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Homework Help: Show That a Function is Contractive?

  1. May 4, 2007 #1
    1. The problem statement, all variables and given/known data
    Show that the following function is contractive on the indicated intervals. Determine the best values of [lamda] in Equation (2).

    abs(x)^(2/3) on abs(x) < or = 1/3


    2. Relevant equations
    A mapping (or function) F is said to be contractive if there exists a number [lamda] less than 1 such that:

    (Equation (2))
    abs(F(x)-F(y)) < or = [lamda]*abs(x-y)

    for all points x and y in the domain F.

    3. The attempt at a solution
    I'm not really sure what to do with this one or how to get [lamda] in this case.
     
  2. jcsd
  3. May 4, 2007 #2
    mean value theorem?

    it seems that for the function that is given, such lamda doesn't exist. consider a interval extremely close to 0, let's say
    [tex](-\epsilon, \epsilon)[/tex]

    you see that
    [tex]\frac{|F(x)-F(y)|}{|x-y|}[/tex]
    goes to infinity, as it approaches the derivative at 0.

    are you sure you have the correct question?
     
  4. May 5, 2007 #3
    Thanks for responding, tim_lou. Yes, I'm sure I have the correct question. :) Maybe it's a trick question and it's actually not contractive?
     
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