A three point extension of Chatterjea's fixed point theorem with at most two fixed points

TL;DR

This paper introduces a new class of three-point Chatterjea type mappings in metric spaces, establishing fixed point theorems and exploring their relationships with other contraction mappings, even without requiring space completeness.

Contribution

It presents a novel three-point extension of Chatterjea's fixed point theorem, differentiates it from existing classes, and proves fixed point results without assuming metric space completeness.

Findings

Established fixed point theorems for generalized Chatterjea type mappings.

Showed these mappings are continuous at fixed points.

Connected Chatterjea type mappings with Kannan type and triangle contraction mappings.

Abstract

In this paper, we introduce a new category of mappings within metric spaces, specifically focusing on three-point analogs of the well-established Chatterjea type mappings. We demonstrate that Chatterjea type mappings and their three-point analogs are different classes of mappings. A fixed point theorem for generalized Chatterjea type mappings is established. It is shown that these mappings are continuous at fixed point. Connections between generalized Chatterjea type mappings, generalized Kannan type mappings, Chatterjea type mappings, and mappings contracting perimeters of triangles are found. Additionally, we derive two fixed point theorems for generalized Chatterjea type mappings in metric spaces, even when completeness is not mandatory.