On the stability of quadratic functional equations (Q938361)

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.