A result concerning derivations in prime rings

A classical result of Herstein asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. It is our aim in this paper to prove the following result, which is in the spirit of Herstein\u27s theorem. Let R be a prime ring with char(R)=0 or 4 < char(R), and let D:R → R be an additive mapping satisfying either the relation D(x3)=D(x2)x+x2D(x) or the relation D(x3)=D(x)x2+xD(x2) for all x R. In both cases D is a derivation