Contracting-mapping principle

contractive-mapping principle, contraction-mapping principle

A theorem asserting the existence and uniqueness of a 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

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c0258104.png" />

(1)

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

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581023.png" />

(2)

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

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025810/c02581030.png" />

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

[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)