Contraction Maps in Pseudometric Structures

In Sect. 1, an extension to semigroup couple metric spaces is given for the fixed point result in Matkowski (Diss Math 127:1–68, 1975). In Sect. 2, we show that the simulation-type contractive maps in quasi-metric spaces introduced by Alsulami et al. (Discrete Dyn Nat Soc 2014, Article ID 269286, 2014) are in fact Meir–Keeler maps. Finally, in Sect. 3, the Brezis–Browder ordering principle (Adv Math 21:355–364, 1976) is used to get a proof, in the reduced axiomatic system (ZF-AC+DC), of a fixed point result [in the complete axiomatic system (ZF)] over Cantor complete ultrametric spaces due to Petalas and Vidalis (Proc Am Math Soc 118:819–821, 1993). The methodological approach we chose consisted in treating each section from a self-contained perspective; so, ultimately, these are independent units of the present exposition.