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Math Help - Contraction Mapping.

  1. #1
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    Post Contraction Mapping.

    I have an exercise to prove something in contraction mapping
    I think it is easy but I'm not familiar with the subject.
    It is REALLY urgent
    Thanks
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    Last edited by mr fantastic; July 29th 2011 at 03:19 PM. Reason: Title.
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  2. #2
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    Re: Contraction Mapping.

    The first problem is quite straightforward: Let \delta\in(|x_{n+1}-x_n|,\epsilon), and notice that |x_{n+1}-x_{n+2}|<\gamma\delta. (This is by definition of the sequence \langle x_k\rangle.) In fact, it's easy to see that by induction |x_{n+i}-x_{n+i+1}|<\gamma^i\delta for i=0,1,2,\cdots. So by the triangle inequality

    |x_n-x_r|\leq \sum_{i=0}^{r-1}|x_{n+i}-x_{n+i+1}|<\sum_{i=0}^{r-1}\gamma^i\delta.

    Letting r\to\infty gives us |x_n-x^*|\leq\sum_{i=0}^\infty\gamma^i\delta=\frac{ \delta  }{1-\gamma}<\frac{\epsilon}{1-\gamma}.

    (Here we have used the identity of a geometric series.)

    I have no idea how to do the second problem, sorry.
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  3. #3
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    Re: Contraction Mapping.

    Thank you very much, last night I tried to solve it and I did as shown in the attached pdf file.
    I would like to know if there is any comment
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  4. #4
    Senior Member
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    Re: Contraction Mapping.

    Looks good to me! I wouldn't have thought to use the Banach fixed point theorem (since I had never heard of it until just now). But it works just fine.
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