Examples of certain contraction mapping

Question

Let $(M,d)$ be a metric space and $T:M\to M$ satisfying $d(T(x),T(y))\le (1/2)d(x,y)$ but $T$ has no fixed points. Can you give an example of such and metric space and a mapping $T$ ?

Answer 1

$M:=\mathbb{R}\setminus \{0\}; d(x,y)=|x-y|.$

Set $T:M \to M$ to be $T(x)=x/2.$

It satisfies the request because $d(T(x),T(y))=|x/2-y/2|=1/2 \cdot|x-y|=1/2 \cdot d(x,y).$

Answer 2

Consider $M=(0,1)$ with the usual metric and $T(x)=x/3$. Can you see what happens?

Answer 3

Let $L$ be the space of real sequences that are zero after some finite index.

Let $d(x,y) = \|x-y\|_\infty$, and $f(x) = (1,{x_1\over 2}, {x_2\over 2},...)$.

Note that $\|f(x)-f(y)\|_\infty = {1 \over 2} \|x-y\|_\infty$.

However, $f$ has no fixed points.