Authors: Bilal Ahmad Wani ¹

• 1 Department of Mathematics, Aligarh Muslim University

Let R be a semiprime ring with unity e and φ, ϕ be automorphisms of R. In this paper it is shown that if R satisfies2D(xn) = D(xn−1)φ(x) + ϕ(xn−1)D(x) + D(x)φ(xn−1) + ϕ(x)D(xn−1)for all x ∈ R and some fixed integer n ≥ 2, then D is an (φ, ϕ)-derivation.Moreover, this result makes it possible to prove that if R admits an additivemappings D, G : R → R satisfying the relations2D(xn) = D(xn−1)φ(x) + ϕ(xn−1)G(x) + G(x)φ(xn−1) + ϕ(x)G(xn−1),2G(xn) = G(xn−1)φ(x) + ϕ(xn−1)D(x) + D(x)φ(xn−1) + ϕ(x)D(xn−1),for all x ∈ R and some fixed integer n ≥ 2, then D and G are (φ, ϕ)--derivations under some torsion restriction. Finally, we apply these purelyring theoretic results to semi-simple Banach algebras.