Can anyone please provide me an example of a Contractive mapping which is not a Contraction mapping.
Definitions:
A mapping $T: M\to M$ is said to be contractive if $d(Tx, Ty)<d(x,y)$ for each $x,y\in M$ with $x\neq y,$
A mapping $T: M\to M$ is said to be contraction if there exist a constant $0\leq k<1$ such that $d(Tx, Ty)\leq k d(x,y)$ for each $x,y\in M$ with $x\neq y,$
$$ \frac{3x + \sqrt{1 + x^2}}{4} $$ is a bijection of the real line with no fixpoint. The bit about contraction is the Mean Value Theorem together with the observation that the derivative is always between $0$ and $1,$ while getting arbitrarily close to $1$ as $x$ goes to $+ \infty$