Lie ternary (σ, τ, ξ)-derivations on Banach ternary algebras

Let a be a banach ternary algebra over a scalar fixd r or c and x be a ternary banach a–module.let σ, τ and ξ be linear mappings on a, a linear mapping d : (a, [ ]a) → (x, [ ]x ) is called a lieternary (σ, τ, ξ)–derivation, if d([a, b, c]) = [[d(a)bc]x ](σ,τ,ξ) − [[d(c)ba]x ](σ,τ,ξ) , for all a, b, c ∈ a, where [abc](σ,τ,ξ) = aτ (b)ξ(c)−σ(c)τ (b)a and [a, b, c] = [abc]a −[cba]a. in this paper, we prove the generalized hyers–ulam–rassias stability of lie ternary (σ, τ, ξ)–derivations on banach ternary algebras and c∗–lie ternary (σ, τ, ξ)–derivations on c∗–ternary algebras for the following euler–lagrange type additive mapping: n σ i=1 f ( n σ j=1 q(xi − xj )) n σ i=1 qxi)=nq n σ i=1 f (xi).