# Thread: Banach Fixed Point Theorem

1. ## Banach Fixed Point Theorem

How do you show that the fixed point is unique?

I would rather have hints and tips so that I can try and work through it! But I just don't know where to start!

2. Originally Posted by Furbylicious
How do you show that the fixed point is unique?

I would rather have hints and tips so that I can try and work through it! But I just don't know where to start!
A contraction is a function $f$ such that $d(f(x),f(y))\leq kd(x,y)$ for $k<1$.

Say that $x_0$ is a fixed point (so that $f(x_0)=x_0$). Assume that it is not unique; that is, there is another fixed point $y_0$.

Now look at $d(x_0,y_0)$. What can you say?

3. What do you mean it's unique? That depends on the function. The function $f(x)=x$ has infinitely many fixed points on any interval.

4. I used proof by contradiction.

Let phi: A->A be a contraction with constant k<1
Assume that there exists 2 fixed points
phi(x1) = x1
phi(x2) = x2

According to the definition

|phi(x1) - phi(x2)|<=k|x1-x2|

as phi(x1) = x1 and phi (x2) = x2 then

|x1-x2|<=k|x1-x2|

which gives

1<= k
which contradicts the original condition that k<1....

Am i fumbling along the right lines, or is it all rubbish?

5. That's correct.