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Generalized fixed point theorems for asymptotically regular mappings in \(b\)-metric spaces via contractive families

Luca Guerrini1
1Department of Management, Polytechnic University of Marche, Piazzale Martelli 8, 60121 Ancona, Italy
Copyright © Luca Guerrini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper introduces a unified framework for fixed point theorems involving asymptotically regular mappings in \(b\)-metric spaces through the concept of contractive families. We establish a general fixed point theorem that encompasses various existing results, including those of Kannan-type and generalized contractive conditions, as special cases. In particular, we demonstrate that the recent results of Nagac and Tas [1] emerge naturally as special cases of our main theorem through appropriate parameter choices. The main result employs coefficient functions and a general auxiliary function with strengthened continuity conditions, providing flexibility that allows the derivation of numerous particular cases. Several corollaries with complete proofs are presented to demonstrate that our results properly generalize and extend well-known theorems in the literature.

Keywords: fixed point, b-metric space, asymptotically regular mapping, Kannan-type contraction