Results 1 to 3 of 3

Math Help - Application of the Arzela-Ascoli Theorem

  1. #1
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943

    Application of the Arzela-Ascoli Theorem

    Here is the problem as it is written in the text:
    _________________

    Let f(t,x) be defined and continuous for a\leq t\leq b and x\in\mathbb{R}^n. The purpose of this exercise is to show that the problem dx/dt=f(t,x), x(a)=x_0, has a solution on the interval t\in[a,c] for some c>a. Perform the operations as follows: Divide [a,b] into n equal parts: a=t_0,t_1,...,t_n=b, and define a continuous function x_n inductively by

    \bigg\{\begin{array}{l}x_n'(t)=f(t_i,x_n(t_i)),~~t  _i<t<t_{i+1}\\x_n(a)=x_0\end{array}

    Put \Delta_n(t)=x_n'(t)-f(t,x_n(t)), so that

    x_n(t)=x_0+\int_a^t f(s,x_n(s))+\Delta_n(s)\,ds

    Use the Arzela-Ascoli Theorem to find a convergent subsequence of the x_n. Show that the limit satisfies dx/dt=f(t,x) and x(a)=x_0.
    ____________________

    The version of the A-A Thm that seems the most applicable here is the following:

    Let A be a compact set. If \mathcal{B}\subset\mathcal{C}(A,\mathbb{R}^n) is (uniformly) equicontinuous and pointwise bounded, then every sequence in \mathcal{B} has a uniformly convergent subsequence.

    Letting x_n=\mathcal{B}, I don't know how to show equicontinuity or pointwise boundedness. I know that each individual x_n is bounded and uniformly continuous, but that doesn't mean that they're pointwise bounded or equicontinuous (because there are infinitely many x_n).

    EDIT: I've figured out what to do after applying the A-A Theorem. However, I still don't know how to prove the criteria for using the A-A Theorem, so I still need help with that.

    Also, I'm not sure where that constant c came from. It pops up and then disappears almost immediately, never to be mentioned again. Maybe it's a typo.

    Please help!
    Last edited by redsoxfan325; October 15th 2009 at 09:59 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by redsoxfan325 View Post
    I'm not sure where that constant c came from. It pops up and then disappears almost immediately, never to be mentioned again. Maybe it's a typo.
    The constant c is an essential ingredient in this theorem. To see why it's important, look at this example: Let f(t,x) = x^2, defined for 0\leqslant t\leqslant 2 and x\in\mathbb R. The initial value problem dx/dt = f(t,x),\ x(0)=1 has the solution x(t) = 1/(1-t) . But that is only defined on the interval [0,1), not on the interval [0,2] on which we defined f(x,t). Admittedly, the function f(t,x) does not explicitly involve t in that example, but it shows that the solution to the problem can blow up unexpectedly in the interval [a,b]. That is why it's going to be necessary to restrict to a shorter interval if you want the function x(t) to exist.

    You can find a solution to the problem here (it's a theorem due to Peano). The proof is quite subtle. It starts by using the continuity of the function f(t,x) to get a neighbourhood of (a,x_0) on which f is bounded.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    Ugh, that's tough. The homework was due earlier today, and I submitted a proof that (based on what I understood of the correct proof in that textbook) should be good enough for half credit. (I had the general idea, but I apparently oversimplified a lot of the not-so-simple steps.) Thanks for explaining the c though!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Application of Rouche's theorem
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 17th 2010, 12:59 PM
  2. Why not Ascoli-Arzela?
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: April 1st 2010, 10:33 PM
  3. Help with Arzela-Ascoli theorem proof
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: March 29th 2010, 11:23 AM
  4. A weaker version of Arzela Ascoli Thm
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 4th 2010, 12:50 PM
  5. Arzelā-Ascoli theorem
    Posted in the Advanced Math Topics Forum
    Replies: 14
    Last Post: April 13th 2006, 04:12 PM

Search Tags


/mathhelpforum @mathhelpforum