Banach Fixed Point theorem states: Let $(X,d)$ be a complete metric space. Suppose that $f:X→X$ is a strong contraction, i.e. there exists $q ∈ [0, 1)$ such that
$d(f(x),f(y))$ $\le$ $q$ $d(x,y)$, then there is a unique point $x_0∈X$ s.t. $f(x_0)=x_0$
My questions are:
1- If we allow $q$ to be equal to $1$, does the theorem fail? Could someone provide an example?
2- If we substitute the strong contraction condition with the following condition: $d(f(x),f(y))$ $<$ $d(x,y)$, does the theorem fail? example?