1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook