Significance of F ‐Contraction After Proinov’s Fixed‐Point Results

Wardowski laid out his idea of an F ‐contraction mapping, which provides an imperative generalization of the Banach contraction principle, back in 2012. Although Wardowski’s work has generated considerable interest in fixed‐point theory, motivating many researchers to extend these results to more general spaces, to investigate broader classes of contraction mappings and to relax some of the original assumptions imposed on the auxiliary function. Wardowski’s fixed‐point theorems, along with those of Jleli and Samet (2014), are merely variations of a 1977 Skof classical fixed‐point determination, as explained by Proinov’s 2020 interpretation in the very well‐known journal in fixed‐point. Additionally, Proinov claimed that one of the key results in his own paper provides an improvement over Wardowski’s original F ‐contraction imposed on the auxiliary functions, they also have a disruptive and controversial impact on the theoretical foundation of many existing results related to F ‐contraction. In this work, we show by means of a counterexample that, in the absence of an essential condition on the auxiliary function, Wardowski’s F ‐contraction is neither a special case of Skof’s fixed‐point theorem nor an immediate consequence of Proinov’s fixed‐point theorem. Moreover, inspired by the concept of generalized F − expanding mapping introduced by Shahid et al. (published in Adv. Differ. Equ., 2021), we introduced ( φ , ψ , χ )−Suzuki expanding mapping, generalized ( φ , ψ )−Suzuki expanding mapping and established fixed‐point results for these generalized classes of mappings. MSC2020 Classification : 47H10, 47H19, 54H25