On Lie ideals with generalized (alpha, alpha)-derivations in prime rings

Let R be a prime ring and a an automorphism on R. An additive mapping F on R is said to be a generalized (alpha,alpha)-derivation on R if there exists an (alpha,alpha)-derivation d on R such that F(xy) = F(x)alpha(y) + a(x) d(y) holds for all x, y is an element of R. In this paper our main objective is to study the following identities: (i) G(xy) +/- F(x) F(y) is an element of Z(R); (ii) G(xy) +/- F(x) F(y) +/- a(yx) = 0; (iii) G(xy) +/- F(x) F(y) +/- a(xy) is an element of Z(R); (iv) G(xy) +/- F(x) F(y) +/- a([ x, y]) = 0; (v) G(xy) +/- F(x) F(y) +/- alpha(x o y) = 0; for all x, y in some suitable subset of R, where G and F are two generalized (alpha,alpha)-derivations on R