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Orthogonal stability of additive type equations

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Suppose that (\(\mathcal{X}, \bot\)) is a symmetric orthogonality module and \({\mathcal{Y}}\) a Banach module over a unital Banach algebra \({\mathcal{A}}\) and \(f : \mathcal{X} \rightarrow {\mathcal{Y}}\) is a mapping satisfying

$$\|f(ax_{1} + ax_{2}) + (-1)^{k+1}f(ax_{1} - ax_{2}) - 2af(x_{k})\| \leq \epsilon$$

, for k = 1 or 2, for some ε ≥ 0, for all a in the unit sphere \({\mathcal{A}}_{1}\) of \({\mathcal{A}}\) and all \(x_{1}, x_{2} \in \mathcal{X}\) with \(x_{1} \bot x_{2}\). Assume that the mapping \(t \mapsto f(tx)\) is continuous for each fixed \(x \in \mathcal{X}\) . Then there exists a unique \({\mathcal{A}}\) -linear mapping \(T : \mathcal{X} \rightarrow {\mathcal{Y}}\) satisfying \(T(ax) = aT(x), a \in \mathcal{A}, x \in \mathcal{X}\) such that

$$\|f(x) - f(0) - T(x)\|\leq\frac{5}{2}\epsilon$$

, for all \(x \in \mathcal{X}\).

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Correspondence to Mohammad Sal Moslehian.

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Manuscript received: August 18, 2005 and, in final form, May 3, 2006.

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Moslehian, M.S., Rassias, T.M. Orthogonal stability of additive type equations. Aequ. math. 73, 249–259 (2007).

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Mathematics Subject Classification (2000).