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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 \(2,3,4, \dots\).

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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