Three point analogue of Ćirić-Reich-Rus type mappings with non-unique fixed points
Ravindra K. Bisht, Evgeniy Petrov
TL;DR
The paper develops a three-point analogue of Ćirić-Reich-Rus type mappings, introducing generalized ĆRR-type inequalities with parameters $\alpha$ and $\lambda$ and proving that fixed points exist (with at most two) under a no-2-cycle condition in complete metric spaces. It further shows that such mappings are continuous at fixed points and extends the theory via asymptotic regularity to relax the parameter constraints, enabling fixed-point results for generalized $F$-ĆRR-type, as well as $\text{B}$-ĆRR and $(\varphi-F)$ variants. The results are broadened to incomplete metric spaces by establishing subsequence convergence criteria and density-based continuity, thereby yielding two main fixed-point theorems in less restrictive settings. Collectively, the work generalizes classical two-point contractive results to tri-point contractions, introduces several generalized mapping classes, and highlights continuity at fixed points as a robust feature of generalized ĆRR-type mappings.
Abstract
In this paper, we introduce a three-point analogue of Ćirić-Reich-Rus type mappings, termed as generalized Ćirić-Reich-Rus type mappings. We demonstrate that these mappings generally exhibit discontinuity within their domain of definition but necessitate continuity at their fixed points. We showcase the existence and non-uniqueness of fixed points for these generalized Ćirić-Reich-Rus type mappings. By imposing additional conditions, specifically asymptotic regularity and continuity, we extend the applicability of fixed-point theorems to a broader class of mappings. Finally, we obtain two fixed point theorems for generalized Ćirić-Reich-Rus type mappings in metric spaces that are not necessarily complete.