On some equations in prime rings

The main purpose of this paper is to prove the following result. Let R be a prime ring of characteristic different from two and let T : R -> R be an additive mapping satisfying the relation T(x(3)) = T(x)x(2) - xT(x)x + x(2) T(x) for all x is an element of R. In this case T is of the form 4T(x) = qx + xq, where q is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.