Let $f:X\rightarrow X$ be a contractive mapping of a complete metric space satisfying
$$d(f(x),f(y))\leq\alpha(d(x,y))d(x,y)$$
where $\alpha:\mathbf{R}^+\rightarrow [0,1)$, and $\alpha(t_n)\rightarrow 1$ implies $t_n\rightarrow 0$

Does $f$ have a fixed point?

Thank you.