Three point analogue of \'Ciri\'c-Reich-Rus type mappings with non-unique fixed points

TL;DR

This paper introduces a new class of three-point mappings extending Cirić-Reich-Rus type, explores their fixed points, and establishes fixed point theorems in metric spaces without requiring completeness.

Contribution

It defines generalized three-point Cirić-Reich-Rus type mappings, analyzes their fixed points, and extends fixed point theorems under weaker conditions.

Findings

Mappings can be discontinuous but have fixed points with continuity at those points.

Fixed points may be non-unique under these mappings.

Fixed point theorems are established in non-complete metric spaces.

Abstract

In this paper, we introduce a three-point analogue of \'Ciri\'c-Reich-Rus type mappings, termed as generalized \'Ciri\'c-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 \'Ciri\'c-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 \'Ciri\'c-Reich-Rus type mappings in metric spaces that are not necessarily complete.