On certain functional equations related to Jordan triple (q, f){(theta, phi)}-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let q{theta} and f{phi} be automorphisms of R. Suppose T:R? R{T:Rrightarrow R} is an additive mapping satisfying the relation T(xyx)=T(x)q(y)q(x)-f(x)T(y)q(x)+f(x)f(y)T(x){T(xyx)=T(x)theta (y)theta (x)-phi (x)T(y)theta (x)+phi (x)phi (y)T(x)}, for all pairs x,y ? R{x,yin R}. In this case T is of the form 2T(x)=qq(x)+f(x)q{2T(x)=qtheta (x)+phi (x)q}, for all x ? R{xin R} and some fixed element q ? Q_s{qin Q_{s}}.