# Firmly nonexpansive but not contractive

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

A contraction mapping or simply a contraction is an operator $T: H\to H$ (here, we’ll consider $H$ to be a normed space and, in particular, a Hilbert space) with the property

\begin{aligned} \|Tx - Ty\| \leq \beta \|x-y\|, \end{aligned}

for some $\beta \in (0,1)$, that is, we say that $T$ is a contraction if it is $\beta$-Lipschitz for some $\beta \in (0,1)$.

We say that an operator $T: H\to H$ is nonexpansive if it is $1$Lipschitz, that is

\begin{aligned} \|Tx - Ty\| \leq \|x-y\|. \end{aligned}

Clearly, every contraction is nonexpansive. We now introduce another notion of nonexpansiveness: An operator $T: H\to H$ is firmly nonexpansive if

\begin{aligned} \|Tx-Ty\|^2 + \|(I-T)x - (I-T)y\|^2 \leq \|x-y\|^2, \end{aligned}

where $I$ is the identity operator, i.e., $Ix = x$ for all $x \in H$. It is clear that not all firmly nonexpansive operators are contractions. Take for example $T=I$. Here, motivated by Exercise 4.15 in the book of Bauschke and Combettes [1], we are looking for an operator $F:\mathbb{R}\to\mathbb{R}$ so that:

(i) $F$ is firmly nonexpansive
(ii) $I-F$ is injective
(iii) there is no $\beta\in [0,1)$ so that $F$ is $\beta$-Lipschitz

We know for $F$ to be FNE, $2F-I$ needs to be nonexpansive because of [Prop 4.2(iii), 1]. For that, it suffices to pick $F$ to be smooth so that the function

\begin{aligned} G(x) = 2F(x) - x \end{aligned}

has derivative $|G'(x)|\leq 1$, or equivalently

\begin{aligned} 0 \leq F'(x) \leq 1, \text{ for all} x\in \mathbb{R}. \end{aligned}

Now, $I-F$ must be an injection. Let us choose $I-F$ to be strictly increasing, that is

\begin{aligned} F'(x) < 1, \text{ for all} x\in \mathbb{R}. \end{aligned}

So, we need to find a $F$ with $0 \leq F'(x) < 1$, but we need to be 1-Lipschitz and no less, so we’ll require that

\begin{aligned} \lim_{x\to\infty}F'(x) = 1. \end{aligned}

We may now choose such a function:

\begin{aligned} F'(x) = 1-\lambda e^{-x^2}, \end{aligned}

for some $\lambda \in (0,1]$. Integrating, we get

\begin{aligned} F(x) = c + x - \frac{\lambda\sqrt{\pi}}{2}\mathrm{erf}(x). \end{aligned}

For $c=0$, the operator $(I-F)(x) = \frac{\lambda\sqrt{\pi}}{2}\mathrm{erf}(x)$ is indeed injective.

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

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