Suppose that \(X\) and \(Y\) are linear spaces and that the mapping \(f:X\to Y\) satisfies the following functional equation: \[ \sum_{1\leq i_1<\dots <i_m\leq n} f\left(\frac{\sum_{j=1}^m x_{i_j}}{m}+\sum_{l=1}^{n-m}x_{k_l}\right) =\frac{n-m+1}{n} \left( \begin{matrix} n \\ m \end{matrix}\right) \sum_{i=1}^n f(x_i) \] (with \(1\leq k_l(\neq i_j, \forall j\in \{1,\dots, m\})\leq n\) in the first sum) for all \(x_1,\dots,x_n\in X\), where \(n,m\in \mathbb{N}\) are fixed integers with \(n\leq 2, 1\leq m\leq n\).
The authors investigate the generalized Hyers-Ulam stability of the above equation using the \(\Delta_{\mu}\)-condition without using the Fatou property in \(\rho\)-complete convex modular algebras. Here the modular \(\rho\) is said to satisfy the \(\Delta_{\mu}\)-condition if there exists \(k>0\) such that \(\rho(\mu x) \leq k\rho(x)\) for all \(x\in X_{\rho}\), \(\mu=n-m+1\). Finally they study an alternative generalized Hyers-Ulam stability of the equation in \(\rho\)-complete convex modular algebras without using neither the Fatou property nor the \(\Delta_{2}\)-condition.