Salvador Romaguera and Pedro Tirado
Remarks on the Quasi-Metric Extension of the Meir-Keeler Fixed Point Theorem With an Application to D3-Systems
Dynamic Systems and Applications 31 (2022) 195-205
https://doi.org/10.46719/dsa202231.03.04
ABSTRACT.
In this paper we first observe that the classical Meir-Keeler fixed point theorem
admits a full extension to complete T1 quasi-metric spaces. Related to this fact we show that the
key example of the paper \On the MeirKeeler theorem in quasi-metric spaces” (J. Fixed Theory Appl.
23:37 (2021)) is not correct. We present a quasi-metric version of a fixed point theorem due to B.
Samet, C. Vetro and H. Yazidi, that involves a Meir-Keeler type contraction and, finally, connections
between our quasi-metric version of the Meir-Keeler theorem and discrete disperse dynamical systems
(D3-systems in short) are discussed.
AMS (MOS) Subject Classification. 47H10, 54H25, 54E50, 54H20.
Key Words and Phrases. Fixed point, complete quasi-metric space, Meir-Keeler theorem,
Samet-Vetro-Yazidi theorem, D3-system.