A Note on the Paper “On Completeness in Metric Spaces and Fixed Point Theorems"

Abstract

In the present note, we show that the assumption of continuity used in the fixed point theorem of Gregori et al. (Results Math. 73 (2018), no. 4, Art. 142, 13) can be relaxed to some weaker version of continuity. More precisely, we prove a fixed point theorem for orbitally continuous and k-continuous mappings in weak G-complete metric space and provide an appropriate example to show that our result is not only valid for continuous mappings but also for some discontinuous mappings. Moreover, we apply our main result to establish a common fixed point theorem for two self-mappings