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

$f$ has a fixed point ??

Thank you  .

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.