Nearly quartic mappings in \(\beta\)-homogeneous \(F\)-spaces
type: Item
instance of: scholarly article
title: Nearly quartic mappings in \(\beta\)-homogeneous \(F\)-spaces
zbMATH Open document ID: 1267.39020
author: Hark-Mahn Kim
author: Ick-Soon Chang
author: Madjid Eshaghi Gordji
author: H. Khodaei
published in: Results in Mathematics
publication date: 2013-02-08T00:00:00Z
review text: The first stability problem concerning group homomorphisms was raised by Ulam in 1940 and affirmatively solved by Hyers. The result of Hyers was generalized by Aoki for approximate additive functions and by Rassias for approximate linear functions. Since then the stability problems of various functional equations and mappings, such as the quadratic equation, cubic equation, quartic equation and mixed equation have been investigated by a number of authors. Let \(X\) be a linear space. A non-negative valued function \(\|.\|\) is a \(F\)-norm if it satisfies the following conditions: {\parindent=10mm \begin{itemize} \item[(\(FN_1\))] \(\|x\|=0\) if and only if \(x=0\); \item[(\(FN_2\))] \(\|\lambda .x\|=|\lambda|.\|x\|\) for all \(x\in X\) and all \(\lambda\), \(|\lambda|=1\); \item[(\(FN_3\))] \(\|x+y\|\leq \|x\|+\|y\|\) for all \(x,y\in X\); \item[(\(FN_4\))] \(\|\lambda_n x\|\to 0\) provided \(\lambda\to 0\); \item[(\(FN_5\))] \(\|\lambda x_n\|\to 0\) provided \(x_n\to 0\). \end{itemize}} Then \((X,\|.\|)\) is called a \(F^*\)-space. An \(F\)- space is a complete \(F^*\)-space. An \(F\)-pseudonorm is called \(\beta\)-homogeneous (\(\beta>0\)) if \(\|tx\|=|t|^{\beta}\|x\|\) for all \(x\in X\) and all \(t\in \mathbb{R}\). Every solution of the equation \[ f(2x_1+x_2)+f(2x_1-x_2)=4f(x_1+x_2)+4f(x_1-x_2)+24f(x_1)-6f(x_2) \] is called a quartic mapping. The main purpose of the present paper is to prove the Hyers-Ulam-Rassias stability of the quartic equation in \(n\)-variables as follows: \[ \begin{multlined} \sum_{k=2}^n\left(\sum_{i_1=2}^k\sum_{i_2=i_1+1}^{k+1}\dots \sum_{i_{n-k+1}=i_{n-k}+1}^n\right)\times f\left(\sum_{i=1,i\neq i_1,\dots,i_{n-k+1}}^n x_i-\sum_{r=1}^{n-k+1}x_{i_r}\right)+f\left(\sum{i=1}^nx_i\right)\\ =2^{n-2}\sum_{1\leq i<j\leq n}\left(f(x_i+x_j)+f(x_i-x_j)\right)-2^{n-5}(n-2)\sum_{i=1}^nf(2x_i),\quad n\geq 3,\end{multlined} \] in \(\beta\)-homogeneous \(F\)-spaces.
reviewed by: Nasrin Eghbali
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 6135264
zbMATH Keywords: quartic mapping
zbMATH Keywords: \(F\)-space
zbMATH Keywords: \(F\)-norm
zbMATH Keywords: \(F\)-pseudonym
zbMATH Keywords: Hyers-Ulam-Rassias stability
MaRDI profile type: Publication
full work available at URL: s00025-011-0215-9
OpenAlex ID: W1994664340
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DOI: 10.1007/S00025-011-0215-9
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description: scientific article; zbMATH DE number 6135264