I am trying to prove existence and uniqueness for the following system. Let $\theta$ be a constant in $(0,1)$, let $i,l=1,...,N$, let $a_{l}$ and $b_{i,l}$ be some constants with $a_l \in (0,1)$ and $b_{i,l}>0$ for all $i,l$. Consider the following system of equations in unknowns $z_{1},...,z_{N}$ and $p_{1},...,p_{N}$:$$\sum_{l}\left(\frac{b_{i,l}}{\sum_{l}b_{i,l}}\right) \left[ \frac{1}{1+p_{l}z_{i}} - \frac{1}{1+z_{i}^{\theta}} \right] = 0,i=1,...,N$$ $$\sum_{i}\left(\frac{b_{i,l}}{\sum_{i}b_{i,l}}\right) \left[ \frac{1}{1+p_{l}z_{i}} - a_{l} \right] = 0,l=1,...,N$$
I am focusing on real and positive solutions.
If $\theta < 1$ then one can proceed as follows. First, construct a mapping $ p \rightarrow p'$ as follows. First take $p$ and find the unique positive solution for $z$ from the first set of equations, leading to functions $z_i(p)$. These functions are well defined because at a (strictly positive) solution the term in square parenthesis is necessarily decreasing. Second, given $z$ then solve for $p$ from the second equation, which yields functions $p_l(z)$, again well defined because the term in square parenthesis is decreasing in $p_l$). Third, construct a mapping from $p$ to $p'$ by using $p_l'(z_1(p),...,z_N(p))$ for $l = 1,...,N$. One can show that if $\theta<0$ then this is a contraction mapping, and it stays in some box, implying that there is a unique solution and one can find it using this iteration.
The previous argument fails when $\theta\in(0,1)$. In this case, my current idea is to consider the system in $p$ given by $f_l(p) = 0, l=1,...,N$ with $$f_l(p) \equiv \sum_{i}\left(\frac{b_{i,l}}{\sum_{i}b_{i,l}}\right) \left[ \frac{1}{1+p_{l}z_{i}(p)} - a_{l} \right] = 0$$ If the Jacobian of $f$ is a P-matrix for all $p$ then one can use the result in Gale and Nikaido (1965, here) to show uniqueness, although not necessarily existence. But I have found counterexamples to the P-matrix property of the Jacobian of $f$.
Any suggestions would be highly appreciated.