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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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