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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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Full Text: Euclid