Research article

Solution of a 3-D cubic functional equation and its stability

  • Received: 05 October 2019 Accepted: 10 February 2020 Published: 13 February 2020
  • MSC : 39B52, 39B72, 39B82, 47H10, 47S40

  • In this paper, we define and find the general solution of the following 3-D cubic functional equation $ \begin{eqnarray*} &&\left(2x_{1}+x_{2}+x_{3}\right) = 3f\left(x_{1}+x_{2}+x_{3}\right)+f\left(-x_{1}+x_{2}+x_{3}\right) +2f\left(x_{1}+x_{2}\right)+2f\left(x_{1}+x_{3}\right) \nonumber \\&& \hspace{3.0cm}-6f\left(x_{1}-x_{2}\right)-6f\left(x_{1}-x_{3}\right)-3f\left(x_{2}+x_{3}\right)+2f\left(2x_{1}-x_{2}\right)\nonumber \\&& \hspace{3.0cm} +2f\left(2x_{1}-x_{3}\right)-18f\left(x_{1}\right)-6f\left(x_{2}\right)-6f\left(x_{3}\right). \end{eqnarray*} $ We also prove the Hyers-Ulam stability of this functional equation in fuzzy normed spaces by using the direct method and the fixed point method.

    Citation: Vediyappan Govindan, Choonkil Park, Sandra Pinelas, S. Baskaran. Solution of a 3-D cubic functional equation and its stability[J]. AIMS Mathematics, 2020, 5(3): 1693-1705. doi: 10.3934/math.2020114

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  • In this paper, we define and find the general solution of the following 3-D cubic functional equation $ \begin{eqnarray*} &&\left(2x_{1}+x_{2}+x_{3}\right) = 3f\left(x_{1}+x_{2}+x_{3}\right)+f\left(-x_{1}+x_{2}+x_{3}\right) +2f\left(x_{1}+x_{2}\right)+2f\left(x_{1}+x_{3}\right) \nonumber \\&& \hspace{3.0cm}-6f\left(x_{1}-x_{2}\right)-6f\left(x_{1}-x_{3}\right)-3f\left(x_{2}+x_{3}\right)+2f\left(2x_{1}-x_{2}\right)\nonumber \\&& \hspace{3.0cm} +2f\left(2x_{1}-x_{3}\right)-18f\left(x_{1}\right)-6f\left(x_{2}\right)-6f\left(x_{3}\right). \end{eqnarray*} $ We also prove the Hyers-Ulam stability of this functional equation in fuzzy normed spaces by using the direct method and the fixed point method.


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    [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. JPN, 2 (1950), 64-66. doi: 10.2969/jmsj/00210064
    [2] V. Govindan, G. Balasubramanian, C. Muthamilarasi, Generalized stability of cubic functional equation, Int. J. Sci. Eng. Res., 2 (2017), 74-82.
    [3] V. Govindan, S. Murthy, General solution and generalized HU-stability of new n-dimensional cubic functional equation in fuzzy ternary Banach algebras using two different methods, Int. J. Pure Appl. Math., 113 (2017), 6.
    [4] V. Govindan, S. Murthy, M. Saravanan, Solution and stability of new type of (aaq,bbq,caq,daq) mixed type functional equation in various normed spaces: using two different methods, Int. J. Math. Appl., 5 (2017), 187-211.
    [5] V. Govindan, S. Murthy, M. Saravanan, Solution and stability of a cubic type functional equation: Using direct and fixed point methods, Kragujevac J. Math., 44 (2018), 7-26.
    [6] V. Govindan, S. Pinelas, S. Murthy, et al. Generalized Hyers-Ulam stability of an AQ functional equation, Int. J. Res. Eng. Appl. Manag., 4 (2018), 6.
    [7] D. H. Hyers, On the stability of the linear functional equation, P. Nat. Acad. Sci. USA, 27 (1941), 222-224. doi: 10.1073/pnas.27.4.222
    [8] K. Jun, H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274 (2002), 867-878. doi: 10.1016/S0022-247X(02)00415-8
    [9] K. Jun, H. Kim, On the Hyers-Ulam-Rassias stability of a general cubic functional equation, Math. Inequal. Appl., 6 (2003), 87-95.
    [10] K. Jun, H. Kim, On the Hyers-Ulam stability of a generalized quadratic and additive functional equation, B. Korean Math. Soc., 42 (2005), 133-148. doi: 10.4134/BKMS.2005.42.1.133
    [11] J. Lee, C. Park, S. Pinelas, et al. Stability of a sexvigintic functional equation, Nonlinear Funct. Anal. Appl., 24 (2019), 911-924.
    [12] Y. Lee, S. Chung, Stability of the Jensen type functional equation, Banach J. Math. Anal., 1 (2007), 91-100. doi: 10.15352/bjma/1240321559
    [13] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy almost quadratic functions, Results Math., 52 (2008), 161-177. doi: 10.1007/s00025-007-0278-9
    [14] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Set. Syst., 159 (2008), 720-729. doi: 10.1016/j.fss.2007.09.016
    [15] S. A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos Soliton. Fract., 42 (2009), 2989-2996. doi: 10.1016/j.chaos.2009.04.040
    [16] S. A. Mohiuddine, Q. M. Lohani, On generalized statistical convergence in intuitionistic fuzzy normed space, Chaos Soliton. Fract., 42 (2009), 1731-1737. doi: 10.1016/j.chaos.2009.03.086
    [17] S. A. Mohiuddine, H. Şevli, Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space, J. Comput. Appl. Math., 235 (2011), 2137-2146. doi: 10.1016/j.cam.2010.10.010
    [18] M. Mursaleen, S. A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Soliton. Fract., 41 (2009), 2414-2421. doi: 10.1016/j.chaos.2008.09.018
    [19] M. Mursaleen, S. A. Mohiuddine, Nonlinear operators between intuitionistic fuzzy normed spaces and Fréchet differentiation, Chaos Soliton. Fract., 42 (2009), 1010-1015. doi: 10.1016/j.chaos.2009.02.041
    [20] M. Mursaleen, S. A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos Soliton. Fract., 42 (2009), 2997-3005. doi: 10.1016/j.chaos.2009.04.041
    [21] M. Mursaleen, S. A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Comput. Appl. Math., 233 (2009), 142-149. doi: 10.1016/j.cam.2009.07.005
    [22] S. Murthy, M. Arunkumar, V. Govindan, General solution and generalized Ulam-Hyers stability of a generalized n-type additive quadratic functional equation in Banach space and Banach algebra: Direct and fixed point methods, Int. J. Adv. Math. Sci., 3 (2015), 25-64.
    [23] A. Najati, Hyers-Ulam-Rassias stability of a cubic functional equation, B. Korean Math. Soc., 44 (2007), 825-840. doi: 10.4134/BKMS.2007.44.4.825
    [24] S. Pinelas, V. Govindan, K. Tamilvanan, Stability of cubic functional equation in random normed space, J. Adv. Math., 14 (2018), 7864-7877. doi: 10.24297/jam.v14i2.7614
    [25] S. Pinelas, V. Govindan, K. Tamilvanan, New type of quadratic functional equation and its stability, Int. J. Math. Trends Tech., 60 (2018), 180-186.
    [26] S. Pinelas, V. Govindan, K. Tamilvanan, Stability of a quartic functional equation, J. Fix. Point Theory A., 20 (2018), 4.
    [27] S. Pinelas, V. Govindan, K. Tamilvanan, Stability of Pinelas-septoicosic functional equation, Open J. Math. Sci., 3 (2019), 11-28.
    [28] T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta. Appl. Math., 62 (2000), 123-130.
    [29] K. Ravi, J. M. Rassias, P. Narasimman, Stability of a cubic functional equation in fuzzy normed space, J. Appl. Anal. Comput., 1 (2011), 411-425.
    [30] R. Saadati, J. Park, On the intuitionistic fuzzy topological spaces, Chaos Soliton. Fract., 27 (2006), 331-344. doi: 10.1016/j.chaos.2005.03.019
    [31] S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, 1960.
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