Commun. Korean Math. Soc. 31 (2016), No. 1, pp. 101–113 http://dx.doi.org/10.4134/CKMS.2016.31.1.101 STABILITY OF (α, β, γ )-DERIVATIONS ON LIE C * -ALGEBRA ASSOCIATED TO A PEXIDERIZED QUADRATIC TYPE FUNCTIONAL EQUATION Nasrin Eghbali and Somayeh Hazrati Abstract. In this article, we considered the stability of the following (α, β, γ)-derivation αD[x, y]= β[D(x),y]+ γ[x, D(y)] and homomorphisms associated to the quadratic type functional equation f (kx + y)+ f (kx + σ(y)) = 2kg(x)+2g(y), x, y ∈ A, where σ 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. 1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [16] in 1940, concerning the stability of group homomorphisms: Let (G 1 , ·) be a group and (G 2 , ∗) be a metric group with metric d(·, ·). Given ε> 0, does there exist δ> 0, such that if a mapping h : G 1 → G 2 satisfies the inequality d(h(x · y),h(x) ∗ h(y)) <δ for all x, y ∈ G 1 , then there exists a homomorphism H : G 1 → G 2 with d(h(x),H (x)) <ε for all x ∈ G 1 ? A C * -algebra A endowed with the Lie product [x, y]= xy − yx on A is called a Lie C * -algebra. Let A be a Lie C * -algebra. A C-linear mapping D : A → A is called a Lie derivation of A if D : A → A satisfies D[x, y]=[D(x),y]+[x, D(y)] Received May 27, 2015. 2010 Mathematics Subject Classification. Primary 46S40; Secondary 39B52, 39B82, 26E50, 46S50. Key words and phrases. (α, β, γ)-derivation, Lie C ∗ -algebra, quadratic functional equa- tion. c 2016 Korean Mathematical Society 101