I have a method that searches for the fixed point of the equation
$$x = A(x)x$$
where x is a vector and A(x) is a square matrix. I am able to convert this into a fixed point method
$$x_{n+1} = A(x_{n})x_{n}$$
and in practice, this converges. I am looking for the general properties of the matrix A(x) that allow me to guarantee convergence and say something more rigorous. The matrix A(x) asymptotically approaches a stationary matrix A with stationary distribution x*. Are there examples of a sequence of matrices approaching a stationary matrix?
Let me know if that makes any sense.