Fixed points of map and norm

In summary, the conversation discusses a problem involving a map \Theta and finding solutions as fixed points of the map. The first part involves setting x(t) and checking if the initial condition holds. The second part involves finding out when \Theta is a contraction, which requires checking for continuity. The conversation also includes a guide for posting mathematical equations.
  • #1
cummings12332
41
0

Homework Statement



QQ截图20121202233027.png



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 ?
 
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  • #2
cummings12332 said:

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 [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.
 
  • #3
micromass said:
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 don't know how to prove here. if it is then i know how to solve the problem now,many thanks
 
Last edited:
  • #4
cummings12332 said:
how to say that [itex]\Theta(f)[/itex] is continuous?

Fundamental theorem of calculus.
 

1. What is a fixed point of a map and norm?

A fixed point of a map and norm is a point in a mathematical space that remains unchanged after the application of a map and norm. In other words, the fixed point is a point that maps to itself under the given map and norm.

2. How are fixed points of map and norm useful in mathematics?

Fixed points of map and norm are useful in many areas of mathematics, including functional analysis, differential equations, and optimization. They provide important insights into the behavior and properties of mathematical systems.

3. Can a map and norm have more than one fixed point?

Yes, a map and norm can have more than one fixed point. In fact, some maps and norms have infinitely many fixed points. The number of fixed points depends on the specific properties of the map and norm.

4. How can fixed points of map and norm be calculated?

The calculation of fixed points of map and norm can be a complex and challenging task. It often involves using advanced mathematical techniques, such as fixed point theorems and iterative methods. In some cases, fixed points can also be found through numerical approximation.

5. What are some real-world applications of fixed points of map and norm?

Fixed points of map and norm have numerous applications in real-world problems, such as optimizing traffic flow, predicting the spread of diseases, and designing efficient algorithms. They are also used in physics, engineering, and economics to model and understand complex systems.

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