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How can you (easily) determine if the unique fixed point of the following function is attractive? I have my own way in which I don`t use inequalities or derivatives but only use algebra. I wonder if you have a faster, easier way.

  1. $f(x) = 1-x^2$ in $[0,1]$ , not attractive

  2. $g(x) = 1+\sqrt{x}$ in $[0,4]$ , attractive

My method applies to monotonic function. It tests if the double composition $f(f(x))$ has only one or more fixed points.

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  • $\begingroup$ umm.. so what does your method say when you pick $f(x) = x/2$ or $f(x)=2x$ ? $\endgroup$
    – mercio
    Jun 14 '13 at 13:01
  • $\begingroup$ It must be assumed that f transforms $[a,b]$ into itself. For the first function you may take $[0,1]$ and you get an attracted fixed-point 0. I will not apply my method for 2x since I don't have a suitable interval. $\endgroup$ Jun 16 '13 at 2:00
  • $\begingroup$ It must be assumed that f transforms $[a,b]$ into itself. For the first function you may take $[0,1]$ and get an attracted fixed-point $0$. I may not apply my method for $2x$ since there is no suitable interval for that. $\endgroup$ Jun 21 '13 at 21:00

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