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
f(x)2=2xN−f(2x)⟹N2+o(1)=−2x2ax+o(x)⟹a=−N−2Assuming f(x)=N−N−2x+bx2+o(x2), then
f(x)2=2xN−f(2x)⟹N2−2N−1x+o(x)=1N−2−2bx+o(x)⟹b=−N−5And so on...
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