Cubic derivations on Banach algebras

  • 81 Accesses

  • 5 Citations


Let A be a Banach algebra and X be a Banach A-bimodule. A mapping D: AX is a cubic derivation if D is a cubic homogeneous mapping, that is, D is cubic and D(λa)=λ 3 D(a) for any complex number λ and all aA, and D(ab)=D(a)⋅b 3+a 3D(b) for all a,bA. In this paper, we prove the stability of a cubic derivation with direct method. We also employ a fixed point method to establish the stability and the superstability of cubic derivations.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.


  1. 1.

    Bodaghi, A., Alias, I.A., Ghahramani, M.H.: Approximately cubic functional equations and cubic multipliers. J. Inequal. Appl. 53 (2011). doi:10.1186/1029-242X-2011-53

  2. 2.

    Bodaghi, A., Alias, I.A., Ghahramani, M.H.: Ulam stability of a quartic functional equation. Abstr. Appl. Anal. 2012, 232630 (2012), 9 pp. doi:10.1155/2012/232630

  3. 3.

    Cădariu, L., Radu, V.: Fixed points and the stability of quadratic functional equations. An. Univ. Timişoara Ser. Mat. Inform. 41, 25–48 (2003)

  4. 4.

    Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: A fixed point approach. Grazer Math. Ber. 346, 43–52 (2004)

  5. 5.

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

  6. 6.

    Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

  7. 7.

    Eshaghi Gordji, M., Bodaghi, A., Park, C.: A fixed point approach to the stability of double Jordan centralizers and Jordan multipliers on Banach algebras. U.P.B. Sci. Bull., Ser. A 73(2), 65–74 (2011)

  8. 8.

    Eshaghi Gordji, M., Kaboli Gharetapeh, S., Savadkouhi, M.B., Aghaei, M., Karimi, T.: On cubic derivations. Int. J. Math. Anal. 4(51), 2501–2514 (2010)

  9. 9.

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

  10. 10.

    Lee, K.Y., Lee, J.R.: Cubic double centralizers and cubic multipliers. Korean J. Math. 17(4), 437–449 (2009)

  11. 11.

    Najati, A.: Hyers–Ulam–Rassias stability of a cubic functional equation. Bull. Korean Math. Soc. 44(4), 825–840 (2007)

  12. 12.

    Jang, S.Y., Park, C.: Approximate -derivations and approximate quadratic -derivations on C -algebras. J. Inequal. Appl. 55 (2011). doi:10.1186/1029-242X-2011-55

  13. 13.

    Park, C.: Fixed points and Hyers-Ulam-Rassias stability of Cauchy–Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007, 50175 (2007)

  14. 14.

    Park, C.G., Hou, J.: Homomorphisms between C -algebras associated with the Trif functional equation and linear derivations on C -algebras. J. Korean Math. Soc. 41(3), 461–477 (2004)

  15. 15.

    Rassias, J.M.: On approximately of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–130 (1982)

  16. 16.

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

  17. 17.

    Ravi, K., Rassias, J.M., Narasimman, P.: Stability of a cubic functional equation in fuzzy normed space. J. Appl. Anal. Comput. 1(3), 411–425 (2011)

  18. 18.

    Turinici, M.: Sequentially iterative processes and applications to Volterra functional equations. Ann. Univ. Mariae Curie-Skl̄odowska, Sect. A 32, 127–134 (1978)

  19. 19.

    Ulam, S.M.: Problems in Modern Mathematics, Chap. VI. Wiley, New York (1940)

Download references


The author sincerely thanks the anonymous referee for a careful reading, constructive comments and fruitful suggestions to improve the quality of the paper and suggesting a related reference.

Author information

Correspondence to Abasalt Bodaghi.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bodaghi, A. Cubic derivations on Banach algebras. Acta Math Vietnam. 38, 517–528 (2013).

Download citation


  • Banach algebra
  • Cubic derivation
  • Stability
  • Superstability

Mathematics Subject Classification (2000)

  • 39B52
  • 47B47
  • 39B72
  • 46H25