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Math Help - Banach Fixed Point Theorem

  1. #1
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    Banach Fixed Point Theorem

    How do you show that the fixed point is unique?

    I would rather have hints and tips so that I can try and work through it! But I just don't know where to start!
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Furbylicious View Post
    How do you show that the fixed point is unique?

    I would rather have hints and tips so that I can try and work through it! But I just don't know where to start!
    A contraction is a function f such that d(f(x),f(y))\leq kd(x,y) for k<1.

    Say that x_0 is a fixed point (so that f(x_0)=x_0). Assume that it is not unique; that is, there is another fixed point y_0.

    Now look at d(x_0,y_0). What can you say?
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    What do you mean it's unique? That depends on the function. The function f(x)=x has infinitely many fixed points on any interval.
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  4. #4
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    I used proof by contradiction.

    Let phi: A->A be a contraction with constant k<1
    Assume that there exists 2 fixed points
    phi(x1) = x1
    phi(x2) = x2

    According to the definition

    |phi(x1) - phi(x2)|<=k|x1-x2|

    as phi(x1) = x1 and phi (x2) = x2 then

    |x1-x2|<=k|x1-x2|

    which gives

    1<= k
    which contradicts the original condition that k<1....


    Am i fumbling along the right lines, or is it all rubbish?
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  5. #5
    Super Member redsoxfan325's Avatar
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    That's correct.
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