Fuzzy normed spaces and stability of a generalized quadratic functional equation

Choonkil Park; K. Tamilvanan; Batool Noori; M. B. Moghimi; Abbas Najati; Choonkil Park; K. Tamilvanan; Batool Noori; M. B. Moghimi; Abbas Najati

doi:10.3934/math.2020458

AIMS Mathematics

2020, Volume 5, Issue 6: 7161-7174. doi: 10.3934/math.2020458

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Research article

Fuzzy normed spaces and stability of a generalized quadratic functional equation

1.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
2.
Department of Mathematics, Government Arts College for Men, Krishnagiri, Tamilnadu 635001, India
3.
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran

Received: 10 June 2020 Accepted: 02 September 2020 Published: 10 September 2020
MSC : 46S40, 39B52, 39B82, 26E50, 46S50

In this paper, we acquire the general solution of the generalized quadratic functional equation $ \begin{aligned} \sum\limits_{1 \leq a \lt b \lt c \leq m}\varphi\left(r_{a}+r_{b}+r_{c}\right)& = (m-2)\sum\limits_{1\leq a \lt b\leq m}\varphi\left(r_{a}+r_{b}\right) \\ &\quad-\left(\frac{m^{2}-3m+2}{2}\right)\nonumber \sum\limits_{a = 1}^{m}\frac{\varphi\left(r_{a}\right)+\varphi\left(-r_{a}\right)}{2} \end{aligned} $ where $m\geqslant 3$ is an integer. We also investigate Hyers-Ulam stability results by means of using alternative fixed point theorem for this generalized quadratic functional equation.
- Hyers-Ulam stability,
- fuzzy Banach spaces,
- quadratic mapping
Citation: Choonkil Park, K. Tamilvanan, Batool Noori, M. B. Moghimi, Abbas Najati. Fuzzy normed spaces and stability of a generalized quadratic functional equation[J]. AIMS Mathematics, 2020, 5(6): 7161-7174. doi: 10.3934/math.2020458

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Abstract

In this paper, we acquire the general solution of the generalized quadratic functional equation $ \begin{aligned} \sum\limits_{1 \leq a \lt b \lt c \leq m}\varphi\left(r_{a}+r_{b}+r_{c}\right)& = (m-2)\sum\limits_{1\leq a \lt b\leq m}\varphi\left(r_{a}+r_{b}\right) \\ &\quad-\left(\frac{m^{2}-3m+2}{2}\right)\nonumber \sum\limits_{a = 1}^{m}\frac{\varphi\left(r_{a}\right)+\varphi\left(-r_{a}\right)}{2} \end{aligned} $ where $m\geqslant 3$ is an integer. We also investigate Hyers-Ulam stability results by means of using alternative fixed point theorem for this generalized quadratic functional equation.

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