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Let 풜 be a standard operator algebra containing the identity operator I on an infinite dimensional complex Hilbert space ℋ which is closed under adjoint operation. Suppose that ϕ : 풜 → 풜 is the second nonlinear mixed Lie triple derivation. Then ϕ is an additive ∗ -derivation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the commutativity of a prime ring ℛ with anti-automorphism ψ and generalized derivation (ℱ , ξ) satisfying certain algebraic identities. Furthermore, we describe the structure of any additive map which is ψ-centralizing on a 2-torsion free prime ring with anti-automorphism ψ. We also provide examples to show that various conditions imposed on the hypotheses of our results are essential. [ABSTRACT FROM AUTHOR]