Document Type : Original Article
Authors
1
Department of Mathematical Sciences and Statistics, Malayer University, Malayer, Iran
2
Department of Mathematical Sciences and Statistics, Malayer University, 65719-95863 Malayer, Iran
3
Department of Mathematics, Factulty of Sciences and Techniques, S. M. Ben abdellah University, Fez, Morocco
10.22060/ajmc.2026.25654.1572
Abstract
Suppose that \( \mathcal{H }\) is an Abelian group and \( \mathcal{E} \) is a vector space, and consider a mapping \( f: \mathcal{H}^{\ell} \to \mathcal{E} \). In this paper, we determine the solutions of a multi-variable cubic-type functional equation in a generalized setting:
\begin{align*}
\begin{split}
&f(\xi_1,\ldots, \xi_{p-1},\xi_p+\lambda\zeta_p,\xi_{p+1},\ldots, \xi_{\ell})
-\lambda f(\xi_1,\ldots, \xi_{p-1},\xi_p+\zeta_p,\xi_{p+1},\ldots, \xi_{\ell})\\
&~+\lambda f(\xi_1,\ldots, \xi_{p-1},\xi_p-\zeta_p,\xi_{p+1},\ldots, \xi_{\ell})
-f(\xi_1,\ldots, \xi_{p-1},\xi_p-\lambda\zeta_p,\xi_{p+1},\ldots, \xi_{\ell})\\
& =2\lambda(\lambda^2-1)f(\xi_1,\ldots, \xi_{p-1},\zeta_p,\xi_{p+1},\ldots, \xi_{\ell}),
\end{split}
\end{align*}
for a fixed positive integer \( \lambda \neq 1 \). In addition, we study the behavior of almost solutions of the above functional equation
with respect to Lipschitz norms. This investigation improves and extends several results obtained in [6, 7, 11, 12, 13, 14, 18, 15, 16, 20, 23].
Keywords
Subjects
