I have a system of two nonlinear equations:
$$\theta_{LB}=\frac{1}{k_{LB}} \left[\frac{k_{LG}}{(1-\delta_L)\Delta q(\theta_{LG})} -(1-\beta_L)(y_{LG}-z_L)-(1-\beta_L)k_{LG}\theta_{LG} \right]$$
$$\theta_{LG}=\frac{1}{k_{LG}} \left[ \frac{k_{LB}}{(1-\delta_L)\Delta q(\theta_{LB})} -(1-\beta_L)(y_{LB}-z_L)-(1-\beta_L)k_{LB}\theta_{LB} \right]$$
where $q(\theta_{Li})=\frac{\theta_{Li}}{(\theta_{Li}^\eta+1)^{\frac{1}{\eta}}}$ with $\eta>0$. The two unknowns are $\theta_{LB}$, $\theta_{LG}$ which are restricted to be greater than $0.$ All other names denote parameters. Some properties: $\Delta \in (0,1)$; $\delta_L \in (0,1)$; $\beta_L \in (0,1)$; $0 < kLB<kLG$; $zL<yLB<yLG.$ I can show that $\theta_{LB}$ is strictly decreasing in $\theta_{LG}$ and is concave up (i.e. the second derivative of $\theta_{LB}$ w.r.t. $\theta_{LG}$ is positive) and $\theta_{LG}$ is decreasing in $\theta_{LB}$ and is concave up.
I know there can be at most two solutions. But it would be great to find parameters values for which there is 1, 0, or 2. I have found numerical examples, but I am looking for a more general solution. Basically wondering if there is a theorem I am unaware of that will help me out here.
Any guidance will be great!