Hi,
I need to prove that f(x) has a fixed point, given that f'(x) >= 2 for all x.
my problem is that I've reached the part in which g(x) = f(x) - x
and g'(x) = f'(x) - 1 and therefore g'(x) >= 1
but now I'm completely stuck. I knwo that I need to use the mean value theorem, but i just...
I'm trying to work out how an existence of a fixed point is linked to the constraint on the differential of that function.
For example, i need to prove f has a fixed point if f'(x)=>2.
I understand that what I have is a monotone increasing function so it is 1-1. All the fixed points are...
I have read the following :
"The usual proof of Brouwer's fixed-point theorem makes use of some machinery from simplical homology theory. First we establish that there does not exist a "retraction" of an n-cell onto its boundary, which is to say, there is no continuous mapping from an n-cell...