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Cubic derivations on Banach algebras

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Abstract

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.

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References

  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)

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Acknowledgements

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.

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Correspondence to Abasalt Bodaghi.

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Bodaghi, A. Cubic derivations on Banach algebras. Acta Math Vietnam. 38, 517–528 (2013). https://doi.org/10.1007/s40306-013-0031-2

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Keywords

  • Banach algebra
  • Cubic derivation
  • Stability
  • Superstability

Mathematics Subject Classification (2000)

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