Here is the poblem: Let $(X,d)$ be a complete metric space and $\omega:\mathbb R_{\geq0}\to\mathbb R_{\geq0}$ is a right-continuous function such that $\omega(0)=0$ and for any $t>0$, one have $0\leq\omega(t)<t$, if $f:X\to X$ is such that $$d(f(x),f(y))\leq\omega(d(x,y))$$ then $f$ admits a unique fixed point.
I think one have to find an appropriate $x\in X$ and prove that $\{f^n(x)\}$ is a Cauchy sequence. But I cannot find a proper estimate of $d(f^n(x),f^m(x))$, for $\omega$ is not monotonic increasing. Indeed there is no way to adjust $\omega$ so that the adjusted function is increasing. The minimal increasing function that is greater than $\omega$ is $\widetilde\omega(x):=\sup_{t\leq x}\omega(x)$ but $\widetilde\omega(t)$ may not be strictly less than $t$ for positive $t$.