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 such that for all x and y in M,

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 is a contractive mapping if there is a constant such that

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 nonempty 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 mapping plays an important role in dynamic programming problems.[2][3]

Firmly non-expansive mapping

A non-expansive mapping with can be strengthened to a firmly non-expansive mapping in a Hilbert space if the following holds for all x and y in :

where

.

This is a special case of averaged nonexpansive operators with .[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 nonempty 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 , then for any initial point , iterating

yields convergence to a fixed point . 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

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 kp < 1 such that p(f(x) - f(y)) kp 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 xn+1 = f(xn), and if (E,P) is Hausdorff, then the fixed point is unique.[8]

See also

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.
  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.
  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. 40 (Second ed.). New York: Springer. pp. 125–134. ISBN 978-0-387-90748-2.
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