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Fixed points of map and norm

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 ???
 
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1. The problem statement, all variables and given/known data

View attachment 53573


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
Shouldn't that f be an x in this case?

but for secound part [0,T] i dont know how to begin can anyone help me ???
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.
 
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[tex]\ 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.
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:

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