Brstjjy

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

d(f(x),f(y))≤kd(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:Mrightarrow N} is a contractive mapping if there is a constant k<1{displaystyle k<1} such that

d′(f(x),f(y))≤kd(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 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 k=1{displaystyle k=1} can be strengthened to a firmly non-expansive mapping in a Hilbert space H if the following holds for all x and y in 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.

Subcontraction map

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)unlessx=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.^[5]

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.^[6]

See also

• Short map


• Contraction (operator theory)


• Transformation

References

Further reading

• 
  Istratescu, Vasile I. (1981). Fixed Point Theory : An Introduction. Holland: D.Reidel. ISBN 90-277-1224-7. provides an undergraduate level introduction.


• 
  Granas, Andrzej; Dugundji, James (2003). Fixed Point Theory. New York: Springer-Verlag. ISBN 0-387-00173-5.
  


• 
  Kirk, William A.; Sims, Brailey (2001). Handbook of Metric Fixed Point Theory. London: Kluwer Academic. ISBN 0-7923-7073-2.
  


• 
  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 0-387-90748-3.