Nearly quartic mappings in \(\beta\)-homogeneous \(F\)-spaces
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.
scientific article; zbMATH DE number 6135264
scientific article