Let $f$ be a continuous map from $X$ to $X$ (compact metric space). I know that we always have a set $A$ such that $f(A)=A$.
I want to know when it has a fixed point.
Similarly, if $f$ is a continuous map from $X$ (complete metric space) to $X$ such that $d(f(x),f(y)) < d(x,y)$, then it has a fixed point. However is there any other condition except this which will ensure $f$ to have a fixed point?
I am in general looking for different conditions under which a fixed point exist for a continuous map.