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On the stability of quadratic functional equations. (English) Zbl 1146.39045

Summary: Let \(X,Y\) be vector spaces and \(k\) a fixed positive integer. It is shown that a mapping \(f(kx+y)+f(kx-y)=2k^{2}f(x)+2f(y)\) for all \(x,y\in X\) if and only if the mapping \(f:X\rightarrow Y\) satisfies \(f(x+y)+f(x-y)=2f(x)+2f(y)\) for all \(x,y\in X\). Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

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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References:

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