I'm looking for a continuous solution to the functional equation
f(2x)=N−2xf(x)2
where N is a constant natural number and x∈R is nonnegative. I don't have much experience with functional equations so I haven't tried anything yet. If it helps I'm mostly interested near x=0. Any ideas?
Answer
This is a simple study of f(x) as x→0.
Let N>0.
First case, if f(0)=0, then
limx→02xf(x)2=N⟹f(x)=√2xN+o(√x)
Second case, if f(0)=N,
Assuming f(x)=N+ax+o(x), then
\begin{align} f(x)^2 = \frac{2x}{N-f(2x)}&\implies N^2+o(1)=-\frac{2x}{2ax+o(x)}\\ &\implies a=-N^{-2} \end{align}Assuming f(x)=N-N^{-2}x+bx^2+o(x^2), then
\begin{align} f(x)^2 = \frac{2x}{N-f(2x)}&\implies N^2-2N^{-1}x+o(x)=\frac{1}{N^{-2}-2bx+o(x)}\\ &\implies b=-N^{-5} \end{align}And so on...
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