How Can Fixed Points Determine Solutions in Differential Equations?

[cummings12332]

Homework Statement

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 don't know how to begin can anyone help me ?

 

[micromass]

Homework Statement

View attachment 53573

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 don't know how to begin can anyone help me ?

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

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

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

Now, you need to find out when $\Theta$ 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.

 

[cummings12332]

Shouldn't that f be an x in this case?
You have a map $\Theta:\mathcal{C}([0,T],\mathbb{R})\rightarrow \mathcal{C}([0,T],\mathbb{R})$ such that

[tex]\Theta(f):[0,T]\rightarrow \mathbb{R}:t\rightarrow$$\ 1+\int_0^t 2\cos(sf^2(s))ds$$<br /> <br /> Strictly speaking, you first need to check that $\Theta(f)$ is in fact continuous before you can say that the codomain of $\Theta$ is $\mathcal{C}([0,T],\mathbb{R})$.<br /> <br /> Now, you need to find out when $\Theta$ is a contraction. Can you tell us what that means??<br /> <br /> Also, here is a LaTeX guide on how to post mathematical equations: <a href="https://www.physicsforums.com/showpost.php?p=3977517&postcount=3" class="link link--internal">https://www.physicsforums.com/showpost.php?p=3977517&postcount=3</a> It would help us a lot if you would use this to make your equations more readable.[/tex]

[tex] <br /> how to say that $\Theta(f)$ is continuous? i just don't know how to prove here. if it is then i know how to solve the problem now,many thanks[/tex]

 

[micromass]

how to say that $\Theta(f)$ is continuous?

Fundamental theorem of calculus.