Approximation of cubic mappings with $n$-variables in $\beta$-normed left Banach modules on Banach algebras
Bull. Korean Math. Soc. 2011 Vol. 48, No. 5, 1063-1078
Printed September 1, 2011
Majid Eshaghi Gordji, Hamid Khodaei, and Abbas Najati
Semnan University, Semnan University, University of Mohaghegh Ardabili
Abstract : Let $M=\{\,1,2,\ldots,n\,\}$ and let $\mathcal V=\{\,I\subseteq M: 1\in I\,\}$. Denote $M\setminus{I}$ by $I^c$ for $I\in \mathcal V.$ The goal of this paper is to investigate the solution and the stability using the alternative fixed point of generalized cubic functional equation \begin{align*} &\ \sum_{I\in\mathcal V}f\Big(\sum_{i\in I}a_ix_i-\sum_{i\in I^c}a_ix_i\Big)\\ =&\ 2^{n-2}a_{1} \sum^{n}_{i=2}a^2_{i}\big[f(x_{1}+x_{i})+f(x_{1}-x_{i})\big] +2^{n-1}a_{1}\Big(a^2_{1}-\sum^{n}_{i=2}a^2_{i}\Big)f(x_{1}) \end{align*} in $\beta$--Banach modules on Banach algebras, where $a_{1},\ldots,a_{n} \in \mathbb{Z}\setminus\{0\}$ with $a_{1}\neq\pm1$ and $a_{n}=1.$
Keywords : cubic functional equation, generalized Hyers--Ulam stability, Banach module
MSC numbers : 39B82, 39B52, 46H25
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