So I have this function $g(x$) which is continuously differentiable on some domain in $R$ such that there exists a value $c$ with $|g'(c)| \lt 1$ and $g(c)=c$ i.e. the fixed point.
I have already shown, using the mean value theorem and Lipschitz condition that there exists a value d such that for any x in $[c-d,c+d]=I$, $|g'(x)| \lt 1$.
I am now told that I need to show that I can find a value $d$ where $g(I)$ is a subset of the domain $I$ itself. I was wondering whether this would mean finding a d small enough where $g(x) = c$ for all $x$ and proving that the supremum of the difference between the functions $g(y)$ and $g(z)$ for some $y,z$ in $I$ was zero.
Any suggestions?
Seany :)
You have a $d$ such that $|g'(x)| <1$ for all $x \in I$. Then $|g(x)-g(c)| = |g'(\xi)||x-c|$ for some $\xi \in I$, and $|g'(\xi)|<1$. Hence you have $|g(x)-c| < |x-c|$, and consequently $g(x) \in I$. Hence $g(I) \subset I$.