I changed my question slightly.
(Tarski Fixed Point Theorem). Let $X=\prod^{N}_{i=1} X_{i}$ where each $X_{i}$ is a compact interval of $\mathbb{R}$. Suppose $\phi : X \rightarrow X$ is an increasing function. Then $\phi$ has a fixed point.
I'm not very sure how I should interpret a function being "increasing" when $X\subset \mathbb{R}^{2}$.
Does that mean when we put a two-dimensional vector $x_{1}$ that is elementwise bigger than $x_{2}$, $\phi(x_{1})$ is also elementwise bigger than $\phi(x_{2})$?