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 1Lipschitz, 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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