Contraction mapping

In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some nonnegative real number $0\leq k<1$ such that for all x and y in M,

d(f(x),f(y))\leq k\,d(x,y).

The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map.

More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M, d) and (N, d') are two metric spaces, then $f:M\rightarrow N$ is a contractive mapping if there is a constant $0\leq k<1$ such that

d'(f(x),f(y))\leq k\,d(x,y)

for all x and y in M.

Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1).

A contraction mapping has at most one fixed point. Moreover, the Banach fixed-point theorem states that every contraction mapping on a non-empty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for iterated function systems where contraction mappings are often used. Banach's fixed-point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.^[1]

Contraction mappings play an important role in dynamic programming problems.^[2]^[3]

Firmly non-expansive mapping[edit]

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

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

where

d(x,y)=\|x-y\|

.

This is a special case of $\alpha$ averaged nonexpansive operators with $\alpha =1/2$ .^[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 ${\text{Fix}}f:=\{x\in {\mathcal {H}}\ |\ f(x)=x\}\neq \varnothing$ , then for any initial point $x_{0}\in {\mathcal {H}}$ , iterating

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

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

Subcontraction map[edit]

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

d(f(x),f(y))\leq d(x,y);

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]

Locally convex spaces[edit]

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]

References[edit]

^ Shifrin, Theodore (2005). Multivariable Mathematics. Wiley. pp. 244–260. ISBN 978-0-471-52638-4.
^ Denardo, Eric V. (1967). "Contraction Mappings in the Theory Underlying Dynamic Programming". SIAM Review. 9 (2): 165–177. Bibcode:1967SIAMR...9..165D. doi:10.1137/1009030.
^ Stokey, Nancy L.; Lucas, Robert E. (1989). Recursive Methods in Economic Dynamics. Cambridge: Harvard University Press. pp. 49–55. ISBN 978-0-674-75096-8.
^ Combettes, Patrick L. (2004). "Solving monotone inclusions via compositions of nonexpansive averaged operators". Optimization. 53 (5–6): 475–504. doi:10.1080/02331930412331327157.
^ ^a ^b Bauschke, Heinz H. (2017). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. New York: Springer.
^ Combettes, Patrick L. (July 2018). "Monotone operator theory in convex optimization". Mathematical Programming. B170: 177–206. arXiv:1802.02694. Bibcode:2018arXiv180202694C. doi:10.1007/s10107-018-1303-3.
^ Goldstein, A.A. (1967). Constructive real analysis. Harper's Series in Modern Mathematics. New York-Evanston-London: Harper and Row. p. 17. Zbl 0189.49703.
^ Cain, G. L., Jr.; Nashed, M. Z. (1971). "Fixed Points and Stability for a Sum of Two Operators in Locally Convex Spaces". Pacific Journal of Mathematics. 39 (3): 581–592. doi:10.2140/pjm.1971.39.581.

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