In this paper, we investigate the following typical form of a class of cubic functional equations:
for some rational number m and some real number k. Furthermore, we provide a systematic program to prove the generalized Hyers-Ulam stability for the class of functional equations via the stability for the typical form in fuzzy normed spaces.
AokiT., On the stability of the linear transformation in Banach spaces, J Math Soc Japan2 (1950), 64–66.
2.
BagT. and SamantaS.K., Finite dimensional fuzzy normed linear spaces, J Fuzzy Math11 (2003), 687–705.
3.
ChengS.C. and MordesonJ.N., Fuzzy linear operator and fuzzy normed linear spaces, Bull Calcutta Math Soc86 (1994), 429–436.
4.
CholewaP.W., Remarkes on the stability of functional equations, Aequationes Math27 (1984), 76–86.
5.
CieplinskiK., Applications of fixed point theorems to the hyers-ulam stability of functional equation-A survey, Ann Funct Anal3(1) (2012), 151–164.
6.
CzerwikS., On the stability of the quadratic mapping in normed spaces, Bull Abh Math Sem Univ Hamburg62 (1992), 59–64.
7.
DiazJ.B. and MargolisB., A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull Amer Math Soc74 (1968), 305–309.
8.
GăvrutaP., A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J Math Anal Appl184 (1994), 431–436.
9.
HyersD.H., On the stability of the linear functional equation, Proc Nat Acad Sci27 (1941), 222–224.
10.
IsacG. and RassiasTh.M., Stability of -additive mappings, Appications to nonlinear analysis, Internat J Math and Math Sci19 (1996), 219–228.
11.
JunK.W. and KimH.M., The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J Math Anal Appl274 (2002), 867–878.
12.
JunK.W. and KimH.M., On the Hyers-Ulam-Rassias stability of a general cubic functional equation, Mathe-matical Inequalities and Applications6 (2003), 289–302.
13.
JunK., KimH. and ChangI., On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, J Comput Anal Appl7 (2005), 21–33.
14.
KramosilI. and MichálekJ., Fuzzy metric and statistical metric spaces, Kybernetica11 (1975), 336–344.
MihetD. and RaduV., On the stability of the additive Cauchy functional equation in random normed spaces, J Math Anal Appl343 (2008), 567–572.
17.
MirmostafaeeA.K., MirzavaziriM. and MoslehianM.S., Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst159 (2008), 730–738.
18.
MirmostafaeeA.K. and MoslehianM.S., Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst159 (2008), 720–729.
19.
MirzavaziriM. and MoslehianM.S., A fixed point approach to stability of a quadratic equation, Bulletin of the Brazilian Mathematical Society37(3) (2006), 361–376.
20.
RassiasJ.M., Solution of the Ulam stability problem for cubic mappings, Glasnik Matematički36 (2001), 63–72.
21.
RassiasTh.M., On the stability of the linear mapping in Banach sapces, Proc Amer Math Sco72 (1978), 297–300.
22.
UlamS.M., Problems in modern mathematics, Science Editions John Wiley and Sons, Inc., New York, 1964.
