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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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