How to confirm that if a function $f\circ f$ is a strong contraction, then $f$ has a fixed point or not?

Question

Suppose that $(X,d)$ is a complete metric space and $f:X \rightarrow X$ is such that $f\circ f$ is a strong contraction. Must $f$ have a fixed point?

So, it is given that $f\circ f$ is a strong contraction. Therefore,

$$d\left(f\circ f(x),f\circ f(y)\right)\leq cd(x,y)$$ for $0<c<1$ and also there exist an $x_0 \in X$ such that $f\circ f(x_0)=x_0$. I want to know whether $f$ has a fixed point or not and please give some hint to prove it.

Thanks!

Answer 1

Hint

By hypothesis $f\circ f$ has unique fixed point $x_0$. We have $f\circ f\circ f(x_0)=f(x_0)$ hence $f(x_0)$ is a fixed point for $f\circ f$. Can you take it from here?