We study an affine equation $\phi(X)=X$ but there is no closed form for "the" solution.
Here $\phi:X\rightarrow cAXA^T-diag(cAXA^T)+I$ ; let $E_i$ be the matrix with all entries $=0$ except the $(i,i)^{th}$ that is $1$. If we stack the square matrices row by row, then $\sum_i E_i\bigotimes E_i:X\rightarrow diag(X)$ (cf. http://en.wikipedia.org/wiki/Kronecker_product). Thus $\phi=cA\bigotimes A-c(\sum_i E_i\bigotimes E_i)(A\bigotimes A)+1=cA\bigotimes A-c\sum_i(E_iA)\bigotimes(E_iA)+1=cA\bigotimes A-c\sum_i A_i\bigotimes A_i+1$ where $1:X\rightarrow I$ and $A_i$ is the matrix with all rows $=0$ except the $i^{th}$ that is the $i^{th}$ row of $A$. There exist numerical methods that solve this type of equation.
EDIT: for $l^2$ norm, $||\sum_i E_i\bigotimes E_i)||=1$ and $||A\bigotimes A||\leq ||A||^2$. Therefore $||D\phi||\leq 2c||A||^2$ ; if $2c||A||^2\leq k<1$, then we can use the Banach fixed point theorem.