View source for Contracting-mapping principle

''contractive-mapping principle, contraction-mapping principle''

A theorem asserting the existence and uniqueness of a [[Fixed point|fixed point]] of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258101.png" /> of a complete metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258102.png" /> (or a closed subset of such a space) into itself, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258103.png" /> the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

holds, for some fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258106.png" />. This principle is widely used in the proof of the existence and uniqueness of solutions not only of equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258107.png" />, but also of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258108.png" />, by changing the equation to the equivalent: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258109.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581010.png" />.

The scheme of application of the principle is usually as follows: Starting from the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581011.png" /> first find a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581012.png" />, usually a closed ball, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581013.png" />, and then prove that on this set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581014.png" /> is a contractive mapping. After this, starting from an arbitrary element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581015.png" />, construct the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581018.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581019.png" />, which converges to some element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581020.png" />. This will be the unique solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581021.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581022.png" /> will be a sequence of approximate solutions.

In general, condition (1) cannot be changed to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

However, if this condition is satisfied on a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581024.png" /> that is mapped into itself by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581025.png" />, then it guarantees the existence of a unique fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581027.png" />.

The following generalization of the contractive-mapping principle holds. Again, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581028.png" /> map a complete metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581029.png" /> into itself and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581030.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581033.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581034.png" /> has a unique fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581035.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,   G.M. Vainikko,   P.P. Zabreiko,   et al.,   "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Lyusternik,   V.I. Sobolev,   "Elements of functional analysis" , Hindushtan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V. Trenogin,   "Functional analysis" , Moscow  (1980)  (In Russian)</TD></TR></table>



====Comments====
This principle is also known as the contraction principle or Banach's fixed-point theorem. It was proved by S. Banach in [[#References|[a1]]]. The generalization discussed at the end of the article above goes by the name generalized contraction mapping in the sense of Krasnosel'skii [[#References|[a5]]], [[#References|[a6]]]. For this and other generalizations of the idea of a contractive mapping, cf. [[#References|[a4]]], Chapt. 3.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Banach,   "Sur les opérations dans les ensembles abstraits et leurs application aux équations intégrales"  ''Fund. Math.'' , '''3'''  (1922)  pp. 7–33</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Willard,   "General topology" , Addison-Wesley  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.I. Istrăţescu,   "Fixed point theory" , Reidel  (1981)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.A. Krasnosel'skii,   "Topological methods in the theory of nonlinear integral equations" , Macmillan  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.A. Krasnosel'skii,   "Positive solutions of operator equations" , Noordhoff  (1964)  (Translated from Russian)</TD></TR></table>