The generalized Hyers-Ulam - Rassias stability of a cubic functional equation

The generalized Hyers-Ulam - Rassias stability of a cubic functional equation

In this paper, we obtain the general solution and the generalized Hyers–Ulam– Rassias stability for a cubic functional equation f(mx + y) + f(mx − y) = mf(x+ y) + mf(x − y) + 2($m^3$ −m)f(x) for a positive integer m $geq$ 1.

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  • [1] Aczel, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, 1989.
  • [2] Baker, J.: The stability of the cosine equation. Proc. Amer. Math. Soc. 80, 411–416 (1980).
  • [3] Cholewa, P. W.: Remarks on the stability of functional equations. Aequationes Math. 27, 76–86 (1984).
  • [4] Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62, 59–64 (1992).
  • [5] Forti, G. L.: Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations. J. Math. Anal. Appl. 295, 127–133 (2004).
  • [6] Grabiec, A.: The generalized Hyers–Ulam stability of a class of functional equations. Publ. Math. Debrecen. 48, 217–235 (1996).
  • [7] Hyers, D. H.: On the stability of the linear functional equation.Proc. Natl. Acad. Sci. U.S.A. 27, 222–224 (1941).
  • [8] Hyers, D. H., Isac, G., and Rassias, Th. M.: Stability of Functional Equations in Several Variables. Birkh¨auser, Basel, 1998.
  • [9] Hyers, D. H., Isac, G., and Rassias, Th. M.: On the asymptoticity aspect of Hyers–Ulam stability of mappings. Proc. Amer. Math. Soc. 126, 425–430 (1998).
  • [10] Hyers, D. H., Rassias, Th. M.: Approximate homomorphisms. Aequationes Math. 44, 125–153 (1992).
  • [11] Jun, K.W., Kim, H. M.: The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274, 867–878 (2002).
  • [12] Jun , K. W., Lee, Y. H.: On the Hyers–Ulam–Rassias stability of a pexiderized quadratic inequality.Math. In eq. Appl. 4, 93–118 (2001).
  • [13] Jung, S. M.: On the Hyers–Ulam–Rassias stability of a quadratic functional equation. J. Math. Anal. Appl. 232, 384–393 (1999).
  • [14] Kannappan, Pl.: Quadratic functional equation and inner product spaces. Results Math. 27, 368–372 (1995).
  • [15] Rassias, Th. M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978).
  • [16] Rassias, Th. M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264–284 (2000).
  • [17] Skof, F.: Proprietaa locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano. 53, 113–129 (1983).
  • [18] Ulam, S. M.: Problems in Modern Mathematics. Chap. VI, Science Ed., Wiley, New York, 1960