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I have a functional equation that is valid for $0\leq a\leq b \leq c$ of the form $$f(b-a)g(c-a) = g(c)\left[f(b) - g(b)\frac{f(a)}{g(a)}\right],$$ with the additional information that $f(x)g(x) > 0, \forall x \in \mathbb{R}$ and $\frac{f(x)}{g(x)}$ is monotonically increasing. I was wondering if, other than the trivial solution $g(x) = k$ and $f(x) = mx$, there are any other functions $g(x)$ that yield a functional equation that is solvable.

Thanks for any help :)

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