How do i find k in the banach fixed point theorem

In summary, to find k in the Banach fixed point theorem for the function f(x)=1+3x-x^2 in the interval [1,2], you can apply the definition of a fixed point where k is a value such that F(k) = k. Then, using algebra, solve the equation F(k) = 1-3k-k^2 = k to find the value of k.
  • #1
sara_87
763
0
how do i find k in the banach fixed point theorem.
so say i have a function f(x)=1+3x-x^2 in the interval [1,2]
then how do i find k?

thank you
 
Mathematics news on Phys.org
  • #2
It looks to be obvious that that function is a contraction, so apply the definition of a fixed point (k [tex]\in[/tex] [1,2] such that F(k) = k).

TeX is being weird so I don't know if what I said shows up properly.
 
Last edited:
  • #3
jhicks said:
It looks to be obvious that that function is a contraction, so apply the definition of a fixed point (k [tex]\in[/tex] [1,2] such that F(k) = k).

TeX is being weird so I don't know if what I said shows up properly.

yes but how do i find that k?
 
  • #4
you know that f(k) = k. It's just algebra. What is f(k)?
 
  • #5
but what do i put as k?
what value of x isit?
 
  • #6
k is what you are trying to find! F(k)= 1- 3k- k3 so F(k)= k is just 1- 3k- k2= k. Solve that equation!
 

Related to How do i find k in the banach fixed point theorem

1. What is the Banach fixed point theorem?

The Banach fixed point theorem is a mathematical theorem that guarantees the existence and uniqueness of a fixed point for certain types of functions. It is an important tool in the study of iterative methods and is widely used in various fields of mathematics and science.

2. How do I apply the Banach fixed point theorem?

In order to apply the Banach fixed point theorem, you need to have a function that satisfies certain conditions, such as being a contraction mapping on a complete metric space. Once you have identified such a function, you can use the theorem to prove the existence of a fixed point.

3. What is the role of "k" in the Banach fixed point theorem?

The variable "k" in the Banach fixed point theorem represents the Lipschitz constant of the function. It is a measure of how much the function can contract the distance between two points in the metric space. A smaller value of "k" indicates a stronger contraction and makes it easier to apply the theorem.

4. How do I find the value of "k" in the Banach fixed point theorem?

Finding the exact value of "k" in the Banach fixed point theorem can be challenging and often requires advanced mathematical techniques. However, in some cases, it can be approximated by using numerical methods or by using theorems that provide upper bounds for its value.

5. Can the Banach fixed point theorem be used to find fixed points in all types of functions?

No, the Banach fixed point theorem can only be used for certain types of functions, such as contraction mappings. It cannot be applied to all types of functions and may not always guarantee the existence of a fixed point.

Similar threads

  • General Math
Replies
1
Views
786
Replies
9
Views
4K
Replies
7
Views
1K
Replies
4
Views
698
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
Replies
4
Views
532
Replies
4
Views
2K
  • General Math
Replies
3
Views
866
Back
Top