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Consider the functional equation

$$ g\left(a\right) = \int_0^1 \frac{e^{c(a,h)+f(h)}}{1+e^{c(a,h)+f(h)}}dh $$

and this holds for all $a$. $g(a)$ and $c(a,h)$ are known functions on a continuous support. Assume that $\frac{\partial g(a)}{\partial a}\not=0$, $\frac{\partial c(a,h)}{\partial a}\not=0$, and $\frac{\partial c(a,h)}{\partial h}\not=0$.

How can I show that there is one unique $f(h)$ that solves this equation? The ultimate goal is to figure out under what situation will $f(h)$ be uniquely pinned down. So any additional assumptions on $g$ and $c$ that's needed can be added.

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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Feb 3, 2023 at 20:08
  • $\begingroup$ This is not true in this generality (try $c,g$ constant). $\endgroup$
    – Christian Remling
    Commented Feb 3, 2023 at 20:38
  • $\begingroup$ Thanks @ChristianRemling! This is a good catch. I added some assumptions on $c$ and $g$ that ensures variations. Any additional assumptions can be added if they are necessary to deliver uniqueness of $f$. $\endgroup$
    – DDCM Lover
    Commented Feb 3, 2023 at 21:17

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