In mathematics, the contraction mapping principle is considered one of the most valuable tools used in studying nonlinear equations, such as algebraic equations, integral equations or differential equations. This mapping principle is a fixed point theorem that ensures that a contraction mapping of a complete metric space contains a unique fixed point that. Thus, we may get this fixed point as the limit of an iteration method described by replicated images under the mapping of a random starting point in the metric space. It is also called a constructive fixed point theorem, and it can be used for the numerical calculation of the fixed point.

Learn: Metric space

Banach Contraction Mapping Principle

In real analysis, the contraction mapping principle is often known as the Banach fixed point theorem.

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

Furthermore, if y ∈ T is randomly chosen, then the iterates {x_n}^∞_n=0, given by x₀ = y and x_n = T(x_n−1), n ≥ 1, have the property that lim_n→∞ x_n = x.

Alternatively, we can state the contraction mapping theorem as follows.

Let (X, d) be a complete metric space and f : X → X be a map 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.

Moreover, 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 called a contraction.

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

Contraction Mapping Theorem Proof

Let x₀ be any point in X, i.e., x₀ ∈ X.

Let us define a sequence (x_n) in X as given below.

x_n+1 = Tx_n for all n ≥ 0

Let’s avoid the parentheses around the argument of a map to simply the notation.

Let Tⁿ be the nth iteration of T.

So, x_n = Tⁿ x₀

Let us prove that (x_n) is a Cauchy sequence.

Suppose n ≥ m ≥ 1 then from the contraction and triangle inequality theorems, we can write as:

d(x_n, x_m) = d(Tⁿ x₀, T^m x₀)

≤ c^m d (T^n-m x₀, x₀)

≤ c^m [d (T^n-m x₀, T^n-m-1 x₀) + d (T^n-m-1 x₀, T^n-m-2 x₀) + d (T^n-m-2 x₀, T^n-m-3 x₀) + …. + d (Tx₀, x₀)]

This can be simplified as:

Thus, we can say that (x_n) is a Cauchy sequence.

As X is complete, (x_n) converges to a limit x ∈ X.

Tx = T lim_n→∞ x_n = lim_n→∞ Tx_n = lim_n→∞ x_n+1 = x

Thus, for any two fixed points c and y, we have

0 ≤ d(x, y) = d(Tx, Ty) ≤ cd(x, y)

As we know, c < 1.

Then, d(x, y) = 0.

From this, we can write x = y.

That means the fixed point is unique.

Contraction Mapping Solved Example

Question:

Prove that every contraction mapping is continuous.

(or)

Show that every contraction mapping on a metric space is uniformly continuous.

Solution:

Let T : X → X be a contraction on a metric space (X, d), with modulus β and x ∈ X.

Let y ∈ X and ε > 0.

Also, assume that δ = ε.

Then, d(x, y) < δ

⇒ d(Tx, Ty) ≤ βδ < ε.

Thus, T is continuous at y since y was arbitrary.

Therefore, T is uniformly continuous on X.

Hence, it is proved that every contraction mapping is continuous.