Results 1 to 5 of 5

Math Help - Cauchy sequence

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    3

    Cauchy sequence

    Hello, everyone Iwant to ask about Cauchy seq.
    How to prove that d(x2m+1,x2m)and d(x2m-1,x2m) are cauchy sequences
    if T:X->X is a convex contraction mapping of order 2 , X is complete metric space ,xn+1=Txn and k=max{d(x1,x0),d(x1,x2)}
    Last edited by mr fantastic; April 29th 2010 at 09:56 PM. Reason: Re-titled.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by math3000 View Post
    Hello, everyone Iwant to ask about Cauchy seq.
    How to prove that d(x2m+1,x2m)and d(x2m-1,x2m) are cauchy sequences
    if T:X->X is a convex contraction mapping of order 2 , X is complete metric space ,xn+1=Txn and k=max{d(x1,x0),d(x1,x2)}
    This is hard to read. Could you try to write it in Latex? I'm not sure what a "convex" contract is or what "order 2" means.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    3
    T is a convex contraction of order 2 means
    1)d(Tx,Ty)≤ad(Tx,Ty)+bd(x,y)
    2)a,b and a+b∈(0,1)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by math3000 View Post
    T is a convex contraction of order 2 means
    1)d(Tx,Ty)≤ad(Tx,Ty)+bd(x,y)
    2)a,b and a+b∈(0,1)
    Ok, now that we have our notation down, what have you tried?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Apr 2010
    Posts
    3
    Quote Originally Posted by Drexel28 View Post
    Ok, now that we have our notation down, what have you tried?
    From the inequality
    d(T^{2m+1}x0,T^{2m}x0)≤ad(T^{2m}x0,T^{2m-1}x0)+bd(T^{2m-1}x0,T^{2m-2}x0) and for m≥1 .I get
    d(Tx0,Tx0)≤ad(Tx0,Tx0)+bd(Tx0,x0) ≤(a+b)k
    d(T5x0,T⁴x0)≤ad(T⁴x0,Tx0)+bd(Tx0,Tx0) ≤(a+b)k and so on.
    and From the inequality
    d(T^{2m-1}x0,T^{2m}x0)≤ad(T^{2m-2}x0,T^{2m-1}x0)+bd(T^{2m-3}x0,T^{2m-2}x0) and for m≥2 .I get
    d(Tx0,T⁴x0)≤ad(Tx0,Tx0)+bd(Tx0,Tx0) ≤(a+b)k
    d(T5x0,T6x)≤ad(T⁴x0,T5x0)+bd(T⁴x0,Tx0) ≤(a+b)k and so on.
    Then I arrived to
    d(T^{2m+1}x0,T^{2m}x0)≤k(a+b)^{m} and also
    d(T^{2m-1}x0,T^{2m}x0)≤k(a+b)^{m}
    I want to show that d(T^{m+1}x0,T^{m}x0)≤ A value
    then I use it to sow that
    ∀m<n d(T^{m}x0,Tⁿx0)≤d(T^{m}x0,T^{m+1}x0)+d(T^{m+1}x0,T ^{m+2}x0)+...+d(Tⁿ-x0,Tⁿx0)

    I want a hint
    because I am trying every thing and there is no clearly answer
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Subsequence of a Cauchy Sequence is Cauchy
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: September 30th 2010, 02:29 AM
  2. cauchy sequence, contractive sequence
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 25th 2010, 07:25 AM
  3. cauchy sequence
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: December 14th 2009, 09:13 PM
  4. cauchy sequence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 3rd 2009, 07:06 PM
  5. Cauchy sequence
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 24th 2006, 02:03 PM

Search Tags


/mathhelpforum @mathhelpforum