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Define the system by: $$ \lim_{n\to \infty} f_n(x) = e^{x} \\ f_n(x) = f(xf_{n-1}(\frac{x}{n})) \\ f(x) = f_0(x) $$ How many solutions $f$ are there?

The solution set is nonempty. Take $f(x) = 1 + x$.

Is this solution unique?

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    $\begingroup$ $$e^x=f_0(xe^x)$$.Now find solutions to this $\endgroup$ – Qwerty Dec 14 '16 at 5:17

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