A generalization of the probability that the commutator of two group elements is equal to a given element

Let $R$ be a 2-torsion free ring and let $U$ be a square closed Lie ideal of $R$‎. ‎Suppose that $\alpha‎, ‎\beta$ are automorphisms of $R$‎. ‎An additive mapping $\delta‎: ‎R \longrightarrow R$ is said to be a Jordan left $(\alpha,\beta)$-derivation of $R$ if $\delta(x^2)=\alpha(x)\delta(x)+\beta(x)\delta(x)$ holds for all $x\in R$‎. ‎In this paper it is established that if $R$ admits an additive mapping $G‎ : ‎R\longrightarrow R$ satisfying $G(u^2)=\alpha(u)G(u)+\alpha(u)\delta(u)$ for all $u\in U$ and a Jordan left $(\alpha,\alpha)$-derivation $\delta$; and $U$ has a commutator which is not a left zero divisor‎, ‎then $G(uv)=\alpha(u)G(v)+\alpha(v)\delta(u)$ for all $u‎, ‎v\in U$‎. ‎Finally‎, ‎in the case of prime ring $R$ it is proved that if $G‎: ‎R \longrightarrow R$ is an additive mapping satisfying $G(xy)=\alpha(x)G(y)+\beta(y)\delta(x)$ for all $x,y \in R $ and a left $(\alpha‎, ‎\beta)$-derivation $\delta$ of $R$ such that $G$ also acts as a homomorphism or as an \linebreak anti-homomorphism on a nonzero ideal $I$ of $R$‎, ‎then either $R$ is commutative or $\delta=0$‎ ~‎on $R$‎.