# How many solutions does this recursive system have?

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?

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