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Math Help - banach contraction

  1. #1
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    banach contraction

    X=C[0,1/2] , F:X---->X defined by
    F(f)(x)=1+f(x)/3+integral of f(s)ds from 0 to x
    how to prove F is a contraction?
    how to find the unique fixed point?

    Thanx
    Last edited by mathmad; January 4th 2010 at 10:18 AM.
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  2. #2
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    To prove it's a contraction you proceed as follows:

    \vert F(f)-F(g) \vert = \vert \frac{f(x)-g(x)}{3} + \int_{0}^{x} f(t)-g(t)dt \vert \leq  \frac{ \vert f(x)-g(x) \vert }{3} + \int_{0}^{x} \vert f(t)-g(t) \vert dt now taking the supremum over x we get that this last is \leq \frac{\Vert f-g \Vert }{3} + \frac{1}{2} \Vert f-g \Vert = \frac{5}{6} \Vert f-g \Vert

    To find the fixed point you could follow th proof of the fixed point theorem and iterate F given a simple initial function (say f=x or some such) or you could try by inspection see if you can find it.

    Edit: For simplicity you could assume f is differentiable, derive the expression F(f)=f and obtain an easy diff. equation, solve, substitute and determine the constant.

    Spoiler:
    The answer is f(x)= \frac{3}{2} e^{\frac{3}{2} x}
    Last edited by Jose27; January 3rd 2010 at 03:12 PM.
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