Functional identities involving additive maps on division rings

Abstract: Let $g$ be an additive map on division ring $D$ and $G_{1}(Y), G_{2}(Y) \neq 0$, $H(Y)$ are generalized polynomials in $D {Y}$. In this paper, we study the functional identity $G_{1}(y)g(y)G_{2}(y) = H(y)$. By application of the result and its implications, we prove that if $D$ is a non-commutative division ring with characteristic different from $2$, then the only possible solution of additive maps $g_{1},g_{2}: D \rightarrow D$ satisfying the identity $g_{1}(y)y^{-m} + y^{n}g_{2}(y^{-1})= 0$ with $(m,n) \neq (1,1)$ are positive integers is $ g_{1} = g_{2} = 0$.