Aequationes mathematicae

, Volume 73, Issue 3, pp 249–259

# Orthogonal stability of additive type equations

• Themistocles M. Rassias
Article

## 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 $$x \in \mathcal{X}$$.

## Mathematics Subject Classification (2000).

Primary 39B55 secondary 46L05, 39B52, 46H25, 39B82

## Keywords.

Hyers-Ulam stability orthogonality orthogonally additive mapping orthogonally quadratic mapping orthogonally additive type equation orthogonality space Banach module C*-algebra