Strongly generalized derivations on C*-algebras

Let A and B be two algebras, let M be a B-bimodule and let n be a positive integer. A linear mapping Dₙ:A → M is called a strongly generalized derivation of order n, if there exist the families ₖ:A → M = 1ⁿ, ₖ:A → M = 1ⁿ, ₖ:A → B = 1ⁿ and ₖ:A → B = 1ⁿ of mappings which satisfy Dₙ(ab) = ∑k = 1ⁿ[Eₖ(a) Fₖ(b) + Gₖ(a)Hₖ(b)] for all a, b ∈ A. In this paper, we prove that every strongly generalized derivation of order one from a C^-algebra into a Banach bimodule is automatically continuous under certain conditions. The main theorem of this paper extends some celebrated results in this regard.