Results 1 to 2 of 2

Math Help - contraction

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    9

    contraction

    let X=C[0,1/2] with standard sup-norm and F:X---->X defined by
    F(f)(x)=1+integral of f(x-t)dt from 0 to x
    how to prove F is a contraction with factor k=1/2?
    and how to find the unique fixed point?
    Last edited by mathmad; January 13th 2010 at 01:47 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    What is t? a constant? if it is then F(f)= 1+xf(x-t) which is clearly a contraction. The fixed point of this function however is not so easy to find (to me at least).

    If however you meant \int_{0}^{x} f(x-t)dt then, arguing as in your last question, one gets it's indeed a contraction and using induction we prove that F^n(1)= \sum_{j=0}^{n} \frac{x^j}{j!} \rightarrow e^x (where F^n means composition n-times)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Contraction Mapping.
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: July 30th 2011, 07:38 AM
  2. contraction
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: May 26th 2011, 02:00 AM
  3. Uniqueness in a contraction
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: May 18th 2010, 08:24 AM
  4. Contraction functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 19th 2009, 06:18 PM
  5. [SOLVED] Contraction
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 2nd 2008, 10:36 AM

Search Tags


/mathhelpforum @mathhelpforum