Home / Maths / Contraction Mapping Principle - Understanding Fixed Point Theorem

Contraction Mapping Principle - Understanding Fixed Point Theorem

Last Updated on Jul 31, 2023
Download As PDF
IMPORTANT LINKS
30 in words 50 in words 70 in words 40 in words Midpoint Formula Square Root 45000 in words Cube Root 1999 in roman numerals 13 in roman numerals 200 in roman numerals 70 in roman numerals Factors of 27 Factors of 16 Factors of 120 Square Root and Cube Root Squares and Square roots Types of Function Graphs Right Triangle Congruence Theorem 80 in Words

The world of mathematics is filled with fascinating concepts, and one such gem is the contraction mapping principle. This principle is an essential tool for studying nonlinear equations like algebraic equations, integral equations, or differential equations. The contraction mapping principle is a fixed point theorem that guarantees that a contraction mapping of a complete metric space holds a unique fixed point. This fixed point can be obtained as the limit of an iteration method described by replicated images under the mapping of a random starting point in the metric space. Owing to its constructive nature, it is also known as a constructive fixed point theorem and can be utilized for the numerical calculation of the fixed point.

Explore More: Metric space

Banach's Take on the Contraction Mapping Principle

In the realm of real analysis, the contraction mapping principle often goes by the name of the Banach fixed point theorem.

Statement: If T : X → X is a contraction mapping on a complete metric space (x, d), then there exists exactly one solution for T(x) = x for x ∈ X.

Additionally, if we randomly choose y ∈ T, then the iterates {xn}∞n=0, given by x0 = y and xn = T(xn−1), n ≥ 1, exhibit the property that limn→∞ xn = x.

Here's another way to state the contraction mapping theorem.

If we have a complete metric space (X, d) and a map f : X → X such that d(f(x), f(x’)) ≤ cd(x, x’) for some 0 ≤ c < 1 and for all x, x’ ∈ X, then f has a unique fixed point in X. For any x0 ∈ X the sequence of iterates, say x0, f(x0), f(f(x0)), . . . converges to the fixed point of f. If d(f(x), f(x’)) ≤ cd(x, x’) for some 0 ≤ c < 1 and all x and x’ in X, then f is considered a contraction.

A contraction essentially shrinks spaces by a constant factor c < 1 for all pairs of points.

Proof of the Contraction Mapping Theorem

Example of Contraction Mapping

Explore Further:

More Articles for Maths

Frequently Asked Questions

The contraction mapping theorem states that every contraction mapping on a complete metric space contains a unique fixed point. This theorem or principle is also called the Banach fixed point theorem.

Contraction mapping is used in the fixed point theorem

In real analysis, a contraction mapping on a metric space (X, d) is a function f from M to M, i.e., f : M → M, such that there exists some non-negative real number for all x, y ∈ X, which means x and y are in X.
Test Series
131.6k Students
NCERT XI-XII Physics Foundation Pack Mock Test
323 TOTAL TESTS | 5 Free Tests
  • 3 Live Test
  • 163 Class XI Chapter Tests
  • 157 Class XII Chapter Tests
Get Started
Report An Error
Exams Tests Super Practice