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Initial Value Problem

  1. May 3, 2012 #1
    1. The problem statement, all variables and given/known data
    For the space of continuous functions C[0,T] suppose we have the metric ρ(x,y) =sup [itex]_{t\in [0,T]}[/itex]e[itex]^{-Lt}[/itex][itex]\left|x(t)-y(t)\right|[/itex] for T>0, L≥0.

    Consider the IVP problem given by

    x'(t) = f(t,x(t)) for t >0,
    x(0) = x[itex]_{0}[/itex]

    Where f: ℝ×ℝ→ℝ is continuous and globally Lipschitz continuous with
    respect to x.

    Find an integral operator such that the operator is a contraction on (C[0,T],ρ) and hence deduce the IVP has a unique solution on C[itex]^{1}[/itex][0,T]

    3. The attempt at a solution

    I was able to show that the metric space (C[0,T],ρ) is complete, but I'm having problems finding an integral operator that is a contraction on the space. I've tried the operator
    (Tx)(t) = [itex]x_{0}[/itex] + [itex]\int^{t}_{0}f(s,(x(s))ds [/itex]but I was not able to get a contraction. Any help would be much appreciated!
    Last edited by a moderator: May 3, 2012
  2. jcsd
  3. May 3, 2012 #2
    try finding T|x-y| and then use the lipschitz continuity?
  4. May 4, 2012 #3
    I have tried this but was unable to show it was a contraction. I'm not to sure if I have the wrong integral operator for this particular question or if I'm trying to show a contraction in the wrong way.
  5. May 4, 2012 #4
    I'm not sure if your operator is contraction, but it does not seem to be a fixed point iteration operator at all ...
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