Identities involving additive maps on division rings

Let $g$ be an additive map on a division ring $D$. In this paper, we study the functional identity $G_1(y)g(y)G_2(y)=H(y)$, where $G_1(Y),G_2(Y)$, $H(Y)$ are generalized polynomials in $D_G[Y]$ such that both $G_1(Y)$ and $G_2(Y)$ are non-zero. By application of this result and its implications, we prove that if $D$ is a non-commutative division ring with $\mathrm{char}(D)\mathrm{\ne }2$, then the only possible solution of additive maps $g_1,g_2:D\to D$ satisfying the identity $g_1(y)y^{-m}+y^n{g}_2(y^{-1})=0$ is $g_1=g_2=0$, where $m$ and $n$ are positive integers with $(m,n)\mathrm{\ne }(1,1)$.