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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- May 26th 2011, 12:06 AMkinkongcontraction
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 - May 26th 2011, 12:31 AMgirdav
We don't know where $\displaystyle T$ is defined.

- May 26th 2011, 01:46 AMkinkong
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)

- May 26th 2011, 01:50 AMgirdav
For the first question, prove by induction that $\displaystyle d(T^n(x),T^n(y))\leq\alpha ^n d(x,y) \forall x,y\in X$ if $\displaystyle \forall x,y\in X, d(Tx,Ty)\leq \alpha d(x,y)$.

- May 26th 2011, 02:00 AMgirdav
For the second question we can put $\displaystyle X=\mathbb{R}$, $\displaystyle d(x,y)=|x-y|$ and $\displaystyle T(x)=\sqrt{1+x^2}$.