Skip to main content
SpringerLink
Log in

On the stability of a radical cubic functional equation in quasi-\({\beta}\)-spaces

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

Abstract

In this paper, we introduce and solve the radical cubic functional equation

$$f\left({\sqrt[3]{x^{3} + y^{3}}}\right)= f(x) + f(y).$$

We also establish stability in quasi-\({\beta}\)-Banach spaces, and then the stability by using subadditive and subquadratic functions for the radical cubic functional equation in (\({\beta}\), p)-Banach spaces is given.

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Czerwik S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hambg. 62, 59–64 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Eshaghi Gordji and H. Khodaei, On the generalized Hyers–Ulam–Rassias stability of quadratic functional equations. Abstr. Appl. Anal. 2009 (2009), Article ID 923476.

  5. Eshaghi Gordji M., Khodaei H.: Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces. Nonlinear Anal. 71, 5629–5643 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Eshaghi Gordji and H. Khodaei, Stability of Functional Equations. Lap Lambert Academic Publishing, Saarbrücken, 2010.

  7. M. Eshaghi Gordji, H. Khodaei, A. Ebadian and G. H. Kim, Nearly radical quadratic functional equations in p-2-normed spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 896032.

  8. M. Eshaghi Gordji, H. Khodaei and H. M. Kim, Approximate quartic and quadratic mappings in quasi-Banach spaces. Int. J. Math. Math. Sci. 2011 (2011), Article ID 734567.

  9. Eshaghi Gordji M., Parviz M.: On the Hyers-Ulam stability of the functional equation \({f(\sqrt{x2 + y2}) = f(x) + f(y)}\). Nonlinear Funct. Anal. Appl. 14, 413–420 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Gǎvruta 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 

  13. Ghazanfari A.G., Alizadeh Z.: On approximate ternary m-derivations and \({\sigma}\)-homomorphisms. J. Fixed Point Theory Appl. 17, 625–640 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Jung S.M.: On the Hyers–Ulam–Rassias stability of a quadratic functional equation. J. Math. Anal. Appl. 232, 384–393 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jung S.M., Popa D., Th. Rassias M.: On the stability of the linear functional equation in a single variable on complete metric groups. J. Global Optim. 59, 165–171 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jung S.M., Th. Rassias M., Mortici C.: On a functional equation of trigonometric type. Appl. Math. Comput. 252, 293–303 (2015)

    MathSciNet  MATH  Google Scholar 

  18. D. Kang, On the stability of generalized quartic mappings in quasi-\({\beta}\)-normed spaces. J. Inequal. Appl. 2010 (2010), Article ID 198098.

  19. Khodaei H., Eshaghi Gordji M., Kim S.S., Cho Y.J.: Approximation of radical functional equations related to quadratic and quartic mappings. J. Math. Anal. Appl. 395, 284–297 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. S. S. Kim, Y. J. Cho and M. Eshaghi Gordji, On the generalized Hyers-Ulam- Rassias stability problem of radical functional equations. J. Inequal. Appl. 2012 (2012), Article ID 186.

  21. Lee S.H., Im S.M., Hawng I.S.: Quartic functional equation. J. Math. Anal. Appl. 307, 387–394 (2005)

    Article  MathSciNet  Google Scholar 

  22. Lee Y.H., JungS.M.Th. Rassias M.: On an n-dimensional mixed type additive and quadratic functional equation. Appl. Math. Comput. 228, 13–16 (2014)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Th. Rassias M.: Solution of a functional equation problem of Steven Butler. Octogon Mathematics Magazine 12, 152–153 (2004)

    Google Scholar 

  25. Rassias J.M., Kim H.M.: Generalized Hyers-Ulam stability for general additive functional equation in quasi-\({\beta}\)-normed spaces. J. Math. Anal. Appl. 356, 302–309 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Th. Rassias M., Semrl P.: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proc. Amer. Math. Soc. 114, 989–993 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. S. M. Ulam, Problems in Modern Mathematics. Chapter VI, Science Editions, Wiley, New York, 1964.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Iran

    Z. Alizadeh & A. G. Ghazanfari

Authors
  1. Z. Alizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. G. Ghazanfari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Z. Alizadeh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alizadeh, Z., Ghazanfari, A.G. On the stability of a radical cubic functional equation in quasi-\({\beta}\)-spaces. J. Fixed Point Theory Appl. 18, 843–853 (2016). https://doi.org/10.1007/s11784-016-0317-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-016-0317-9

Mathematics Subject Classification

Keywords

Search

Navigation