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.


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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