Generalizations of metric spaces: from the fixed-point theory to the fixed-circle theory

Abstract

This paper is a research survey about the fixed-point (resp. fixed-circle) theory on metric and some generalized metric spaces. We obtain new generalizations of the well-known Rhoades’ contractive conditions, Ćiri ć’s fixed-point result and Nemytskii-Edelstein fixed-point theorem using the theory of an Sb-metric space. We present some fixed-circle theorems on an Sb -metric space as a generalization of the known fixed-circle (fixed-point) results on a metric and an S-metric space.