I am studying a paper on quasi-metric spaces for the complexity space, and I found that Banach's famous result about fixed points in complete metric spaces can be extended to bicomplete quasi-metric spaces in an analogous way. Wanting to understand how to get there, I found the following paper, which studies theorems of fixed points in bicomplete quasi-metric spaces.
Without going into much detail I came across the following theorem
Theorem 2.7. Let $(X,d)$ be a bicomplete quasi-metric space, $T: X \to X$, and $\eta, \psi:\mathbb{R}^+ \to \mathbb{R}^+$ be functions such that $\psi$ is non-decreasing and continuous, $\eta$ is lower semicontinuous, $\eta^{-1}(0) = \psi^{-1}(0) = \{ 0\}$ and $$ \psi(d(Tx,Ty)) \leq \psi(d(x,y)) - \eta(d(x,y)) $$ for all $x,y \in X$. Then $T$ has a unique fixed point.
and with the following corollary
Corollary 2.9. Let $(X,d)$ be a bicomplete quasi-metric space, $T: X \to X$, $\psi:\mathbb{R}^+ \to \mathbb{R}^+$ be a non-decreasing and continuous function with $\psi^{-1} = \{ 0\}$ and $\alpha \in [0,1)$ be a constant such that $$ \psi(d(Tx,Ty)) \leq \alpha \psi(d(x,y)) $$ for all $x,y \in X$. Then $T$ has a unique fixed point.
The article says that the corollary arises from applying Theorem 2.7 with $\eta:\mathbb{R}^+ \to \mathbb{R}^+$ given by $\eta (t) = (1-c)t$ for all $t \in \mathbb{R}^+$, but I have thought long and hard and I don't understand how to arrive at the inequality of Theorem 2.7, or where $c$ comes from.
If someone could explain to me how the corollary arises, I would appreciate it.