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.

In this paper, we obtain all possible general solutions of the sum form functional equations $$\begin{array}{cc}\hfill \sum _{i=1}^k\sum _{j=1}^{\ell }f\left(p_i{q}_j\right)=& \sum _{i=1}^{k}g\left(p_i\right)\sum _{j=1}^{\ell }h\left(q_j\right)\hfill \\ \hfill \text{and}\sum _{i=1}^k\sum _{j=1}^{\ell }F\left(p_i{q}_j\right)=& \sum _{i=1}^{k}G\left(p_i\right)+\sum _{j=1}^{\ell }H\left(q_j\right)+\lambda \sum _{i=1}^{k}G\left(p_i\right)\sum _{j=1}^{\ell }H\left(q_j\right)\hfill \end{array}$$ valid for all complete probability distributions $(p_1,...,p_k)$, $(q_1,...,q_{\ell })$, $k\ge 3$, $\ell \ge 3$ fixed integers; $\lambda \in ℝ$, $\lambda \ne 0$ and $F$, $G$, $H$, $f$, $g$, $h$ are real valued mappings each having the domain $I=[0,1]$, the unit closed interval.

It is shown that every almost linear Pexider mappings $f$, $g$, $h$ from a unital $C^{*}$-algebra $𝒜$ into a unital $C^{*}$-algebra $ℬ$ are homomorphisms when $f\left(2^{n}uy\right)=f\left(2^{n}u\right)f\left(y\right)$, $g\left(2^{n}uy\right)=g\left(2^{n}u\right)g\left(y\right)$ and $h\left(2^{n}uy\right)=h\left(2^{n}u\right)h\left(y\right)$ hold for all unitaries $u\in 𝒜$, all $y\in 𝒜$, and all $n\in \mathbb{Z}$, and that every almost linear continuous Pexider mappings $f$, $g$, $h$ from a unital $C^{*}$-algebra $𝒜$ of real rank zero into a unital $C^{*}$-algebra $ℬ$ are homomorphisms when $f\left(2^{n}uy\right)=f\left(2^{n}u\right)f\left(y\right)$, $g\left(2^{n}uy\right)=g\left(2^{n}u\right)g\left(y\right)$ and $h\left(2^{n}uy\right)=h\left(2^{n}u\right)h\left(y\right)$ hold for all $u\in \{v\in 𝒜\mid v=v^{*}\text{and}v\text{is}\text{invertible}\}$, all $y\in 𝒜$ and all $n\in \mathbb{Z}$. Furthermore, we prove the Cauchy-Rassias stability of $*$-homomorphisms between unital $C^{*}$-algebras, and $ℂ$-linear...

In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive