# contraction

• May 26th 2011, 12:06 AM
kinkong
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
• May 26th 2011, 12:31 AM
girdav
We don't know where $T$ is defined.
• May 26th 2011, 01:46 AM
kinkong
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 AM
girdav
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 $\forall x,y\in X, d(Tx,Ty)\leq \alpha d(x,y)$.
• May 26th 2011, 02:00 AM
girdav
For the second question we can put $X=\mathbb{R}$, $d(x,y)=|x-y|$ and $T(x)=\sqrt{1+x^2}$.