On certain functional equations related to Jordan triple \((\theta,\varphi)\)-derivations on semiprime rings. (Q629864)

Recently, more and more researchers are interested in describing the form of special additive mappings on associative rings or matrix algebras. The authors of this article are to describe the form of the so-called ``function'' related to Jordan triple \((\theta,\varphi)\)-derivations on semiprime rings. Let \(R\) be a \(2\)-torsion free semiprime ring with symmetric Martindale ring of quotients \(Q_s\) and let \(\theta\) and \(\varphi\) be automorphisms of \(R\). Suppose \(T\colon R\to R\) is an additive mapping satisfying the relation \(T(xyx)=T(x)\theta(y)\theta(x)-\varphi(x)T(y)\theta(x)+\varphi(x)\varphi(y)T(x)\), for all pairs \(x,y\in R\). In this case \(T\) is of the form \(2T(x)=q\theta(x)+\varphi(x)q\), for all \(x\in R\) and some fixed element \(q\in Q_s\).