Some notes on firm nonexpansiveness and contractiveness. We give an example of a firmly nonexpansive operator *F* so that *I*–*F *is injective and *F* is not a contraction.

A contraction mapping or simply a contraction is an operator (here, we’ll consider to be a normed space and, in particular, a Hilbert space) with the property

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 [1], 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.

[1] H.H. Bauschke and P.L. Combettes, *Convex analysis and monotone operator theory in Hilbert spaces,* Springer, 2011.