help with function iteration

Discussion in 'Math' started by lital, Dec 2, 2010.

  1. lital

    Thread Starter New Member

    Dec 2, 2010
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    0
    hi all, i need help with this
    prove or disprove :
    if F:R -> [a,b] is contractive on [a,b] then F has a unique fixed point,
    which can be obtained by function iteration staring an any real value.
     
  2. Papabravo

    Expert

    Feb 24, 2006
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    1,798
    What is the meaning of "contractive". I'm not familiar with that term.
    Also in this context what is the meaning of a "fixed point"
     
  3. lital

    Thread Starter New Member

    Dec 2, 2010
    2
    0
    Contraction mapping mean that |f(x)-f(y)| < k|x-y| when 0<k<1.
    fixed point mean that for some s , f(s)=s
     
  4. Georacer

    Moderator

    Nov 25, 2009
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    1,266
    That might be true, given that you need g'(x)<1, x in [a,b] in order for the function to be contractive. Since all the potential fixed points lie on the line f(x)=x which has f'=1, our function g can intercept f only once.

    Intercepting it twice would mean that in some interval its derivative would be >1 and that is unacceptable, since g is contractive in the whole space [a,b] not in some subspaces.
     
  5. Papabravo

    Expert

    Feb 24, 2006
    10,175
    1,798
    So a contractive mapping is one in which the points in the domain are brought closer together in the range. In differential equations a fixed point is one where the velocity goes to zero.

    On Wikipedia there is a development of the Banach Fixed Point Theorem that you might find useful. I'm close to being able to follow it.

    http://en.wikipedia.org/wiki/Banach_fixed_point_theorem
     
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