## 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

### Keywords:

Hyers-Ulam-Rassias stability; Banach spaces
Full Text:

### References:

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