Research article

Hyers-Ulam stability of an n-variable quartic functional equation

  • Received: 31 August 2020 Accepted: 17 November 2020 Published: 20 November 2020
  • MSC : 32B72, 32B82, 39B52

  • In this note we investigate the general solution for the quartic functional equation of the form $ \begin{align*} \left( 3n+4 \right)f\left( \sum\limits_{i = 1}^{n}{{{x}_{i}}} \right)+\sum\limits_{j = 1}^{n}{f\left( -n{{x}_{j}}+\sum\limits_{i = 1,i\ne j}^{n}{{{x}_{i}}} \right)}& = \left( {{n}^{2}}+2n+1 \right)\sum\limits_{i = 1,i\ne j\ne k}^{n}{f\left( {{x}_{i}}+{{x}_{j}}+{{x}_{k}} \right)}\\ &-\frac{1}{2}\left( 3{{n}^{3}}-2{{n}^{2}}-13n-8 \right)\sum\limits_{i = 1,i\ne j}^{n}{f\left( {{x}_{i}}+{{x}_{j}} \right)} \\ & +\frac{1}{2}\left( {{n}^{3}}+2{{n}^{2}}+n \right)\sum\limits_{i = 1,i\ne j}^{n}{f\left( {{x}_{i}}-{{x}_{j}} \right)}\\ &+\frac{1}{2}\left( 3{{n}^{4}}-5{{n}^{3}}-7{{n}^{2}}+13n+12 \right)\sum\limits_{i = 1}^{n}{f\left( {{x}_{i}} \right)} \\ \end{align*} $ $\left(n\in \mathbb{N}, \, \, n \gt 4 \right)$ and also investigate the Hyers-Ulam stability of the quartic functional equation in random normed spaces using the direct approach and the fixed point approach.

    Citation: Vediyappan Govindan, Inho Hwang, Choonkil Park. Hyers-Ulam stability of an n-variable quartic functional equation[J]. AIMS Mathematics, 2021, 6(2): 1452-1469. doi: 10.3934/math.2021089

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  • In this note we investigate the general solution for the quartic functional equation of the form $ \begin{align*} \left( 3n+4 \right)f\left( \sum\limits_{i = 1}^{n}{{{x}_{i}}} \right)+\sum\limits_{j = 1}^{n}{f\left( -n{{x}_{j}}+\sum\limits_{i = 1,i\ne j}^{n}{{{x}_{i}}} \right)}& = \left( {{n}^{2}}+2n+1 \right)\sum\limits_{i = 1,i\ne j\ne k}^{n}{f\left( {{x}_{i}}+{{x}_{j}}+{{x}_{k}} \right)}\\ &-\frac{1}{2}\left( 3{{n}^{3}}-2{{n}^{2}}-13n-8 \right)\sum\limits_{i = 1,i\ne j}^{n}{f\left( {{x}_{i}}+{{x}_{j}} \right)} \\ & +\frac{1}{2}\left( {{n}^{3}}+2{{n}^{2}}+n \right)\sum\limits_{i = 1,i\ne j}^{n}{f\left( {{x}_{i}}-{{x}_{j}} \right)}\\ &+\frac{1}{2}\left( 3{{n}^{4}}-5{{n}^{3}}-7{{n}^{2}}+13n+12 \right)\sum\limits_{i = 1}^{n}{f\left( {{x}_{i}} \right)} \\ \end{align*} $ $\left(n\in \mathbb{N}, \, \, n \gt 4 \right)$ and also investigate the Hyers-Ulam stability of the quartic functional equation in random normed spaces using the direct approach and the fixed point approach.
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    © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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