On the stability of an \(n\)-dimensional quadratic and additive functional equation. (English) Zbl 1093.39026

The functional equation \[ f(\sum_{i=1}^nx_i) + (n-2) \sum_{i=1}^nf(x_i) = \sum_{1 \leq i < j \leq n} f(x_i + x_j), \quad (n >2) \] is called \(n\)-dimensional quadratic and additive functional equation. The authors show that a mapping \(f: X \to Y\) between real vector spaces satisfies the above equation if and only if there exist mappings \(B: X \times X \to Y\) and \(A: X \to Y\) such that \(f(x) = B(x, x) + A(x)\) for all \(x \in X\), where \(B\) is symmetric biadditive and \(A\) is additive.
They also establish the generalized Hyers-Ulam-Rassias stability of the above equation for the even or odd case in the \(n\) variables. The paper is nice and readable despite its involves technicalities.


39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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