Abstract
Let R be a prime ring, F be a generalized derivation associated with a derivation d of R and m, n be the fixed positive integers. In this paper we study the case when one of the following holds: (i) F (x) omicron(m) F (y) = (x omicron y)(n), (ii) F (x) omicron(m) d (y) = d (x omicron y)(n) for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on non-commutative Banach algebras.