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

d ( f ( x ) , f ( y ) ) ≤ k d ( x , y ) . {\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 f : M → N {\displaystyle f:M\rightarrow N} is a contractive mapping if there is a constant 0 ≤ k < 1 {\displaystyle 0\leq k<1} such that d ′ ( f ( x ) , f ( y ) ) ≤ k d ( x , y ) {\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]

A non-expansive mapping with k = 1 {\displaystyle k=1} can be generalized to a firmly non-expansive mapping in a Hilbert space H {\displaystyle {\mathcal {H}}} if the following holds for all x and y in H {\displaystyle {\mathcal {H}}} : ‖ f ( x ) − f ( y ) ‖ 2 ≤ ⟨ x − y , f ( x ) − f ( y ) ⟩ , {\displaystyle \|f(x)-f(y)\|^{2}\leq \,\langle x-y,f(x)-f(y)\rangle ,} where d ( x , y ) = ‖ x − y ‖ . {\displaystyle d(x,y)=\|x-y\|.} This is a special case of α {\displaystyle \alpha } averaged nonexpansive operators with α = 1 / 2 {\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 Fix ⁡ f := { x ∈ H   |   f ( x ) = x } ≠ ∅ , {\displaystyle \operatorname {Fix} f:=\{x\in {\mathcal {H}}\ |\ f(x)=x\}\neq \varnothing ,} then for any initial point x 0 ∈ H {\displaystyle x_{0}\in {\mathcal {H}}} , iterating x n + 1 = f ( x n ) , ∀ n ∈ N {\displaystyle x_{n+1}=f(x_{n}),\quad \forall n\in \mathbb {N} } yields convergence to a fixed point x n → z ∈ Fix ⁡ f {\displaystyle x_{n}\to z\in \operatorname {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 d ( f ( x ) , f ( y ) ) ≤ d ( x , y ) ; {\displaystyle d(f(x),f(y))\leq d(x,y);} d ( f ( f ( x ) ) , f ( x ) ) < d ( f ( x ) , x ) unless x = f ( x ) . {\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
• Comparametric equation
• Blackwell's contraction mapping theorem
• CLRg property

• 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.