General question: How does one identify if a system of recursive equations (a slightly more complicated system than the standard $Ax=b$) has a unique solution?
Specific example: My system of two equations in the variables $x$ and $y$:
$x_t = x_{t-1} + a_t + b_t y_t$
$y_t = min \{1, \frac{x_{t-1} + a_t}{b_t} \}$
and a boundary condition $x_0 = 0$.
PS: I already have a solution for this system and think it is unique. But I'd like to know a formal way to prove this is indeed unique.