Contracting-mapping principle
contractive-mapping principle, contraction-mapping principle
A theorem asserting the existence and uniqueness of a fixed point of a mapping of a complete metric space
(or a closed subset of such a space) into itself, if for any
the inequality
![]() | (1) |
holds, for some fixed ,
. This principle is widely used in the proof of the existence and uniqueness of solutions not only of equations of the form
, but also of equations
, by changing the equation to the equivalent:
, where
.
The scheme of application of the principle is usually as follows: Starting from the properties of first find a closed set
, usually a closed ball, such that
, and then prove that on this set
is a contractive mapping. After this, starting from an arbitrary element
, construct the sequence
,
,
belonging to
, which converges to some element
. This will be the unique solution of the equation
, and
will be a sequence of approximate solutions.
In general, condition (1) cannot be changed to
![]() | (2) |
However, if this condition is satisfied on a compact set that is mapped into itself by
, then it guarantees the existence of a unique fixed point
for
.
The following generalization of the contractive-mapping principle holds. Again, let map a complete metric space
into itself and let
![]() |
for , where
for
. Then
has a unique fixed point in
.
References
[1] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) |
[2] | M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian) |
[3] | L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Hindushtan Publ. Comp. (1974) (Translated from Russian) |
[4] | V. Trenogin, "Functional analysis" , Moscow (1980) (In Russian) |
Comments
This principle is also known as the contraction principle or Banach's fixed-point theorem. It was proved by S. Banach in [a1]. The generalization discussed at the end of the article above goes by the name generalized contraction mapping in the sense of Krasnosel'skii [a5], [a6]. For this and other generalizations of the idea of a contractive mapping, cf. [a4], Chapt. 3.
References
[a1] | S. Banach, "Sur les opérations dans les ensembles abstraits et leurs application aux équations intégrales" Fund. Math. , 3 (1922) pp. 7–33 |
[a2] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) |
[a3] | S. Willard, "General topology" , Addison-Wesley (1970) |
[a4] | V.I. Istrăţescu, "Fixed point theory" , Reidel (1981) |
[a5] | M.A. Krasnosel'skii, "Topological methods in the theory of nonlinear integral equations" , Macmillan (1964) (Translated from Russian) |
[a6] | M.A. Krasnosel'skii, "Positive solutions of operator equations" , Noordhoff (1964) (Translated from Russian) |
Contracting-mapping principle. V.I. Sobolev (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contracting-mapping_principle&oldid=12147