Revisit to Suzuki's Metric Completenes

Abstract

There have appeared a large number of articles related to the metric completeness. They are mainly concerned with generalizations of the Banach contraction on metric spaces and their modified artificial spaces. A well-known fixed point theorem of Suzuki in 2008 for the so-called Suzuki type map on a complete metric space with a lengthy proof is very popular and has a large number of followers. Such results on metric spaces are consequences of our generalized forms of the Banach contraction principle for weak contractions or the Rus-Hicks-Rhoades maps on quasi-metric spaces. In this paper, we collect the works of Suzuki and his colleagues on metric completeness and give short proofs and improvements for them. Moreover, we collect some positive but incorrect comments given by many followers of Suzuki.