3
$\begingroup$

As we know :

$T:X\longrightarrow X$ is a nonexpansive mapping iff $\|Tx-Ty\|\leq\|x-y\|,$ $\forall x,y\in X$


So my question is I want some nonexpansive mappings, I know $\sin(x)$ , $\cos(x)$ and if I'm not wrong $\frac{1}{x}$ for $x\geq1$

Thanks.

  • 3
    Any differentiable function on $\mathbb R$ whose derivative is bounded by $1$ is an example.2011-09-22
  • 0
    Thanks, but this is obvius from definition,only thing that I want is a nice and beautiful examples..2011-09-22
  • 3
    @Vahid..I don't understand your objection: you want something nice... ok let's thik about something differentiable at least... but then we fall in what Srivatsan said.. I can propose you arctan(x) as a nice example.. but how do you prove that this function works?? Uh it is differentiable and $\arctan'(x)=\frac{1}{1+x^2}\leq 1$... :)2011-09-22
  • 1
    Uh ok... what about $x\mapsto |x|?$ This is not differentiable... but it seems to me as obvious as any 1-Lipschitz Function...2011-09-22

5 Answers 5