Matrix equation involving difference arguments

Question

Suppose we have a matrix equation for a $2\times 2$ matrix $R(x_n)$ that depends on a complex parameter $x_n\in\mathbb{C}$, with $n\in\mathbb{Z}$ being a discrete index. The equation to solve is of the form: \begin{eqnarray} R(x_n)=R_0(x_n) + R_0(x_n)\sum_k L(y_k)*R(x_n-y_k), \end{eqnarray} where $*$ denotes the element-wise product between components of the matrices $L(y_k)$ and $R(x_n-y_k)$; $L(y_k)$ is a known $2\times 2$ matrix dependent on another complex parameter $y_k\in\mathbb{C}$,$k\in\mathbb{Z}$. Are there any general methods to solve such matrix equations? I could think this goes in the direction of matrix convolution or similar, but I have no clue on how to continue solving for the system because all components are eventually coupled in a non-local way. Any good references on how to solve these systems?