Remarks on the fixed point theory for quasi-metric spaces

[EN] Motivated by a recent and interesting article by S. Park [Results in Nonlinear Analysis 6 (2023) No. 4, 116¿127], we recall several different notions of quasi-metric completeness that appear in the literature and revise how they influence on the fixed point theory in quasi-metric spaces. In particular, we point out that there are several classical fixed point theorems that cannot be directly transferred to the quasi-metric setting without extra conditions, when Park's approach is considered. We also recall some emblematic examples that can help to clarify some aspects of the fixed point theory for these spaces.