Let $f:[0,1]\to[0,1]$ be a smooth, convex (downward) function satisfying $$ f(0)=f(1)=1,\quad \lim_{x\to 0}f'(x)=-\infty,\quad \lim_{x\to 1}f'(x)=+\infty. $$
I am confident to be able to argue that $f$ has exactly two fixed points in $[0,1]$ (one of them being $1$, of course.)
I would like to show that for any starting value $x\in (0,1)$, the sequence of function iterates $f(x), f(f(x)),\ldots$ converges to the fixed point which is not $1$.
I know from the convexity of $f$ that there exist $0 I was thinking to try and argue that for any starting value the iterates $f^i(x)$ would eventually lie in $(x_-,x_+)$ and to then apply Banach's fixed point theorem. My questions are: Thank you. Edit: Thanks to the efforts of richard and froggie it now seems that convergence of the iterates cannot be guaranteed under the conditions specified above. I would therefore like to add the following assumptions: ($p$ denotes the fixed point which is not $1$) I think that with these additional assumptions it should be possible to prove convergence of the function iterates from every starting point.