Páles Z.: A fixed point approach to stability of functional equations. Nonlinear Anal. 74(17), 6728–6732 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brzdȩk, J., Ciepliñski, K.: Hyperstability and Superstability. Abstract and Applied Analysis. Hindawi Publishing Corporation (2013)

  11. Brzdȩk J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hung. 141, 1-2–58-67 (2013)

    MathSciNet  Google Scholar 

  12. Brzdȩk J.: A hyperstability result for the Cauchy equation. Bull. Aust. Math. Soc. 89(01), 33–40 (2014)

    Article  MathSciNet  Google Scholar 

  13. Brzdȩk J.: Remarks on stability of some inhomogeneous functional equations. Aequ. Math. 89(1), 83–96 (2015)

    Article  MathSciNet  Google Scholar 

  14. Cholewa P.W.: Remarks on the stability of functional equations. Aequ. Math. 27(1), 76–86 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Czerwik S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg. 62(1), 59–64 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Djoković D.Z̆.: A representation theorem for (X 1−1)(X 2−1)...(X n −1) and its applications. Ann. Pol. Math. 22(2), 189–198 (1969)

    MathSciNet  Google Scholar 

  17. Drygas, H.: Quasi-inner products and their applications, pp. 13–30. Springer, The Netherlands (1987)

  18. Ebanks B.R., Kannappan P.L., Sahoo P.K.: A common generalization of functional equations characterizing normed and quasi-inner-product spaces. Can. Math. Bull 35, 332–327 (1992)

  19. Fai̇̆ziev V.A., Sahoo P.K.: On Drygas functional equation on groups. Int. J. Appl. Math. Stat. 7, 59–69 (2007)

    MathSciNet  Google Scholar 

  20. Forti G.L.: Hyers-Ulam stability of functional equations in several variables. Aequ. Math. 50(1-2), 143–190 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Frechet M.: Une définition fonctionnelles des polynômes. Nouv. Ann. 9, 145–162 (1909)

    Google Scholar 

  22. Gajda Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14(3), 431–434 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gǎvruta P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184(3), 431–436 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gilányi A.: A characterization of monomial functions. Aequ. Math. 54, 289–307 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gselmann E.: Hyperstability of a functional equation. Acta Math. Hung. 124(1-2), 179–188 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hensel K.: Über eine neue Begrundung der Theorie der algebraischen Zahlen. Jahresber. Deutsch. Math. Verein 6, 83–88 (1897)

    MATH  Google Scholar 

  27. 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 

  28. Hyers D.H.: Transformations with bounded n-th differences. Pac. J. Math. 11, 591–602 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hyers D.H., Rassias Th.M.: Approximate homomorphisms. Aequ. Math. 44, 125–153 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hyers, D.H., Isac, G.I., Rassias, Th.M.: Stability of functional equations in several variables. Birkhäuser, Basel (1998)

  31. Jun K.-W., Lee Y.-H.: A generalization of the Hyers-Ulam-Rassias stability of Jensen’s equation. J. Math. Anal. Appl. 238, 305–315 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Jung C.F.K.: On generalized complete metric spaces. Bull. AMS 75, 113–116 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  33. Jung S.-M.: Stability of the quadratic equation of Pexider type. Abh. Math. Sem. Univ. Hamburg 70, 175–190 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Jung S.-M., Sahoo P.K.: Hyers-Ulam stability of the quadratic equation of Pexider type. J. Korean Math. Soc. 38, 645–656 (2001)

    MathSciNet  MATH  Google Scholar 

  35. Jung S.-M., Sahoo P.K.: Stability of a functional equation of Drygas. Aequ. Math. 64(3), 263–273 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Maksa G., Páles Z.: Hyperstability of a class of linear functional equations. Acta Math. 17(2), 107–112 (2001)

    MathSciNet  MATH  Google Scholar 

  37. Mazur, S., Orlicz, W.: Grundlegende Eigenschaften der Polynomischen Operationen. Erst Mitteilung, Studia Math. 5, 50–68 (1934a)

  38. Mazur, S., Orlicz, W.: Grundlegende Eigenschaften der Polynomischen Operationen. Zweite Mitteilung, Ibidem 5, 179–189 (1934b)

  39. Piszczek, M.: Remark on hyperstability of the general linear equation. Aequ. Math. (2013)

  40. Piszczek, M., Szczawińska, J.: Hyperstability of the Drygas functional equation. J. Funct. Spaces Appl. 2013 (2013)

  41. Rassias J.M.: Solution of a problem of Ulam. J. Approx. Theory 57, 268–273 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rassias J.M.: On the Ulam stability of mixed type mappings on restricted domains. J. Math. Anal. Appl. 281, 747–762 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rassias J.M., Rassias M.J.: On the Ulam stability of Jensen and Jensen type mappings on restricted domains. J. Math. Anal. Appl. 281, 516–524 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sirouni M., Kabbaj S.: A fixed point approach to the hyperstability of Drygas functional equation in metric spaces. J. Math. Comput. Sci. 4(4), 705–715 (2014)

    Google Scholar 

  45. Smajdor W.: On set-valued solutions of a functional equation of Drygas. Aequ. Math. 77, 89–97 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Stetkær H.: Functional equations on abelian groups with involution. II. Aequ. Math 55, 227–240 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  48. Szabo Gy.: Some functional equations related to quadratic functions. Glasnik Math. 38, 107–118 (1983)

    MathSciNet  MATH  Google Scholar 

  49. Ulam, S.M.: A Collection of mathematical problems. Interscience Publ. New York, 1961. Problems in Modern Mathematics. Wiley, New York (1964)

  50. Yang D.: Remarks on the stability of Drygas equation and the Pexider-quadratic equation. Aequ. Math. 68, 108–116 (2004)

    MathSciNet  MATH  Google Scholar 

  51. Zhang D.: On the hyperstability of generalised linear functional equation in several variables. Bull. Aust. Math. Soc. 92, 259–267 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Al Jouf University, Sakakah, Saudi Arabia

    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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Almahalebi, M. On the hyperstability of \({\sigma}\)-Drygas functional equation on semigroups. Aequat. Math. 90, 849–857 (2016). https://doi.org/10.1007/s00010-016-0422-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-016-0422-2

Mathematics Subject Classification

Keywords

Search

Navigation