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On three-point generalizations of Banach and Edelstein fixed point theorems

Christian Bey, Evgeniy Petrov, Ruslan Salimov

TL;DR

This paper addresses three-point generalizations of fixed-point theory in metric spaces by introducing $(F,G)$-contracting mappings, which generalize triangle-perimeter contractions, and by formulating Edelstein-type contractions for triangle perimeters. It proves that $(F,G)$-contracting mappings are continuous and satisfy a fixed-point theorem (with potential uniqueness under additional hypotheses), thereby extending Banach-type results to a broader three-point framework. It also develops an Edelstein-type fixed-point theorem for mappings contracting perimeters of triangles, showing existence of a fixed point (at most two) in complete spaces and providing a constructive convergence argument via decreasing perimeters, along with discussion of accumulation points and compact cases. Collectively, the results broaden the fixed-point toolkit for nonlinear three-point mappings and deepen connections between triangle-perimeter contraction and Edelstein-type phenomena.

Abstract

Let $X$ be a metric space. Recently in~[1] it was considered a new type of mappings $T\colon X\to X$ which can be characterized as mappings contracting perimeters of triangles. These mappings are defined by the condition based on the mapping of three points of the space instead of two, as it is adopted in many fixed-point theorems. In the present paper we consider so-called $(F,G)$-contracting mappings, which form a more general class of mappings than mappings contracting perimeters of triangles. The fixed-point theorem for these mappings is proved. We prove also a fixed-point theorem for mappings contracting perimeters of triangles in the sense of Edelstein.

On three-point generalizations of Banach and Edelstein fixed point theorems

TL;DR

This paper addresses three-point generalizations of fixed-point theory in metric spaces by introducing -contracting mappings, which generalize triangle-perimeter contractions, and by formulating Edelstein-type contractions for triangle perimeters. It proves that -contracting mappings are continuous and satisfy a fixed-point theorem (with potential uniqueness under additional hypotheses), thereby extending Banach-type results to a broader three-point framework. It also develops an Edelstein-type fixed-point theorem for mappings contracting perimeters of triangles, showing existence of a fixed point (at most two) in complete spaces and providing a constructive convergence argument via decreasing perimeters, along with discussion of accumulation points and compact cases. Collectively, the results broaden the fixed-point toolkit for nonlinear three-point mappings and deepen connections between triangle-perimeter contraction and Edelstein-type phenomena.

Abstract

Let be a metric space. Recently in~[1] it was considered a new type of mappings which can be characterized as mappings contracting perimeters of triangles. These mappings are defined by the condition based on the mapping of three points of the space instead of two, as it is adopted in many fixed-point theorems. In the present paper we consider so-called -contracting mappings, which form a more general class of mappings than mappings contracting perimeters of triangles. The fixed-point theorem for these mappings is proved. We prove also a fixed-point theorem for mappings contracting perimeters of triangles in the sense of Edelstein.
Paper Structure (3 sections, 11 theorems, 80 equations, 1 figure)

This paper contains 3 sections, 11 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mappings with controlled contraction
  3. Edelstein type fixed point theorem

Key Result

Theorem 1.3

Let $X$ be a metric space and let $T\colon X\to X$ be a mapping satisfying for all $x\neq y$, $x,y \in X$. Assume that there exists $x\in X$ such that the sequence of iterates $(T^nx)$ contains a subsequence $(T^{n_k}x)$ convergent to a point $\xi \in X$. Then $\xi$ is a unique fixed point of $T$.

Figures (1)

  • Figure 1: The points of the space $(X,d)$ with distances between them.

Theorems & Definitions (32)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3: Edelstein, 1962
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 22 more