On some additive mappings on division rings

Let $D$ be a division ring such that char$(D) neq 2$  and $alpha,beta:Drightarrow D$ be automorphisms of $D$. The main purpose of this paper is to characterizes additive maps                              $f$ and $g$ satisfying the identity $f(x)alpha(x^{-1}) + beta(x)g(x^{-1}) = 0$ for all $0 neq xin D.$ As an application, we describe the structure of an additive map $f$ satisfying the identity                            $f(x)alpha(y)+beta(x)f(y) =l$ for all $x,yin D$ such that $xy=a,$ where $l,ain D$ and $a$ is nonzero. With this, many known results can be either generalized or deduced. In particular, we generalized the results proved incite{C1} and cite{C2}, respectively.