Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as $$f_i(\mathbf{x})=\tau_i\prod^n_{k=1}(1-a_{ik}x_k)$$ in which $\tau_i\in(0,1)$, $x_i\in[0,\tau_i]$, and $a_{ik}$ is either $0$ or $1$ for all $i,k\in\{1,2,\cdots,n\}$.
Now, my question is that how to prove $F$ has a unique fixed point $\mathbf{x}^\ast$ such that for each $x^\ast_i\in[0,\tau_i]$.