Fixed Point Iteration Method, an example

Question

I am seeking an example of a bounded function $g:I\to \mathbb{R}$, $I\subset \mathbb{R}$ a non empty segment, such that

1. $|g'(x)|>1$
2. the subsequence $(x_n)$ defined by $$ x_0\in I;\forall n\in \mathbb{N}\;x_{n+1}=g(x_n) $$ is a convergent sequence and its limit $l$ satisfies $l\in I$ and $g(l)=l$
3. $g(I)\subset I$

Answer 1

If you assume that $g$ is of class $C^1$ and $|g'(x)|>1, x \in I$, $g$ will satisfy either $g'(x)>1, x \in I$ or $g'(x) < -1, x \in I$. In both cases you cannot have $g(I)\subset I$.

If you think for instance that $g'(x)>1, x \in I$, you have that $g(b)-g(a) = g'(\xi) (b-a)> b-a$. But if $g(b)-g(a)>b-a$, you cannot have $g([a,b])\subset [a,b]$.