Communications of the
Korean Mathematical Society CKMS

In this article, we considered the stability of the following $(\alpha ,\beta ,\gamma)$-derivation $$\alpha D[x,y]=\beta [D(x),y]+\gamma [x,D(y)] $$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+\sigma (y))=2kg(x)+2g(y),\,\,\,\, x,y\in A,$$ where $\sigma $ is an involution of the Lie C$^{*}$-algebra $A$ and $k$ is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

Keywords: $(\alpha ,\beta ,\gamma)$-derivation, Lie $C^{*}$-algebra, quadratic functional equa\-tion

MSC numbers: Primary 46S40; Secondary 39B52, 39B82, 26E50, 46S50