$c$-class functions

The aim of this paper is to shed some light on the recently introduced generalized Sehgal-Guseman-like contractions with the help of D-functions from other aspects. For this purpose, we used the recently established connection of S-metric and b-metric spaces and thus switched to a new form of generalized Sehgal-Guseman-like contraction. Using that connection as well as the notion of a D-function, we were able to significantly improve recent results on generalized Sehgal-Guseman-like contractions and remove some of the confusion readers had when reading recent work on this topic. At the end of the paper, we listed all 120 possible generalized Sehgal-Guseman-like contractions within both S-metric and b-metric spaces. We admit that for many of them, we do not yet know whether they have a (unique) fixed point. This opens the way for further research of generalized Sehgal-Guseman-like contractions in the mentioned frameworks of certain generalized metric spaces.

‎In this paper‎, ‎we use the concept of $C$-class functions introduced‎ ‎by Ansari [4] to prove the existence and uniqueness of‎ ‎common fixed point for self-mappings in modular spaces of integral‎ ‎inequality‎. ‎Our results extended and generalized previous known‎ ‎results in this direction‎.