There have appeared thousands of works about the Banach contraction and its extensions on metric spaces (X, d). Recently, many works on the Rus-Hicks-Rhoades (RHR) maps T : X → X satisfying d(T x, T 2x) ≤ α d(x, T x) on x ∈ X with α ∈ [0, 1) also appeared. In the present article, we show that the so-called Suzuki type maps are RHR maps and the proofs of results of Suzuki and his colleagues can be simplified within a few lines based on our recent works on quasi-metric spaces.