On the general quadratic functional equation (Q2493340)

The author introduces a generalized quadratic functional equation of the form \[ f\!\left( \sum_{i=1}^p a_i x_i \right) + \sum_{1 \leq i < j \leq p} f(a_j x_i - a_i x_j) = m \sum_{i=1}^p f(x_i), \] where \({\displaystyle m = \sum_{i=1}^p a_i^2}\), and proves that the above equation is stable in the sense of Hyers, Ulam, and Rassias. More precisely, let \(X\) be a normed space and \(Y\) a Banach space. Assume that \({\displaystyle 0 < r = \sum_{i=1}^p r_i \neq 2}\) and \(p \geq 2\). If a mapping \(f : X \to Y\) satisfies \(f(0) = 0\) and \[ \left\| f\!\left( \sum_{i=1}^p a_i x_i \right) + \sum_{1 \leq i < j \leq p} f(a_j x_i - a_i x_j) - m \sum_{i=1}^p f(x_i) \right\| \leq c \prod_{i=1}^p \| x_i \|^{r_i} \] for all \(x_1 , \ldots, x_p \in X\), then there exist a unique quadratic mapping \(Q : X \to Y\) and a number \(K(r, m) > 0\) such that \[ \| f(x) - Q(x) \| \leq K(r,m) c \| x \|^r \] for all \(x \in X\).