Hyers-Ulam stability of an <i>n</i>-variable quartic functional equation

Vediyappan Govindan; Inho Hwang; Choonkil Park; Vediyappan Govindan; Inho Hwang; Choonkil Park

doi:10.3934/math.2021089

AIMS Mathematics

2021, Volume 6, Issue 2: 1452-1469. doi: 10.3934/math.2021089

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Research article

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

1.
Department of Mathematics, DMI St John Baptist University, Mangochi, Malawi
2.
Department of Mathematics, Incheon National University, Incheon 22012, Korea
3.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

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*} $ $\left(n\in \mathbb{N}, \, \, n > 4 \right)$ 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.
- quartic functional equation,
- fixed point method,
- Hyers-Ulam stability,
- random normed space,
- direct method
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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Abstract

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*} $ $\left(n\in \mathbb{N}, \, \, n > 4 \right)$ 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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