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!

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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

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




Similar Discussions: Fixed points of map and norm
  1. Fixed points (Replies: 2)

  2. Fixed point iteration (Replies: 2)

Loading...