Additive $ \rho $-functional inequalities in non-Archimedean 2-normed spaces

Zhihua Wang; Choonkil Park; Dong Yun Shin; Zhihua Wang; Choonkil Park; Dong Yun Shin

doi:10.3934/math.2021116

AIMS Mathematics

2021, Volume 6, Issue 2: 1905-1919. doi: 10.3934/math.2021116

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Research article

Additive $ \rho $-functional inequalities in non-Archimedean 2-normed spaces

1.
School of Science, Hubei University of Technology, Wuhan, Hubei 430068, P. R. China
2.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
3.
Department of Mathematics, University of Seoul, Seoul 02504, Korea

Received: 22 October 2020 Accepted: 01 December 2020 Published: 03 December 2020
MSC : 39B72, 39B62, 12J25

In this paper, we solve the additive $ \rho $-functional inequalities: $ \begin{align*} \|f(x+y)-f(x)-f(y)\| \leq \|\rho(2f(\frac{x+y}{2})-f(x)-f(y))\|, \\ \|2f(\frac{x+y}{2})-f(x)-f(y)\| \leq \|\rho(f(x+y)-f(x)-f(y))\|, \end{align*} $ where $ \rho $ is a fixed non-Archimedean number with $ |\rho| < 1 $. More precisely, we investigate the solutions of these inequalities in non-Archimedean $ 2 $-normed spaces, and prove the Hyers-Ulam stability of these inequalities in non-Archimedean $ 2 $-normed spaces. Furthermore, we also prove the Hyers-Ulam stability of additive $ \rho $-functional equations associated with these inequalities in non-Archimedean $ 2 $-normed spaces.
- additive $ \rho $-functional equation,
- additive $ \rho $-functional inequality,
- Hyers-Ulam stability,
- non-Archimedean 2-normed spaces
Citation: Zhihua Wang, Choonkil Park, Dong Yun Shin. Additive $ \rho $-functional inequalities in non-Archimedean 2-normed spaces[J]. AIMS Mathematics, 2021, 6(2): 1905-1919. doi: 10.3934/math.2021116

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Abstract

In this paper, we solve the additive $ \rho $-functional inequalities: $ \begin{align*} \|f(x+y)-f(x)-f(y)\| \leq \|\rho(2f(\frac{x+y}{2})-f(x)-f(y))\|, \\ \|2f(\frac{x+y}{2})-f(x)-f(y)\| \leq \|\rho(f(x+y)-f(x)-f(y))\|, \end{align*} $ where $ \rho $ is a fixed non-Archimedean number with $ |\rho| < 1 $. More precisely, we investigate the solutions of these inequalities in non-Archimedean $ 2 $-normed spaces, and prove the Hyers-Ulam stability of these inequalities in non-Archimedean $ 2 $-normed spaces. Furthermore, we also prove the Hyers-Ulam stability of additive $ \rho $-functional equations associated with these inequalities in non-Archimedean $ 2 $-normed spaces.

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