I am trying to prove the uniqueness of the solution $(x,y)$ to the following system of non-linear equations, under certain conditions, if required
\begin{align} &w(1-\gamma x) y = \mu x \tag{1}\\ & x \left( 1 + \mu (1- a x(1-e^{-\gamma y})\right) + y = 1 \tag{2} \end{align}
where $0<x,y<1$, $w>0,\gamma>0,\mu>0$, and $0<a<1$.
(The uniqueness can be seen from some numerical examples.)
I have tried two approaches.
Approach 1: I substitute $y = \frac{\mu x}{w(1-\gamma x)}$ into equation (2) and obtain \begin{align} x = \frac{1-y}{1 + \mu (1- a x(1-e^{-\gamma y}))} \triangleq f_y(x). \end{align} The LHS $x$ is increasing with $x$. If the RHS $f_y(x)$ is decreasing with $x$. Then, under some boundary conditions, it can be shown that $x\in(0,1)$ exists. However, the problem is that, the function $f_y(x)$ is too complicated due to the term $e^{-\gamma y}$. I could not obtain a nice expression of $f'$ and determine its sign. So, I turned to the second approach.
Approach 2: For a given $x$, I first show that for equation (2), there exists a unique solution $y$, under the conditions $x(1+\mu)<1$ and $\mu a \gamma x^2<e^{\gamma}$. Then, I try to use contraction mapping to show the uniqueness. Specifically, for a given $x$, I first obtain the unique solution $y$ by equation (2) and then, substitute this $y$ into equation (1) to obtain a unique $x$.
Define $T_1: x \rightarrow y$ according to equation (2) and $T_2: y \rightarrow x$ according to equation (1).
One question is that: if I can show $T(x)\triangleq T_2(T_1(x))$ is a contraction mapping, can I say the uniqueness of $(x,y)$ is proven?
Supposing this method is correct, I derive the derivative of $T$ over $x$ as follows:
\begin{align}
|\frac{\partial T}{\partial x}| = |\frac{\partial x} {\partial y}| |\frac{\partial y} {\partial x}|
=|\frac{1+ \mu (1-2a(1-e^{-\gamma y})x)}{x^2\mu a \gamma e^{-\gamma y} - 1} | |\frac{w\mu}{(\mu + w\gamma y)^2}|
\end{align}
If $|\frac{\partial T}{\partial y}|<1$, then can we complete the proof?
I was trying to find an upper bound of $|\frac{\partial T}{\partial y}|$ and show it is smaller than 1.
However, as the expression of $|\frac{\partial T}{\partial y}|$ is too complicated, so far, it does not work.
Sorry for putting too many details here. (This problem comes from my current research on showing the accuracy of a mean-field approximation for a continuous-time markov chain.) I just wanted to make my problem clearer. I would much appreciate that you could comment on these two approaches. Moreover, are there any other methods for showing the uniqueness? I would love to try them as well.
Thanks!
A partial approach.
Assuming that the equations are$$\begin{align} &w(1-\gamma x) y = \mu x \tag{1}\\ & x \left( 1 + \mu (1- a x(1-e^{-\gamma y})\color{red}{)}\right) + y = 1 \tag{2} \end{align}$$
from $(1)$ extract $x$ as a function of $y$ and rearrange $(2)$ as $$e^{\gamma y}=\frac {A y^2} {B+C y+D y^2+E y^3}$$ $$A=a \mu w^2 \qquad B=-\mu^2 \qquad C=\mu (\mu +w (\mu +1-2 \gamma ))$$ $$D=w (w (\gamma (\mu +1-\gamma )-a \mu )+2 \gamma \mu )\qquad E=w^2\gamma^2$$ If there is only one solution to the cubic greater than $1$ we are safe.
I tried to work the cubic but I gave up.