The main purpose of this research is to characterize generalized (left)
derivations and Jordan (*,*)-derivations on Banach algebras and rings using
some functional identities. Let A be a unital semiprime Banach algebra and let
F,G : A ? A be linear mappings satisfying F(x) =-x2G(x-1) for all x ?
Inv(A), where Inv(A) denotes the set of all invertible elements of A. Then
both F and G are generalized derivations on A. Another result in this regard
is as follows. Let A be a unital semiprime algebra and let n > 1 be an
integer. Let f,g : A ? A be linear mappings satisfying f (an) = nan-1g(a)
= ng(a)an-1 for all a ? A. If g(e) ? Z(A), then f and g are generalized
derivations associated with the same derivation on A. In particular, if A is
a unital semisimple Banach algebra, then both f and 1 are continuous linear
mappings. Moreover, we define a (*,*)-ring and a Jordan (*,*)-derivation. A characterization of Jordan (*,*)-derivations is presented as follows. Let R be an n!-torsion free (*,*)-ring, let n > 1 be an integer and let d : R ? R be an additive mapping satisfying d(an) = ?nj =1 a*n-jd(a)a* j-1 for all a ? R. Then d is a Jordan (*,*)-derivation on R. Some other functional identities are also investigated.