Generalized Jordan derivations on semiprime rings and its applications in range inclusion problems. (Q644586)

The paper considers various generalizations of Jordan derivations. For instance, a generalized Jordan triple derivation of a ring \(R\) is defined as an additive map \(d\colon R\to R\) satisfying \(d(xyx)=d(x)yx+xf(y)x+xyf(x)\), where \(f\) is a Jordan triple derivation. It is shown that on \(2\)-torsion free semiprime rings such maps are necessarily generalized derivations. Some related extensions of higher derivations are also studied, and applications to Banach algebras are given.