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Thread: Fixed point syntax!

  1. #1
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    Fixed point syntax!

    Hi everyone!

    I am a little bit confused how write this:

    My map is: $\displaystyle F(f)(x)=\frac{1}{2}f(x)-\frac{3}{5}$. I must calculated a fixed point. I know how to do that and I now how much it is. But problem is in syntax. Is that the right one: $\displaystyle F(f(x_0))=f(x_0)$ and the result in that form $\displaystyle f(x_0)=-\frac{6}{5}$ ? Or is there $\displaystyle f_0(x)$ and not $\displaystyle f(x_0)$

    Thanks in advance
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  2. #2
    MHF Contributor

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    It is $\displaystyle f_0(x)$. Your map, F, is acting on functions, not numbers. A "fixed point" of F is a function, $\displaystyle f_0(x)$ such that $\displaystyle F(f_0)= f_0$. Since $\displaystyle F(f)(x)$ is defined as $\displaystyle \frac{1}{2}f(x)- \frac{3}{5}$ a fixed point is a function $\displaystyle f_0(x)$ such that $\displaystyle \frac{1}{2}f(x)- \frac{3}{5}= f(x)$ for all x. Of course, from that $\displaystyle \frac{1}{2}f(x)= -\frac{3}{5}$ so that the function $\displaystyle f_0$ is defined by $\displaystyle f_0(x)= -\frac{6}{5}$ for all x.
    Last edited by HallsofIvy; Dec 22nd 2010 at 02:58 AM.
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