Consider the vector-valued map $f$ with components $$ f_{i}(\mathbf{r}) = p_i + \sum_{\ell \neq i} p_\ell r_{-\ell} r_i $$
where $i = \pm 1, \pm 2 \dots \pm N$ for some $N \geq 2$. We also have $p_i \in (0, 1)$ and $\sum_{i} p_{i} = 1$.
Fact 1: The map $f$ has a unique fixed point in $[0, 1)^{2N}$.
Note that $f_i(\mathbf{1}) = 1$, which is why the end point is excluded. Fact 1 can be proved by solving $f_i(\mathbf{r}) = r_i$ directly (see here) and showing that each $r_i$ will be in $[0, 1)$. My question is
Can Fact 1 be proved via some kind of fixed point theorem, or any means which do not require explicitly finding the fixed point?
I've tried to use the contraction mapping theorem, but taking the domain $[0, t]^{2N}$ for some $t \in (0, 1)$ didn't work since it's possible for $f$ to map points to $(t, 1)^{2N}$. Eventually, iterates of $f$ will not do this, but computing $f^{(5)}(\mathbf{r})$ doesn't sound promising. It's also not hard to numerically find cases where $|f_i(\mathbf{x}) - f_{i}(\mathbf{y})| > |\mathbf{x} - \mathbf{y}|$ in the Euclidean and maximum norm which kills the contraction condition.