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APPROXIMATION OF CUBIC MAPPINGS WITH n-VARIABLES IN β-NORMED LEFT BANACH MODULE ON BANACH ALGEBRAS

  • Gordji, Majid Eshaghi (Department of Mathematics Semnan University, Center of Excellence in Nonlinear Analysis and Applications (CENAA) Semnan University) ;
  • Khodaei, Hamid (Department of Mathematics Semnan University, Center of Excellence in Nonlinear Analysis and Applications (CENAA) Semnan University) ;
  • Najati, Abbas (Department of Mathematics University of Mohaghegh Ardabili)
  • Received : 2010.04.03
  • Published : 2011.09.30
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Abstract

Let M = {1, 2, ${\ldots}$, n} and let V = {$I{\subseteq}M:1{\in}I$}. Denote M\I by $I^c$ for $I{\in}V$. The goal of this paper is to investigate the solution and the stability using the alternative fixed point of generalized cubic functional equation $ \sum\limits_{I{\in}V}f(\sum\limits_{i{\in}I}a_ix_i-\sum\limits_{i{\in}I^c}a_ix_i)=2{^{n-2}{a_1}}\sum\limits_{i=2}^na_i^2[f(x_1+x_i)+f(x_1-x_i)]+2{^{n-1}{a_1}(a^2_1-\sum\limits_{i=2}^2a^2_i)f(x_1)$ in ${\beta}$-Banach modules on Banach algebras, where $a_1,{\ldots},a_n{\in}\mathbb{Z}{\backslash}\{0\}$ with $a_1{\neq}={\pm}1$ and $a_n=1$.

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References

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