for some , that is, we say that is a contraction if it is -Lipschitz for some .
We say that an operator is nonexpansive if it is –Lipschitz, that is
Clearly, every contraction is nonexpansive. We now introduce another notion of nonexpansiveness: An operator is firmly nonexpansive if
where is the identity operator, i.e., for all . It is clear that not all firmly nonexpansive operators are contractions. Take for example . Here, motivated by Exercise 4.15 in the book of Bauschke and Combettes , we are looking for an operator so that:
(i) is firmly nonexpansive
(ii) is injective
(iii) there is no so that is -Lipschitz
We know for to be FNE, needs to be nonexpansive because of [Prop 4.2(iii), 1]. For that, it suffices to pick to be smooth so that the function
has derivative , or equivalently
Now, must be an injection. Let us choose to be strictly increasing, that is
So, we need to find a with , but we need to be 1-Lipschitz and no less, so we’ll require that
We may now choose such a function:
for some . Integrating, we get
For , the operator is indeed injective.
 H.H. Bauschke and P.L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, Springer, 2011.