The generalized cubic functional equation and the stability of cubic Jordan \(*\)-derivations

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In the current work, we obtain the general solution of the following generalized cubic functional equation

$$\begin{aligned}&f(x+my)+f(x-my)\\&\quad =2\left( 2\cos \left( \frac{m\pi }{2}\right) +m^2-1\right) f(x)-\frac{1}{2}\left( \cos \left( \frac{m\pi }{2}\right) +m^2-1\right) f(2x)\\&\qquad +m^2\{f(x+y)+f(x-y)\} \end{aligned}$$

for an integer \(m \ge 1\). We prove the Hyers–Ulam stability and the superstability for this cubic functional equation by the directed method and a fixed point approach. We also employ the mentioned functional equation to establish the stability of cubic Jordan \(*\)-derivations on \(C^*\)-algebras and \(JC^*\)-algebras.

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  1. 1.

    An, J., Cui, J., Park, C.: Jordan \(*\)-derivations on \(C^*\)-algebras and \(JC^*\)-algebras, Abst. Appl. Anal. 2008. Article ID 410437 (2008)

  2. 2.

    Bodaghi, A., Alias, I.A.: Approximate ternary quadratic derivations on ternary Banach algebras and \(C^*\)-ternary rings. Adv. Differ. Equ. 2012. Article No. 11 (2012)

  3. 3.

    Bodaghi, A., Alias, I.A., Ghahramani, M.H.: Approximately cubic functional equations and cubic multipliers. J. Inequal. Appl. 2011. Article No. 53 (2011)

  4. 4.

    Bodaghi, A., Alias, I.A., Ghahramani, M.H.: Ulam stability of a quartic functional equation, Abst. Appl. Anal. 2012. Article ID 232630. doi:10.1155/2012/232630

  5. 5.

    Bodaghi, A., Eshaghi Gordji, M., Paykan, K.: Approximate multipliers and approximate double centralizers: a fixed point approach. An. St. Univ. Ovidius Constanta 20(3), 21–32 (2012)

  6. 6.

    Bodaghi, A., Zabandan, G.: On the stability of quadratic (\(*\)-) derivations on (\(*\)-) Banach algebras. Thai J. Math (to appear)

  7. 7.

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

  8. 8.

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

  9. 9.

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

  10. 10.

    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)

  11. 11.

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

  12. 12.

    Eshaghi Gordji, M., Najati, A.: Approximately \(J^*\)-homomorphisms: a fixed point approach. J. Geom. Phys. 6, 809–814 (2010)

  13. 13.

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

  14. 14.

    Hyers, D.H., Isac, G., Rassias, ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

  15. 15.

    Koh, H., Kang, D.: On the stability of a generalized cubic functional equation. Bull. Korean Math. Soc. 45(4), 739–748 (2008)

  16. 16.

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

  17. 17.

    Najati, A.: The generalized Hyers–Ulam–Rassias stability stability of a cubic functional equation. Turk. J. Math. 31, 395–408 (2007)

  18. 18.

    Jang, S.Y., Park, C.: Approximate \(*\)-derivations and approximate quadratic \(*\)-derivations on \(C^*\)-algebras. J. Inequal. Appl. 2011. Article No. 55 (2011)

  19. 19.

    Jun, K.W., Kim, H.M.: The generalized Hyers–Ulam–Russias stability of a cubic functional equation. J. Math. Anal. Appl. 274(2), 267–278 (2002)

  20. 20.

    Jun, K.W., Kim, H.M.: On the Hyers–Ulam–Rassias stability of a general cubic functional equation. Math. Inequal. Appl. 6(2), 289–302 (2003)

  21. 21.

    Park, C.: Homomorphisms between Poisson \(JC^*\)-algebras. Bull. Braz. Math. Soc. 36, 79–97 (2005)

  22. 22.

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

  23. 23.

    Park, C., Bodaghi, A.: On the stability of \(*\)-derivations on Banach \(*\)-algebras, Adv. Differ. Equ. 2012. Article No. 138 (2012)

  24. 24.

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

  25. 25.

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

  26. 26.

    Saadati, R., Vaezpour, S.M., Park, C.: The stability of the cubic functional equation in various spaces. Math. Commun. 16(3), 131–145 (2011)

  27. 27.

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

  28. 28.

    Yang, S.Y., Bodaghi, A., Mohd Atan, K.A.: Approximate cubic \(*\)-derivations on Banach \(*\)-algebras. Abst. Appl. Anal. 2012. Article ID 684179. doi:10.1155/2012/684179

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The authors express their sincere thanks to the reviewers for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially.

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Correspondence to Abasalt Bodaghi.

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Bodaghi, A., Moosavi, S.M. & Rahimi, H. The generalized cubic functional equation and the stability of cubic Jordan \(*\)-derivations. Ann Univ Ferrara 59, 235–250 (2013).

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  • Banach algebra
  • Cubic derivation
  • Cubic functional equation
  • Hyers–Ulam stability
  • Superstability

Mathematics Subject Classification (2010)

  • 39B52
  • 39B72
  • 46L05
  • 47B47