Some critical remarks on recent results concerning $\digamma-$contractions in $b$-metric spaces

Abstract

This paper aims to correct recent results on a generalized class of $\digamma-$contractions in the context of $b-$metric spaces. The significant work consists of repairing some novel results involving $\digamma-$contraction within the structure of $b$-metric spaces. Our objective is to take advantage of the property $(F1)$ instead of the four properties viz. $(F1)$, $(F2)$, $(F3)$ and $(F4)$ applied in the results of Nazam \textit{et al.} [``Coincidence and common fixed point theorems for four mappings satisfying $(\alpha_s,F)-$contraction", Nonlinear Anal: Model. Control., vol. 23, no. 5, pp. 664--690, 2018]. Our approach of proving the results utilizing only the condition $(F1)$ enriches, improves, and condenses the proofs of a multitude of results in the existing state-of-art.