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

Shunya Hashimoto, Misako Kikkawa, Shuji Machihara, Aqib Saghir

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 Ćirić [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.

On the weakest conditions for the existence of fixed points of Kannan and Chatterjea type 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 Ćirić [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.
Paper Structure (5 sections, 7 theorems, 49 equations)

This paper contains 5 sections, 7 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. The CJM version of a Kannan-type contraction mapping
  3. The weakest condition for a Kannan-type contraction mapping
  4. The case of Chatterjea type contractions
  5. Discussion on the weakest conditions

Key Result

Theorem 2.1

Let $(X,d)$ be a complete metric space and suppose that the mapping $T: X\to X$ satisfies the following. Then, $T$ has a unique fixed point.

Theorems & Definitions (14)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Remark 3.1
  • proof : Proof of Theorem \ref{['3.1']}
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Proposition 5.1
  • ...and 4 more