Suppose you have a discrete system, whose evolution is governed by the following equations:
$\mathbf{x}[k+1] = f_1(\mathbf{F}[k], \mathbf{x}[k])$
$\mathbf{F}[k+1] = f_2(\mathbf{F}[k], \mathbf{x}[k+1], \mathbf{x}[k])$
where $\mathbf{x}$ is a vector and $\mathbf{F}$ is a matrix. What is the best way to show they both converge to some fixed point? Shall I look at the difference between two consecutive steps, $k+1$ and $k$, or is there a more effective way to solve it? For simplicity, suppose $f_1$ and $f_2$ are linear (and/or affine) functions.