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
    f(x)2=2xNf(2x)N2+o(1)=2x2ax+o(x)a=N2


  • Assuming f(x)=NN2x+bx2+o(x2), then
    f(x)2=2xNf(2x)N22N1x+o(x)=1N22bx+o(x)b=N5


  • And so on...



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