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Math Help - Help with proof about uniform contraction

  1. #1
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    Help with proof about uniform contraction

    Anyone have a proof for this? Or a starting point?

    Prove that for every uniform contraction f there exists a unique x* in R such that f(x*)=x*.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by paupsers View Post
    Anyone have a proof for this? Or a starting point?

    Prove that for every uniform contraction f there exists a unique x* in R such that f(x*)=x*.
    A contraction is a function f such that d(f(x),f(y))\leq kd(x,y) for all x,y and some k<1.

    For uniqueness, assume (for a contradiction) f(x^*)=x^* and f(y^*)=y^*. What can you say about d(x^*,y^*)?

    As for existence, define a sequence \{x_n\} such that x_{n+1}=f(x_n). Try to find a formula in terms of k and d(x_1,x_0) for d(x_n,x_m) and use this to show that \{x_n\} is Cauchy (and therefore converges to some point x^*). Then use the continuity of f to show that f(x^*)=x^*.

    What you are proving is known as the Contraction Mapping Principle or alternatively, the Banach Fixed Point Theorem.
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