Suppose $f_1$ and $f_2$ are two probability density functions on support $[0,1]$ (i.e. $f_1(x)=f_2(x)=0$ for any $x\not\in[0,1]$). Let $\varphi(x)$ denote a known probability density function on support $[0,1]$ and let $k\in(0,1)$ be a known constant. If
$$ \int_0^{z(x)} c f_1(x+(1-k)t)f_2(x-kt)\,\mathrm{d}t = \varphi(x), $$
where the upper limit $z(x)=\min\left(\frac{1-x}{1-k},\frac{x}{k}\right)$ and $c$ is a constant that makes $\int_0^1 \varphi(x)\,\mathrm{d}x=1$,
- Is the pair of density functions $(f_1,f_2)$ satisfying the above integral equation unique?
- If yes, how to prove it? If no, is there any counterexample?
Thank you.