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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 39B52 Functional equations for functions with more general domains and/or ranges
##### Keywords:
Hyers-Ulam-Rassias stability; Banach spaces
Full Text:
##### References:
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