on generalized left (alpha, beta)-derivations in rings

let $r$ be a 2-torsion free ring and $u$ 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 $xin r$. in this paper it is established that if $r$ admits an additive mapping $g : rlongrightarrow r$ satisfying $g(u^2)=alpha(u)g(u)+alpha(u)delta(u)$ for all $uin u$ 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 $u, vin u$. 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 $r$.