So I am considering this problem:
$f(x)$ is a probability density function for $x\in[0,1]$. $$g(x)+h(f(x))=g(x')+h(f(x')), \forall x,x'\in[0,1]$$ where $g(\cdot)$ and $h(\cdot)$ are continuous and strictly increasing. I am trying to determine what additional assumptions I have to make about $g$ and $h$ to guarantee existance of a solution $f(x)$.
I tried to solve pointwise for $f(x)$ by $$g(x)+h(f(x))=C$$ and then use $$\int_0^1f(x)dx=1$$ to determine C. But I realize the pointwise equation may not have a solution itself and making assumptions based on the pointwise equation may be putting too many restrictions to the original problem than necessary. I would really appreciate suggestions on how to consider this problem or what literature I should look into. Also please let me know if I have not made the problem clear enough. Thanks!
Since $g$ and $h$ are both strictly increasing, each one has a well-defined inverse on its range, so as long as the domains work out, we can express $$ f(x) = h^{-1}(C - g(x)) $$ where $C = h(1) + g(1)$. If the range of $g$ is contained in $[a,b]$, say, then this expression for $f$ is well-defined as long as, say, the range of $h$ contains $[C-b,C-a]$. You then need $$ h^{-1}(C - g(x)) \geq 0 $$ for all $x \in [0,1]$, and $$ \int_0^1 h^{-1}(C - g(x)) \, dx = 1 $$