Hyers–Ulam–Rassias stability of a finite variable cubic functional equation in matrix paranormed spaces

Abstract

We introduce the finite variable cubic functional equation of the form

$$\begin{aligned} \sum _{a=1}^{m}\phi \left( -t_{a}+\sum _{b=1;a \ne b}^{m}t_{b}\right) -\sum _{a=1}^{m}\phi \left( 2t_{a}\right) =\left( m-6\right) \sum _{1 \le a< b< c \le m}\phi \left( t_{a}+t_{b}+t_{c}\right) \\ +\left( -m^{2}+9m-14\right) \sum _{1\le a<b\le m}\phi \left( t_{a}+t_{b}\right) \\ +\left( \frac{m^{3}-11 m^{2}+28 m-36}{2}\right) \sum _{a=1}^{m}\phi \left( t_{a}\right) \end{aligned}$$

where \(m \ge 4\) is a fixed integer, and we establish the Hyers–Ulam–Rassias stability results in paranormed spaces and matrix paranormed spaces.