The fixed points of contractions of $f$-quasimetric spaces

The recent articles of Arutyunov and Greshnov extend the Banach and Hadler Fixed-Point Theorems and the Arutyunov Coincidence-Point Theorem to the mappings of $(q_1,q_2)$-quasimetric spaces. This article addresses similar questions for $f$-quasimetric spaces.
Given a function $f\colon\mathbb R_+^2\to\mathbb R_+$ with $f(r_1,r_2)\to0$ as $(r_1,r_2)\to(0,0)$, an $f$-quasimetric space is a nonempty set $X$ with a possibly asymmetric distance function $\rho\colon X^2\to\mathbb R_+$ satisfying the $f$-triangle inequality: $\rho(x,z)\leq f(\rho(x,y),\rho(y,z))$ for $x,y,z\in X$. We extend the Banach Contraction Mapping Principle, as well as Krasnoselskii's and Browder's Theorems on generalized contractions, to mappings of $f$-quasimetric spaces.