Bulletin of the
Korean Mathematical Society BKMS

In this paper, we generalize the stability for an $n$-dimensional cubic functional equation in Banach space to set-valued dynamics. Let $n\ge 2$ be an integer. We define the $n$-dimensional cubic set-valued functional equation given by \begin{eqnarray*} &\qquad\quad f(2\sum_{i=1}^{n-1}x_{i}+x_{n})\oplus f(2\sum_{i=1}^{n-1}x_{i}-x_{n})\oplus 4\sum_{i=1}^{n-1}f(x_{i})\\ &=16f(\sum_{i=1}^{n-1}x_{i})\oplus 2\sum_{i=1}^{n-1}(f(x_{i}+x_{n})\oplus f(x_{i}-x_{n})). \end{eqnarray*} We first prove that the solution of the $n$-dimensional cubic set-valued functional equation is actually the cubic set-valued mapping in \cite{CKY14}. We prove the Hyers-Ulam stability for the set-valued functional equation.

Keywords: Hyers-Ulam stability, $n$-dimensional cubic set-valued functional equation

MSC numbers: Primary 39B82, 47H04, 47H10, 54C60