# How do you prove a contraction mapping?

How do you prove a contraction mapping? If X is a complete metric space and f : X → X is a mapping such that some iterate fN : X → X is a contraction,

## How do you prove a contraction mapping?

If X is a complete metric space and f : X → X is a mapping such that some iterate fN : X → X is a contraction, then f has a unique fixed point. Moreover, the fixed point of f can be obtained by iteration of f starting from any x0 ∈ X. Proof. By the contraction mapping theorem, fN has a unique fixed point.

What do you mean by contraction mapping?

In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some nonnegative real number such that for all x and y in M, The smallest such value of k is called the Lipschitz constant of f.

How do you prove a function is a contraction?

A function f : X → X is called a contraction if there exists k < 1 such that for any x, y ∈ X, kd(x, y) ≥ d(f(x),f(y)). +b) is a contraction if a, c > 1. For a fixed point, we want f(x, y)=(x, y). The Contraction Theorem will specify that the metric space must be complete.

### How do you prove a fixed point theorem?

Let f be a continuous function on [0,1] so that f(x) is in [0,1] for all x in [0,1]. Then there exists a point p in [0,1] such that f(p) = p, and p is called a fixed point for f. Proof: If f(0) = 0 or f(1) = 1 we are done .

Is every f contraction is contractive mapping?

Remark 2.1 From (F1) and (2) it is easy to conclude that every F-contraction T is a contractive mapping, i.e. Thus every F-contraction is a continuous mapping.

Is Sinx a contraction?

The function f (x) = sin x is an example of a weak contraction which is not a contraction, since limx→0(sin x − sin 0)/(x − 0) = 1 and therefore there exists no 0 < k < 1 with | sin x − sin 0| ≤ k|x − 0| for all x = 0.

## What are contractions examples?

A contraction is a word made by shortening and combining two words. Words like can’t (can + not), don’t (do + not), and I’ve (I + have) are all contractions.

What is the importance of fixed point theory?

Metric fixed point theory for important classes of mapping gained respectability and prominence to become a vast field of specialization partly and not only because many results have constructive proofs, but also because it sheds a revealing light on the geometry of normed spaces, not to mention its many applications …

What is an example of a fixed point?

Examples: A continous function that maps [0,1] into itself has a fixed point. A continuous function that maps a disk into itself has a fixed point. A continuous function that maps a spherical ball into itself necessarily has a fixed point.

### What is F contraction?

A mapping T : X → X is called an F-contraction of. Hardy–Rogers-type if there exist τ > 0 and F ∈ F such that. τ + F(d(Tx, Ty)) ≤ F(α · d(x, y) + β · d(x, Tx) + γ · d(y, Ty) + δ · d(x, Ty) + L · d(y, Tx))

What is a contractive sequence?

The sequence. a0,a1,a2,… (1) in a metric space (X,d) is called contractive, iff there is a real number r∈(0,1) r ∈ ( 0 , 1 ) such that for any positive integer n the inequality.

Is COSX a contraction?

To show cosx is a contraction mapping on [0,1], we will use the mean-value theorem: for any differentiable function f, f(x)−f(y) = f (t)(x−y) for some t between x and y, so bounding the derivative of f will give us a contraction constant. Since sine is increasing on [0,1], |sint| = sint ≤ sin 1 ≈ . 84147.