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Sunday, 8 April 2007

ca.analysis and odes - Are there any nonlinear solutions to f(x+1)f(x)=f(x)?

Yes, there exist nonlinear solutions.



Multiplying by ex+1 and setting g(x):=exf(x) transforms the question into finding a solution to g(x+1)=eg(x) not of the form ex(ax+b).



Start with any Cinfty function on mathbbR whose Taylor series centered at 0 and 1 are identically 0, but which is nonzero somewhere inside (0,1). Restrict it to [0,1]. Let g(x) on [0,1] be this. Using g(x+1):=eg(x) for xin[0,1] extends g(x) to a Cinfty function g(x) on [0,2], which can then be extended to [0,3], and so on. In the other direction, use g(x):=intx0e1g(t+1)dt to define g(x) for xin[1,0], and then for xin[2,1], and so on. These piece together to give a Cinfty function g(x) on all of mathbbR. The corresponding f(x) satisfies f(0)=0 and f(1)=0 but is not identically 0, so it is not linear.

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