convergence of vectors Let $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$.
Consider sequences of vectors 
$$x_{i+1} = F(x_i, y_i)$$
$$y_{i+1} = G(y_i, x_i)$$
I have two related questions.
What properties should functions $F$ and $G$ have so that sequences
$x_1, x_2, \dots$, $y_1, y_2, \dots$ converges?
And given some functions $F$ and $G$, how to determine whether the sequences converge or not.
Any thoughts or links to literature are welcome!

If the functions $F$ and $G$ are continuous they should behave in a way that the function $H = (F,G)$ has a fixed point.
 This is because if a limit $(x,y)$ exists then by taking limits on the equations we arrive to the functional equation 
$$(x,y) = (F(x,y), G(x,y)) = H(x,y)$$. 
To see if there is a fixed point you can use Banach fixed point theorem which guarantees the existence of a fixed point if $H$ is a contraction. You can check also Brower's fixed point theorem.
