On maps characterized by action on equal products

Let D be a division ring with characteristic different from 2. In this paper we will describe an additive map f satisfying the identity f(x)y+xf(y)=l for every x,y∈D such that xy=a, where l,a∈D and a is nonzero. Additionally, for nonzero m,k∈D, we will describe the additive bijection f satisfying the identity f(x)f(y)=m for every x,y∈D such that xy=k; this description is the solution to a problem posed by Chebotar, Ke, Lee, and Shiao in 2005.