I have a two-equation system of implicit functions, \begin{align} 0 & = f(x,y) = a y^\alpha \left(\bar{x} - x\right)^\gamma - x \\ 0 & = g(x,y) = a x^\beta \left(\bar{y} - y\right)^\gamma - y \end{align} with $x \in \left[0,\bar{x}\right]$, $y \in \left[0,\bar{y}\right]$, $a>0$, and $\alpha,\beta, \gamma > 0$. The parameters $\bar{x}$, $\bar{y}$, $a$, $\alpha$, $\beta$, and $\gamma$ all are fixed beforehand. The equations satisfy the contraction property (because $f(0,y) > 0$, $f(\bar{x},y) < 0$, and $f_x<0$ and $g(x,0)>0$, $g(x,\bar{y})<0$, and $g_y<0$), so I know that unique solutions for the endogenous variables, $x$ and $y$, exist.
I am trying to derive some properties of $x$ and $y$. Is there a way to determine an upper bound of $x+y$ as a function of the parameters $\bar{x}$, $\bar{y}$, $a$, $\alpha$, $\beta$, and $\gamma$?