Contraction mapping

A non-expansive mapping with ${\displaystyle k=1}$ can be generalized to a firmly non-expansive mapping in a Hilbert space ${\displaystyle {\mathcal {H}}}$ if the following holds for all x and y in ${\displaystyle {\mathcal {H}}}$:

${\displaystyle \|f(x)-f(y)\|^{2}\leq \,\langle x-y,f(x)-f(y)\rangle .}$

where

${\displaystyle d(x,y)=\|x-y\|}$.

This is a special case of ${\displaystyle \alpha }$ averaged nonexpansive operators with ${\displaystyle \alpha =1/2}$.[4] 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 combinations, but not compositions.[5] This class includes proximal mappings of proper, convex, lower-semicontinuous functions, hence it also includes orthogonal projections onto non-empty closed convex sets. The class of firmly nonexpansive operators is equal to the set of resolvents of maximally monotone operators.[6] 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 guarantee global convergence to a fixed point, provided a fixed point exists. More precisely, if ${\displaystyle {\text{Fix}}f:=\{x\in {\mathcal {H}}\ |\ f(x)=x\}\neq \varnothing }$, then for any initial point ${\displaystyle x_{0}\in {\mathcal {H}}}$, iterating

${\displaystyle (\forall n\in \mathbb {N} )\quad x_{n+1}=f(x_{n})}$

yields convergence to a fixed point ${\displaystyle x_{n}\to z\in {\text{Fix}}f}$. This convergence might be weak in an infinite-dimensional setting.[5]

A subcontraction map or subcontractor is a map f on a metric space (M, d) such that

${\displaystyle d(f(x),f(y))\leq d(x,y);}$

${\displaystyle d(f(f(x)),f(x))<d(f(x),x)\quad {\text{unless}}\quad x=f(x).}$

If the image of a subcontractor f is compact, then f has a fixed point.[7]

In a locally convex space (E, P) with topology given by a set P of seminorms, one can define for any p ∈ P a p-contraction as a map f such that there is some k_p < 1 such that p(f(x) − f(y)) ≤ k_p p(x − y). If f is a p-contraction for all p ∈ P and (E, P) is sequentially complete, then f has a fixed point, given as limit of any sequence x_n+1 = f(x_n), and if (E, P) is Hausdorff, then the fixed point is unique.[8]

• Short map
• Contraction (operator theory)
• Transformation

• Istratescu, Vasile I. (1981). Fixed Point Theory : An Introduction. Holland: D.Reidel. ISBN 978-90-277-1224-0. provides an undergraduate level introduction.
• Granas, Andrzej; Dugundji, James (2003). Fixed Point Theory. New York: Springer-Verlag. ISBN 978-0-387-00173-9.
• Kirk, William A.; Sims, Brailey (2001). Handbook of Metric Fixed Point Theory. London: Kluwer Academic. ISBN 978-0-7923-7073-4.
• Naylor, Arch W.; Sell, George R. (1982). Linear Operator Theory in Engineering and Science. Applied Mathematical Sciences. Vol. 40 (Second ed.). New York: Springer. pp. 125–134. ISBN 978-0-387-90748-2.

• Bullo, Francesco (2022). Contraction Theory for Dynamical Systems. Kindle Direct Publishing. ISBN 979-8836646806.