Stability problem of Hyers-Ulam-Rassias for generalized forms of cubic functional equation

Let n ≥ 2 be an integer number. In this paper, we investigate the generalized Hyers-Ulam-Rassias stability in Banach spaces and also Banach modules over a Banach algebra and a C*-algebra and the stability using the alternative fixed point of an n-dimensional cubic functional equation in Banach spaces: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f\left( {2\sum\limits_{j = 1}^{n - 1} {x_j + x_n } } \right) + f\left( {2\sum\limits_{j = 1}^{n - 1} {x_j - x_n } } \right) + 4\sum\limits_{j = 1}^{n - 1} {f(x_j ) = 16f} \left( {\sum\limits_{j = 1}^{n - 1} {x_j } } \right) + 2\sum\limits_{j = 1}^{n - 1} {(f(x_j + x_n ) + f(x_j - x_n ))} . $$\end{document}