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I want to solve a problem for an increasing function $g(x)$, for $x \in [0,1]$ and with $g(0) = 0$ and $g(1) = 1$.
The solution will be solution to the following equation $\forall x$, $f_1(x) = f_2(g(x))$.
(with $f_1$ and $f_2$ known)

However, $f_2$ may not be invertible (it is piecewise monotone though). Thus for some $x$, you have multiple solution $g(x)$, that we'll denote $(g^1(x), g^2(x), ..., g^k(x))$ (given the structure of the problem, you always have a countable number of solutions).

However, overall, there is a unique increasing solution $g^*()$ of $[0,1] \rightarrow [0,1]$.
My problem is, I don't know how to impose the monotonicity constraint when solving a system, such that I should be able to pin down the unique solution 'directly' (I don't know if that's even possible). Or in other words, I don't know tricks to write the system with monotonicity embedded, solving for the complete function $g()$, instead of solving pointwise $x$ by $x$.

I was trying to build a contraction mapping operator (maybe it's not the way to go), but I need to embed the monotonicity constraint into the contraction operator and I don't know how to do that. (Because taken at a given $x$, our system can yield several solution and thus no contraction mapping is feasible).

PS: Just to be clear, in this particular case I know how to pin down $g()$ entirely exploiting piecewise monotonicity (and thus building piecewise invertibility). My question is really about tricks to include the monotonicity into a system (as an additional equation? constraint?), and if possible use contraction mapping with it.

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