I want to prove that there exists $f:[0,1] \to [0,1]$ such that $f(0)=0$, $$ \frac{d w(y-f(y))}{d y} = g(y) \frac{d v(f(y))}{d y}, \forall y \in [0,1], $$ where $w:[0,1] \to [0,1]$ and $v:[0,1] \to [0,1]$ are continuous strictly increasing functions with $w(0)=v(0)=0$ and $v(1)=w(1)=1$, and $g:[0,1] \to (0,\infty)$ is a continuous function.

If it simplifies the problem, I am particularly interested in the case where $$ w(x)=\frac{x^a}{[x^a+(1-x)^a]^{1/a}}, v(x)=\frac{x^b}{[x^b+(1-x)^b]^{1/b}}, $$ for some $a,b \in (0,1]$.

For example, if $a=b=1$, then $v(x)=w(x)=x$, and the solution is $f(y) = \int_0^y \frac{1}{1+g(x)} dx$.

I don't know how to approach this problem with nonlinear $w$ and $v$ functions. I only need to prove that $f$ exists (and if there is a way to compute it numerically, it would be even better).