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Hyers-Ulam stability of an $$n$$-Apollonius type quadratic mapping. (English) Zbl 1148.39025
Let $$X$$ and $$Y$$ be linear spaces. The author examines the solutions and the Hyers-Ulam stability of $$n$$-Apollonius type functional equation:
$\sum_{i=1}^{n}Q(z-x_i)=\frac1n\sum_{1\leq i<j\leq n}^{n}Q(x_i-x_j)+nQ\left(z-\frac1n\sum_{i=1}^{n}x_i\right)\,,$ where $$z,x_i\in X (i=1\,\dots,n)$$ and $$Q:X\to Y$$. He proves that the solutions are the same as those of the quadratic functional equation:
$Q(x+y)+Q(x-y)=2Q(x)+2Q(y)\,.$ It was cleared up that this equation is stable in the sense of Hyers-Ulam.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B22 Functional equations for real functions 39B52 Functional equations for functions with more general domains and/or ranges
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