Geodesic
Nonlinear stability of a quadratic functional equation with complex involution
Archivum mathematicum, Tome 47 (2011) no. 2, pp. 111-117.

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Let $X, Y$ be complex vector spaces. Recently, Park and Th.M. Rassias showed that if a mapping $f : X \rightarrow Y$ satisfies \begin{eqnarray} f(x+i y)+ f(x-iy) = 2 f(x) - 2f(y) \end{eqnarray} for all $x$, $y\in X$, then the mapping $f \colon X \rightarrow Y$ satisfies $f(x+y) + f(x-y) = 2 f(x) + 2 f(y)$ for all $x$, $y \in X$. Furthermore, they proved the generalized Hyers-Ulam stability of the functional equation () in complex Banach spaces. In this paper, we will adopt the idea of Park and Th. M. Rassias to prove the stability of a quadratic functional equation with complex involution via fixed point method.
Classification : 39B72, 47H10
Keywords: quadratic mapping; fixed point; quadratic functional equation; generalized Hyers-Ulam stability 
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Saadati, Reza; Sadeghi, Ghadir. Nonlinear stability of a quadratic functional equation with complex involution. Archivum mathematicum, Tome 47 (2011) no. 2, pp. 111-117. http://geodesic.mathdoc.fr/item/ARM_2011__47_2_a4/