1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Fixed points of map and norm

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

    QQ截图20121202233027.png


    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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook