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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.

MSC:
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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