Jensen type quadratic-quadratic mapping in Banach spaces
Bull. Korean Math. Soc. 2006 Vol. 43, No. 4, 703-709
Printed December 1, 2006
Choonkil Park, Seong-Ki Hong, and Myoung-Jung Kim
Hanyang University, Chung-nam National University, Chung-nam National University
Abstract : Let $X, Y$ be vector spaces. It is shown that if an even mapping $f : X \rightarrow Y$ satisfies $f(0)=0$ and $$\aligned &f\left(\frac{x+y}{2}+z\right) + f\left(\frac{x+y}{2}-z\right) + f\left(\frac{x-y}{2}+z\right)\\ &\ + f\left(\frac{x-y}{2}-z\right) = f(x)+f(y)+4f(z) \endaligned \tag 0.1 $$ for all $x, y, z\in X$, then the mapping $f : X \rightarrow Y$ is quadratic. Furthermore, we prove the Cauchy--Rassias stability of the functional equation {\rm (0.1)} in Banach spaces.
Keywords : Cauchy--Rassias stability, quadratic mapping, functional equation
MSC numbers : 39B52
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