Let $[a,b],\,\, [c,d]$ be two bounded closed intervals of $\mathbb{R}$ such that $$[a,b]\cap[c,d]\not=\emptyset$$ and $f:[a,b]\to[c,d]$ be a bijective, smooth function.
We know that, if $[a,b]=[c,d],$ then $f$ has a fixed point.
My question is:
If $[a,b]\not=[c,d],$ which condition(s) guaranteed the existence of a fixed point of $f$ ?
Some hints:
Smoothness and bijectivity together imply monotonicity. Distinguish the four cases $$c<a<d<b,\quad c<a<b<d,\quad a<c<d<b,\quad a<c<b<d\ ,$$ (maybe equality signs have to be considered extra), and for each of these cases a monotonically increasing, resp., decreasing $f$ satisfying the conditions. Draw figures.