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Homework Help: Analysis: fixed point, contraction mapping

  1. Mar 30, 2010 #1
    Let [tex]p,q : \mathbb{C} \to \mathbb{C}[/tex] be
    defined by
    [tex]
    \begin{align*}
    p(z) =& z^7 + z^3 - 9z - i, \\
    q(z) =& \frac{z^7 + z^3 - i}{9}
    \end{align*}
    [/tex]

    1. Prove that [tex]p[/tex] has a zero at [tex]z_0[/tex] if and only if [tex]z_0[/tex] is a
    fixed point for [tex]q[/tex].

    If [tex]z_0[/tex] is a fixed point for [tex]q[/tex] then
    [tex]
    \begin{align*}
    q(z_0) = \frac{z_0^7 + z_0^3 - i}{9} =& z_0 \\
    z_0^7 + z_0^3 - i =& 9 z_0 \\
    z_0^7 + z_0^3 - 9 z_0 - i =& p(z_0) = 0
    \end{align*}
    [/tex]
    Hence [tex]z_0[/tex] is a zero for [tex]p[/tex].

    If [tex]p[/tex] has a zero at [tex]z_0[/tex] then
    [tex]
    \begin{align*}
    p(z_0) = z_0^7 + z_0^3 - 9 z_0 - i =& 0 \\
    z_0^7 + z_0^3 - i =& 9 z_0 \\
    \frac{z_0^7 + z_0^3 - i}{9} =& q(z_0) = z_0
    \end{align*}
    [/tex]
    Hence [tex]z_0[/tex] is a fixed point for [tex]q[/tex].

    2. Hence or otherwise show that [tex]p[/tex] has exactly one zero in
    the closed unit disk [tex]D = \{ z \in \mathbb{C} : |z| \leq 1 \}[/tex].

    How can I solve this?

    3. Where are the other zeros?

    Does this follow part 2?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 30, 2010 #2
    Show that q is a contraction map from D into D, i.e. that [tex]|q(w)-q(z)|<\alpha|w-z|[/tex] with [tex]\alpha<1[/tex]. Then use a theorem about contraction maps.
     
  4. Mar 30, 2010 #3
    What metric should I use for complex number space? The question did not mention any metric function at all.
     
  5. Apr 1, 2010 #4
    In [tex]D[/tex] the derivative [tex]q'[/tex] is not less than 1.

    [tex]
    \begin{align*}
    |q'(z)| =
    |\frac{7z^6 + 3z^2}{9}| \leq \frac{1}{9}
    (|7z^6| + |3z^2|) \leq \frac{1}{9} (7 + 3) = \frac{10}{9}.
    \end{align*}
    [/tex]

    Is [tex]q : D \to D[/tex] is a contraction mapping in D?
     
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