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

Jun, Kil-Woung; Kim, Hark-Mahn

Math. Inequal. Appl. 9, No. 1, 153-165 (2006).

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.

Reviewer: Mohammad Sal Moslehian (Leeds)

MSC:

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

Keywords:

\(n\)-dimensional quadratic and additive functional equation; Banach space; generalized Hyers-Ulam-Rassias stability; additive mapping; quadratic mapping