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.
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"
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
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.
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