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
A non-expansive mapping with can be generalized to a firmly non-expansive mapping in a Hilbert space if the following holds for all x and y in :
This is a special case of averaged nonexpansive operators with . 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. 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. 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.
A subcontraction map or subcontractor is a map f on a metric space (M, d) such that
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.
^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.
^ abBauschke, 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. S2CID 49409638.
^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.
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.