I am working with a system of equations that has an additive-separable structure with certain "symmetry". In particular, consider the $3X3$ system (although in general I am looking for solutions that may apply to $nXn$ case). My system looks the following way:
$$K f_1(x_1)=a_0(1-x_2)(1-x_3)+a_1(1-x_2)x_3+a_2(1-x_3)x_2+a_3x_3x_2$$ $$K f_2(x_2)=a_4(1-x_1)(1-x_3)+a_5(1-x_1)x_3+a_6(1-x_3)x_1+a_7x_3x_1$$ $$K f_3(x_3)=a_8(1-x_2)(1-x_1)+a_9(1-x_2)x_1+a_{10}(1-x_1)x_2+a_{11}x_1x_2$$
Where $K>0$ is a constant and $f_1,f_2,f_3$ are continuous and increasing in their arguments, and $x_1,x_2,x_3\in [0,1]$.
I am looking for suggestions for solving this system. I can clearly establish existence recurring to the Implicit Function Theorem. But I was wondering if the separable structure could allow me to use a useful trick in order to get explicit solutions for $x_1$, $x_2$, and $x_3$.
I don't mind simplifying assumptions, for instance letting $f_i(x_i)=c_i x_i$ for $i=1,2,3$.
More information:
The parameters $a_{k=0,\dots 11}$ depend on some primitives of my problem. In fact, it is the case that $a_0=a_4=a_8$ and so a candidate trick is to consider $(1-x_1)Kf_1(x_1)-(1-x_2)Kf_2(x_2)$ and other variations.
UPDATE:
I also managed to impose some structure in my coefficients and obtain a system that looks like this:
$$K c_1 x_1=(1-x_2)(1-x_3)+\beta_1 \gamma_1(1-x_2)x_3+\beta_1 \gamma_2(1-x_3)x_2+\beta_1\gamma_3x_3x_2$$ $$K c_2 x_2=(1-x_1)(1-x_3)+\beta_2 \gamma_4(1-x_1)x_3+\beta_2(1-\gamma_2)(1-x_3)x_1+\beta_2\gamma_5x_3x_1$$ $$K c_3 x_3=(1-x_2)(1-x_1)+\beta_3(1-\gamma_1)(1-x_2)x_1+\beta_3(1-\gamma_4)(1-x_1)x_2+\beta_3 (1-\gamma_3-\gamma_5)x_1x_2$$
For $c_i>0$, $\gamma_i\in[0,1]$ and $\beta_i \in (0,1]$ for all $i=1,\dots$. An interesting particular case may be $c_i=c>0$ for $i=1,2,3$, but I would be interested in its generalization.