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Mean value theorem help

  1. May 7, 2009 #1
    1. The problem statement, all variables and given/known data

    If h1 and h2 are contractions on a set B with contraction constants δ1 and δ2 prove that the composite function h2 ° h 1 is also a contraction on B and find a contraction constant.


    2. Relevant equations


    |f(a) - f(b)| ≤ δ |a-b|

    f '(c) = (f(a)-f(b))/(a-b)


    (g°f) '(c) = g '(f(c))x f '(c)


    3. The attempt at a solution

    So far i'm pretty sure i have to use the mean value theorem and the chain rule. using the mean value on the composite function i get :

    |h2(h1(a)) - h2(h1(b))| = |h2 °h1 )' (c)| |a-b|

    i get stuck here, i think i should now use the chain rule for the derivaitve term out the front of the equality to somehow make an inequality. am i on the right track?
     
    Last edited: May 7, 2009
  2. jcsd
  3. May 7, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: contractions

    I don't see why you should use derivatives at all. If [itex]h(x)= h_2(h_1(x))[/itex] then [itex]|h(a)- h(b)|= |h_2(h_1(a))- h_2(h_1(b))|\le \delta_2|h_1(a)- h_1(b)|[/itex] and repeat.
     
  4. May 7, 2009 #3
    Re: contractions


    but then arent you assumeing that h(x)= h_2(h_1(x)) is a contration?
     
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