Contraction Map: Proving $f(x)$ is a Contraction

This is also known as the Lipschitz condition. In summary, to prove that $f$ is a contraction map, we need to show that it satisfies the Lipschitz condition.
  • #1
onie mti
51
0
I am given this function

$f(x)=\langle (1/9) \cos(x_1+ \sin(x_2)) , (1/6) \arctan(x_1+ x_2) \rangle$

where $x_1= \langle 0,-1 \rangle$.

may I please get hints on how to prove that this function is a contraction map
 
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  • #2
?Hint: To prove that $f$ is a contraction map, you need to show that it satisfies the following property: for all $x,y \in X$, there exists some constant $k \in [0,1)$ such that $\|f(x)-f(y)\| \leq k\|x-y\|$. In other words, the distance between the images of two points must be less than or equal to $k$ times the distance between the two original points.
 

FAQ: Contraction Map: Proving $f(x)$ is a Contraction

What is a Contraction Map?

A Contraction Map is a function in mathematics that maps a metric space onto itself and satisfies the condition that the distance between the images of any two points in the space is always less than the distance between the two original points. This means that the function "contracts" the space, bringing points closer together.

How do I prove that a function $f(x)$ is a Contraction Map?

To prove that a function $f(x)$ is a Contraction Map, you must show that it satisfies the definition of a Contraction Map. This means that you must show that the distance between the images of any two points in the space is always less than the distance between the two original points.

What are the steps to prove that a function $f(x)$ is a Contraction Map?

The steps to prove that a function $f(x)$ is a Contraction Map are as follows:

  1. Choose two arbitrary points in the space, $x_1$ and $x_2$.
  2. Apply the function $f(x)$ to both points to get $f(x_1)$ and $f(x_2)$.
  3. Calculate the distance between the images of the two points, $d = |f(x_1) - f(x_2)|$.
  4. Calculate the distance between the two original points, $d' = |x_1 - x_2|$.
  5. Show that $d < d'$, which proves that $f(x)$ is a Contraction Map.

What are the benefits of proving that a function $f(x)$ is a Contraction Map?

Proving that a function $f(x)$ is a Contraction Map can have many benefits, including:

  • It can help to prove the existence of a unique fixed point for the function, which can be useful in solving equations.
  • It can be used to show that the function is continuous, which is an important property in many mathematical applications.
  • It can help to prove the convergence of a sequence of points that are iteratively mapped by the function.

Can any function $f(x)$ be a Contraction Map?

No, not all functions can be Contraction Maps. In order for a function to be a Contraction Map, it must satisfy the definition of a Contraction Map, which includes the condition that the distance between the images of any two points in the space is always less than the distance between the two original points. If this condition is not met, then the function cannot be a Contraction Map.

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