during an algorithmn, I have to solve an equation of the form $$AXD-XBD=C$$ with $A\in\mathbb{R}^{n\times n}$,$X\in\mathbb{R}^{n\times m}$,$B\in\mathbb{R}^{m\times m}$,$D\in\mathbb{R}^{m\times p}$ and $C\in\mathbb{R}^{n \times p}$.
The unknown $X$ has to be solved but I don't know how, because I just know sylvester equations. Can someone help?
If the null space of $D$ is not contained in the null space of $C$, there is obviously no solution.
Now suppose the null space of $D$ is contained in the null space of $C$. Let $D^+$ be the Moore-Penrose pseudoinverse of $D$. Solve the Sylvester equation $AX - XB = C D^+$. Then $AXD - XBD = C D^+ D = C $ (since the range of $I - D^+ D$ is the null space of $D$).