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!