Editing Contraction mapping (section)

==Firmly non-expansive mapping==
A non-expansive mapping with <math>k=1</math> can be generalized to a '''firmly non-expansive mapping''' in a [[Hilbert space]] <math>\mathcal{H}</math> if the following holds for all ''x'' and ''y'' in <math>\mathcal{H}</math>:
:<math>\|f(x)-f(y) \|^2 \leq \, \langle x-y, f(x) - f(y) \rangle.</math>
where
:<math>d(x,y) = \|x-y\|</math>.

This is a special case of <math>\alpha</math> averaged nonexpansive operators with <math>\alpha = 1/2</math>.<ref>{{cite journal |title=Solving monotone inclusions via compositions of nonexpansive averaged operators |first=Patrick L. |last=Combettes |year=2004 |journal=[[Optimization (journal)|Optimization]] |volume=53 |issue=5–6 |pages=475–504 |doi=10.1080/02331930412331327157 |s2cid=219698493 }}</ref> A firmly non-expansive mapping is always non-expansive, via the [[Cauchy–Schwarz inequality]].

The class of firmly non-expansive maps is closed under [[convex combination]]s, but not compositions.<ref name=":0">{{Cite book|title=Convex Analysis and Monotone Operator Theory in Hilbert Spaces|last=Bauschke|first=Heinz H.|publisher=Springer|year=2017|location=New York}}</ref> This class includes [[Proximal operator|proximal mappings]] of proper, convex, lower-semicontinuous functions, hence it also includes orthogonal [[Projection (mathematics)|projections]] onto non-empty closed [[convex set]]s. The class of firmly nonexpansive operators is equal to the set of resolvents of maximally [[Monotonic function#Monotonicity in functional analysis|monotone operators]].<ref>{{Cite journal|last=Combettes|first=Patrick L.|date=July 2018|title=Monotone operator theory in convex optimization|journal=Mathematical Programming|volume=B170|pages=177–206|arxiv=1802.02694|doi=10.1007/s10107-018-1303-3|bibcode=2018arXiv180202694C|s2cid=49409638}}</ref> Surprisingly, while iterating non-expansive maps has no guarantee to find a fixed point (e.g. multiplication by -1), firm non-expansiveness is sufficient to [[convergence proof techniques|guarantee global convergence]] to a fixed point, provided a fixed point exists. More precisely, if <math>\operatorname{Fix}f := \{x \in \mathcal{H} \ | \ f(x) = x\} \neq \varnothing</math>, then for any initial point <math>x_0 \in \mathcal{H}</math>, iterating

<math> (\forall n \in \mathbb{N})\quad x_{n+1} = f(x_n) </math>

yields convergence to a fixed point <math> x_n \to z \in \operatorname{Fix} f</math>. This convergence might be [[Weak convergence (Hilbert space)|weak]] in an infinite-dimensional setting.<ref name=":0" />