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Math Help - contraction

  1. #1
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    contraction

    please friends can u help me with the question below
    Q. If T is contraction, show that T^n (n\inN) is a contraction. If T^n is a contraction for all n>1, show that T need not be a contraction.
    Thanks
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  2. #2
    Super Member girdav's Avatar
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    We don't know where T is defined.
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  3. #3
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    T:X-->X is the contraction on X if there is a positive \alpha <1 such that d(tx,ty)\leqslant \alpha d(x,y)
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  4. #4
    Super Member girdav's Avatar
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    For the first question, prove by induction that d(T^n(x),T^n(y))\leq\alpha ^n d(x,y) \forall x,y\in X if \forall x,y\in X, d(Tx,Ty)\leq \alpha d(x,y).
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  5. #5
    Super Member girdav's Avatar
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    For the second question we can put X=\mathbb{R}, d(x,y)=|x-y| and T(x)=\sqrt{1+x^2}.
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