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Math Help - [SOLVED] Fixed Point

  1. #1
    Super Member redsoxfan325's Avatar
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    [SOLVED] Fixed Point

    Let M be a compact metric space and \Phi:M\longrightarrow M be such that d(\Phi(x),\Phi(y))<d(x,y) for all x,y\in M, x\neq y.

    Show that \Phi has a unique fixed point. [Hint: Minimize d(\Phi(x),x).]

    I'm not really sure how to use the hint, or even start the problem. This is in the chapter on the contraction mapping principle, so it seems like I have to get it so that I can apply the CMP. Any suggestions would be most welcome. I don't really want the whole problem solved. If you solve the whole problem, at least put some of it in a spoiler, because I'd like to solve as much of this on my own as I can.
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  2. #2
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    Suppose f:M \rightarrow [0, \infty ) is such that f(x)=d( \Phi(x),x) then f is is continous. M is compact. If in y f attains a minimum, what can you say about f(y)?
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  3. #3
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Jose27 View Post
    Suppose f:M \rightarrow [0, \infty ) is such that f(x)=d( \Phi(x),x) then f is is continous. M is compact. If in y f attains a minimum, what can you say about f(y)?
    So f is a (Lipschitz) continuous function on a compact set. It has a min, max, and it's uniformly continuous.

    What do you mean by "If in y f attains a minimum"?

    y is a point. I'm not sure what you mean. Did you mean that if f(y) is the minimum value of f?
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  4. #4
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    Quote Originally Posted by redsoxfan325 View Post

    y is a point. I'm not sure what you mean. Did you mean that if f(y) is the minimum value of f?
    Yes
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  5. #5
    Super Member redsoxfan325's Avatar
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    Thank you for your help. I have solved the problem.
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