Sunday, 13 July 2014

Solve functional equation f(2x)=Nfrac2xf(x)2




I'm looking for a continuous solution to the functional equation



f(2x)=N2xf(x)2



where N is a constant natural number and xR 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 x0.



Let N>0.




First case, if f(0)=0, then
limx02xf(x)2=Nf(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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