On the weakest conditions for the existence of fixed points of Kannan and Chatterjea type contractions

TL;DR

This paper investigates the minimal conditions needed for fixed point theorems involving Kannan and Chatterjea contractions, extending Suzuki's approach and establishing optimality of these conditions for convergence.

Contribution

It extends Suzuki's approach to Kannan and Chatterjea mappings, identifying optimal conditions for fixed point existence and Picard sequence convergence.

Findings

Established the equivalence of known conditions for these mappings.

Proved the optimality of the new conditions for convergence.

Extended the CJM condition framework to Kannan and Chatterjea contractions.

Abstract

In this paper, we study the weakest possible conditions for fixed point theorems involving two classes of mappings defined by Kannan and Chatterjea. Our approach relies on the so-called CJM condition, which was originally introduced by \'Ciri\'c [5], and later, Suzuki [18] showed that the CJM condition is necessary for the existence of fixed points and the convergence of all Picard sequences of Banach type mappings. Our aim is to extend Suzuki's approach to the case of Kannan and Chatterjea mappings. In particular, in the first case, we discuss the equivalence of previously known conditions and establish that our conditions are optimal for ensuring that all Picard sequences converge to a fixed point of a mapping.