Orthogonal stability of additive type equations

Summary.

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 $x \in \mathcal{X}$.