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Fixed point theorem for generalized Chatterjea type mappings

Ovidiu Popescu, Cristina Maria Păcurar

TL;DR

The paper introduces generalized Chatterjea type mappings, a three-point analogue of Chatterjea mappings, extending triangle-perimeter contraction ideas and linking to Kannan-type mappings. A key contribution is a fixed point theorem: in a complete metric space with at least three points, if a generalized Chatterjea type mapping avoids 2-periodic points, it possesses a fixed point, with at most two fixed points; the proof uses a triple-distance contraction to show a Cauchy forward orbit and convergence to a fixed point. The authors also provide explicit examples of generalized Chatterjea type mappings that are not Chatterjea or Kannan mappings and may be discontinuous, illustrating the independence of the new class. This broadens fixed point theory by enabling discontinuous, triangle-based contraction mappings and clarifies the relationships among Chatterjea, Kannan, and Petrov-type results.

Abstract

We introduce a new type of mappings in metric space which are three-point analogue of the well-known Chatterjea type mappings, and call them generalized Chatterjea type mappings. It is shown that such mappings can be discontinuous as is the case of Chatterjea type mappings and this new class includes the class of Chatterjea type mappings. The fixed point theorem for generalized Chatterjea type mappings is proven.

Fixed point theorem for generalized Chatterjea type mappings

TL;DR

The paper introduces generalized Chatterjea type mappings, a three-point analogue of Chatterjea mappings, extending triangle-perimeter contraction ideas and linking to Kannan-type mappings. A key contribution is a fixed point theorem: in a complete metric space with at least three points, if a generalized Chatterjea type mapping avoids 2-periodic points, it possesses a fixed point, with at most two fixed points; the proof uses a triple-distance contraction to show a Cauchy forward orbit and convergence to a fixed point. The authors also provide explicit examples of generalized Chatterjea type mappings that are not Chatterjea or Kannan mappings and may be discontinuous, illustrating the independence of the new class. This broadens fixed point theory by enabling discontinuous, triangle-based contraction mappings and clarifies the relationships among Chatterjea, Kannan, and Petrov-type results.

Abstract

We introduce a new type of mappings in metric space which are three-point analogue of the well-known Chatterjea type mappings, and call them generalized Chatterjea type mappings. It is shown that such mappings can be discontinuous as is the case of Chatterjea type mappings and this new class includes the class of Chatterjea type mappings. The fixed point theorem for generalized Chatterjea type mappings is proven.
Paper Structure (2 sections, 4 theorems, 48 equations, 1 figure)

This paper contains 2 sections, 4 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalized Chatterjea type mappings

Key Result

Theorem 1.1

Let $(X,d)$, $|X|\geq 3$ be a complete metric space and let $T:X\to X$ be a mapping contracting perimeters of triangles on $X$. Then, $T$ has a fixed point if and only if $T$ does not possess periodic points of prime period $2$. The number of fixed points is at most $2$.

Figures (1)

  • Figure 1: A generalized Chatterjea mapping

Theorems & Definitions (13)

  • Definition 1.1: Petrov Petrov
  • Theorem 1.1: Petrov Petrov
  • Definition 1.2: Petrov and Bisht Petrov-Kannan
  • Theorem 1.2: Petrov and Bisht Petrov-Kannan
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.1
  • Example 2.1
  • Theorem 2.1
  • proof
  • ...and 3 more