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!

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




Similar Discussions: Initial Value Problem
  1. Initial value problem (Replies: 7)

  2. Initial-value problem (Replies: 2)

Loading...