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

Vediyappan Govindan; Choonkil Park; Sandra Pinelas; S. Baskaran; Vediyappan Govindan; Choonkil Park; Sandra Pinelas; S. Baskaran

doi:10.3934/math.2020114

AIMS Mathematics

2020, Volume 5, Issue 3: 1693-1705. doi: 10.3934/math.2020114

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Research article

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

1.
Department of Mathematics, Sri Vidya Mandir Arts and Science College, Uthangarai, Tamil Nadu-636902, India
2.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
3.
Departmento de Cie ncias Exatas e Engenharia, Academia Militar, Portugal
4.
Department of Mathematics, Sri Meenakshi GGHSS,Tirupattur, Tamil Nadu-635601, India

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.
- cubic functional equation,
- fuzzy normed space,
- Hyers-Ulam stability,
- 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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Abstract

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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