Let \(f: X\to Y\) be a mapping between real vector spaces. The functional equation \[ f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)\tag{1} \] is called cubic equation and its solution a cubic function. The authors investigate a generalized cubic equation \[ \begin{split} 4f\left(\sum_{j=1}^{n-1}x_j+mx_n\right)+4f\left(\sum_{j=1}^{n-1}x_j-mx_n\right)+m^2\sum_{j=1}^{n-1}f(2x_j) \\ =8f\left(\sum_{j=1}^{n-1}x_j\right)+4m^2\sum_{j=1}^{n-1}(f(x_j+x_n)+f(x_j-x_n))\end{split}\tag{2} \] and prove that equations (1) and (2) are equivalent.
Next, the stability of the generalized cubic equation is considered for \(X\) and \(Y\) being a normed and Banach spaces, respectively. Under the assumption that the norm of the difference of the left and right hand sides of (2) is bounded by some control mapping \(\phi(x_1,\dots,x_n)\), being subject to several assumptions, it is proved that \(f\) can be uniquely approximated by a cubic mapping \(C\), i.e., that \(\|f(x)-C(x)\|\leq\Phi(x)\), where \(\Phi\) in some way depends on \(\phi\). Three results are given; first two are obtained directly and the last one using a fixed point theorem.