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Math Help - Contraction functions

  1. #1
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    Contraction functions

    A function f from reals to reals is called a contraction of R if and only if there exists a constant r in [0,1) such that x and x' in R we have

    |f(x) - f(x')| <_ r|x - x'| ( less than or equal)

    Now let f be a contraction of R, with constant r.

    a) let x in R be an arbritrary and define a sequence x_n to be f(x_n-1), for each n in the naturals. Show that |x_n+1 - x_n| <_ r^n|x_1 - x_0|.

    b) prove X_n is a cauchy sequence.

    c)Let p = lim x_n asn n approaches infinity, and prove that f(p) = 0.

    in part c) is basically showing every constraction point has a fixed point.
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  2. #2
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    a) Let x_0 be given, and f have the necessary properties. Then |x_2-x_1|=|f(x_1)-f(x_0)|\le r|x_1-x_0| which satisfies the condition for n=1. So assume that the statement is true for some integer n\ge{1}. Then
    |x_{n+1}-x_n|=|f(x_n)-f(x_{n-1})|\le r|x_n-x_{n-1}|\le r^n|x_1-x_0|.
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