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Homework Help: Fixed points of map and norm

  1. Dec 2, 2012 #1
    1. The problem statement, all variables and given/known data


    3. The attempt at a solution

    set x(t)=1+∫2cos(s(f^2(s)))ds(from 0 to t) then check x(0)=1+∫2cos(s(f^2(s)))ds(from 0 to 0)=1 then the initial condition hold, by FTC, we have dx(t)/dt=2cos(tx^(t)), then solutions can be found as fixed points of the map

    but for secound part [0,T] i dont know how to begin can anyone help me ???
  2. jcsd
  3. Dec 2, 2012 #2
    Shouldn't that f be an x in this case?

    You have a map [itex]\Theta:\mathcal{C}([0,T],\mathbb{R})\rightarrow \mathcal{C}([0,T],\mathbb{R})[/itex] such that

    [tex]\Theta(f):[0,T]\rightarrow \mathbb{R}:t\rightarrow 1+\int_0^t 2\cos(sf^2(s))ds[/tex]

    Strictly speaking, you first need to check that [itex]\Theta(f)[/itex] is in fact continuous before you can say that the codomain of [itex]\Theta[/itex] is [itex]\mathcal{C}([0,T],\mathbb{R})[/itex].

    Now, you need to find out when [itex]\Theta[/itex] is a contraction. Can you tell us what that means??

    Also, here is a LaTeX guide on how to post mathematical equations: https://www.physicsforums.com/showpost.php?p=3977517&postcount=3 It would help us a lot if you would use this to make your equations more readable.
  4. Dec 3, 2012 #3
    how to say that [itex]\Theta(f)[/itex] is continuous??? i just dont know how to prove here. if it is then i know how to solve the problem now,many thanks
    Last edited: Dec 3, 2012
  5. Dec 3, 2012 #4
    Fundamental theorem of calculus.
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