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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