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Let $X,Y$ be complex vector spaces. Recently, Park and Th.M. Rassias showed that if a mapping $f:X\to Y$ satisfies $$\begin{array}{c}\hfill f(x+iy)+f(x-iy)=2f\left(x\right)-2f\left(y\right)\end{array}$$ for all $x$, $y\in X$, then the mapping $f:X\to Y$ satisfies $f(x+y)+f(x-y)=2f\left(x\right)+2f\left(y\right)$ 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.

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in multi-Banach algebras and derivations on multi-Banach algebras for the additive functional equation $$\sum _{i=1}^{m}f\left(m{x}_i+\sum _{j=1,j\ne i}^m{x}_j\right)+f\left(\sum _{i=1}^m{x}_i\right)=2f\left(\sum _{i=1}^{m}m{x}_i\right)$$ for each $m\in \mathbb{N}$ with $m\ge 2$.