Let $\left(X,d\right)$ be a complete metric space and suppose that $f:X\to X$ satisfies
$d\left(fx,fy\right)\le\beta\left(d\left(x,y\right)\right)d\left(x,y\right)$ for all x,y $\in$ X where $\beta$ is a decreasing function on ${R}^{+}$ to $[0,1)$. Then $f$ has a unique fixed point.
The...