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 real number ${\displaystyle 0\leq k<1}$ such that for all x and y in M,

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

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 ${\displaystyle f:M\rightarrow N}$ is a contractive mapping if there is a constant ${\displaystyle 0\leq k<1}$ such that

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

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

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 ${\displaystyle \operatorname {Fix} f:=\{x\in {\mathcal {H}}\ |\ f(x)=x\}\neq \varnothing }$, 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 ${\displaystyle x_{n}\to z\in \operatorname {Fix} f}$. This convergence might be weak in an infinite-dimensional setting.^[5]

Subcontraction map

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]

Locally convex spaces

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]

See also

• Short map
• Contraction (operator theory)
• Transformation
• Comparametric equation
• Blackwell's contraction mapping theorem
• CLRg property

References

1. Shifrin, Theodore (2005). Multivariable Mathematics. Wiley. pp. 244–260. ISBN 978-0-471-52638-4.
2. 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.
3. 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.
4. Combettes, Patrick L. (2004). "Solving monotone inclusions via compositions of nonexpansive averaged operators". Optimization. 53 (5–6): 475–504. doi:10.1080/02331930412331327157. S2CID 219698493.
5. Bauschke, Heinz H. (2017). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. New York: Springer.
6. 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. S2CID 49409638.
7. 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.
8. 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.

Further reading

• 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 979-8-8366-4680-6.