Why is the contraction constant important in the Banach contraction principle?

In summary: If you take a contractive map from X to X, then at each point x0, the distance between x0 and y0 will be less than the distance between x0 and any other point in X. Because contractive maps are "tight" (meaning that they preserve distances), this means that the rate of convergence of f^n(x0) to the fixed point will be slower than if we didn't take the contractive map.
  • #1
ozkan12
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0
It was important in the proof of BCP that the contraction constant h be strictly less than 1. That gave us control over the rate of convergence of f^n (x0) to the fixed point since h^n goes to 0 as n goes to infinity...If we consider f is contractive mapping instead of a contraction, then we lose that control and indeed a fixed point need not exist...But I didnt understand this case...how is contraction constant h important ? I don't understand...please talk about this
 
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  • #2
ozkan12 said:
It was important in the proof of BCP that the contraction constant h be strictly less than 1. That gave us control over the rate of convergence of f^n (x0) to the fixed point since h^n goes to 0 as n goes to infinity...If we consider f is contractive mapping instead of a contraction, then we lose that control and indeed a fixed point need not exist.
As far as I know, the term "contractive mapping", as distinct from "contraction", or "contraction mapping", does not have a universally accepted meaning in English. You'll have to give us its definition. Do you mean a "nonexpansive map"?

ozkan12 said:
how is contraction constant h important ?
You said it yourself (I am guessing): if $h\ge 1$, a fixed point need not exist.

ozkan12 said:
I don't understand...
What exactly?
 
  • #3
no I say contractive mapping...the definition of contractive mapping: let f:X to X be self mapping on (X,d) metric space
if d(fx,fy)<d(x,y) then f is contractive mapping and note that in this concept 'h' contraction constant equal to 1 and inequality is strict inequality
 
  • #4
ozkan12 said:
let f:X to X be self mapping on (X,d) metric space
if d(fx,fy)<d(x,y) then f is contractive mapping
OK. So, what is your question?
 
  • #5
It was important in the proof of BCP that the contraction constant h be strictly less than 1. That gave us control over the rate of convergence of f^n (x0) to the fixed point since h^n goes to 0 as n goes to infinity...If we consider f is contractive mapping instead of a contraction, then we lose that control and indeed a fixed point need not exist...

how this happened ? that is why if we choose contractive map we lost control of rate of convergence of f^n(x0)

x0 is arbitrary point in (X,d) metric space
 

FAQ: Why is the contraction constant important in the Banach contraction principle?

What is the Banach contraction principle?

The Banach contraction principle, also known as the Banach fixed point theorem, is a mathematical theorem used to prove the existence and uniqueness of a fixed point for a mapping in a complete metric space.

How does the Banach contraction principle work?

The principle states that if a mapping on a complete metric space satisfies a certain contraction condition, then it has a unique fixed point. This means that the mapping will continuously "contract" the space, bringing points closer and closer to the fixed point until they eventually converge to it.

What is the importance of the Banach contraction principle?

The Banach contraction principle is important in various fields of mathematics, such as functional analysis, differential equations, and optimization. It provides a powerful tool for proving the existence and uniqueness of solutions to problems that involve mappings in metric spaces.

What is the contraction condition in the Banach contraction principle?

The contraction condition in the Banach contraction principle is a mathematical inequality that the mapping must satisfy. It states that the distance between the images of any two points in the space must always be smaller than the distance between the points themselves, multiplied by a constant less than 1.

What are some applications of the Banach contraction principle?

The Banach contraction principle has various applications in fields such as economics, physics, and computer science. It can be used to solve optimization problems, study the behavior of dynamical systems, and prove the convergence of numerical algorithms. It also has applications in the field of artificial intelligence, for example in proving the existence of Nash equilibria in game theory.

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