# help with function iteration

Discussion in 'Math' started by lital, Dec 2, 2010.

1. ### lital Thread Starter New Member

Dec 2, 2010
2
0
hi all, i need help with this
prove or disprove :
if F:R -> [a,b] is contractive on [a,b] then F has a unique fixed point,
which can be obtained by function iteration staring an any real value.

2. ### Papabravo Expert

Feb 24, 2006
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1,850
What is the meaning of "contractive". I'm not familiar with that term.
Also in this context what is the meaning of a "fixed point"

3. ### lital Thread Starter New Member

Dec 2, 2010
2
0
Contraction mapping mean that |f(x)-f(y)| < k|x-y| when 0<k<1.
fixed point mean that for some s , f(s)=s

4. ### Georacer Moderator

Nov 25, 2009
5,151
1,266
That might be true, given that you need g'(x)<1, x in [a,b] in order for the function to be contractive. Since all the potential fixed points lie on the line f(x)=x which has f'=1, our function g can intercept f only once.

Intercepting it twice would mean that in some interval its derivative would be >1 and that is unacceptable, since g is contractive in the whole space [a,b] not in some subspaces.

5. ### Papabravo Expert

Feb 24, 2006
10,340
1,850
So a contractive mapping is one in which the points in the domain are brought closer together in the range. In differential equations a fixed point is one where the velocity goes to zero.

On Wikipedia there is a development of the Banach Fixed Point Theorem that you might find useful. I'm close to being able to follow it.

http://en.wikipedia.org/wiki/Banach_fixed_point_theorem