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On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings. (English) Zbl 1009.39024
The following Hyers-Ulam stability property is established: If \(f:X\to Y\), \(X,Y\) real normed linear spaces, is a mapping for which there exists a constant \(\varepsilon>0\), such that \[ \left\|f\left(\sum^n_{i=1} a_ix_i\right)+ \sum_{1\leq i<j\leq n}f(a_jx_i-a_ix_j)-m\sum^n_{i=1} f(x_i)\right \|\leq \varepsilon \] holds for every \((x_1,x_2,\dots, x_n)\in X^n\), where: \(a=(a_1,\dots,a_n)\in\mathbb{R}^n\) is fixed, \(a\neq 0\), \(1<m=\sum^n_{i=1} a_i^2\neq [1+{k \choose 2}]/k\), \(k=2,3,\dots,\) then there exists a unique quadratic mapping with respect to \(a\), \(Q:X\to Y\), such that \(\|f(x)-Q(x)\|\leq c\), for every \(x\in X\); \(c\) is determined by a formula, in function of \(m,n, \varepsilon\).
A mapping \(Q:X\to Y\) is called quadratic with respect to \(a\) if \[ Q\left( \sum^n_{i=1}a_ix_i\right) +\sum_{1\leq i<j\leq n}Q(a_jx_i -a_ix_j)=m \sum^n_{i=1} Q(x_i) \] holds for every \((x_1,x_2,\dots, x_n)\in X^n\), and fixed \(a=(a_1, \dots,a_n)\) \(\mathbb{R}^n\), \(a\neq 0\), such that \(0<m=\sum^n_{i=1} a_i^2= [1+{n\choose 2}]/n\), where \(n\) is arbitrary but fixed and equals to \(2,3,4, \dots\).

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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