Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Periodic points of mappings contracting total pairwise distance

Evgeniy Petrov

TL;DR

This paper addresses the existence of periodic points for mappings contracting total pairwise distance on $n$ points in metric spaces, providing a genuine generalization of the Banach contraction principle. It defines $S(x_1,\dots,x_n)=\sum_{1\le i<j\le n} d(x_i,x_j)$ and proves continuity of such maps under the contraction condition $S(Tx_1,\dots,Tx_n)\le \alpha S(x_1,\dots,x_n)$. The main result shows that in a complete metric space $(X,d)$ with $|X|\ge n$, any $T:X\to X$ contracting total pairwise distance on $n$ points has a periodic point of prime period in $\{1,\dots,n-1\}$ (at most $n-1$ points); when $n=2$ this yields the Banach fixed-point theorem, and when $n=3$ it yields a fixed-point theorem for mappings contracting triangle perimeters. The paper also provides constructive examples (including a mapping with two fixed points on a space of cardinality $|X|=\mathfrak{c}$) and discusses corollaries relating to accumulation points and broader contraction properties.

Abstract

We consider a new type of mappings in metric spaces so-called mappings contracting total pairwise distance on $n$ points. It is shown that such mappings are continuous. A theorem on the existence of periodic points for such mappings is proved and the classical Banach fixed-point theorem is obtained like a simple corollary as well as the fixed-point theorem for mappings contracting perimeters of triangles. Examples of mappings contracting total pairwise distance on $n$ points and having different properties are constructed.

Periodic points of mappings contracting total pairwise distance

TL;DR

This paper addresses the existence of periodic points for mappings contracting total pairwise distance on points in metric spaces, providing a genuine generalization of the Banach contraction principle. It defines and proves continuity of such maps under the contraction condition . The main result shows that in a complete metric space with , any contracting total pairwise distance on points has a periodic point of prime period in (at most points); when this yields the Banach fixed-point theorem, and when it yields a fixed-point theorem for mappings contracting triangle perimeters. The paper also provides constructive examples (including a mapping with two fixed points on a space of cardinality ) and discusses corollaries relating to accumulation points and broader contraction properties.

Abstract

We consider a new type of mappings in metric spaces so-called mappings contracting total pairwise distance on points. It is shown that such mappings are continuous. A theorem on the existence of periodic points for such mappings is proved and the classical Banach fixed-point theorem is obtained like a simple corollary as well as the fixed-point theorem for mappings contracting perimeters of triangles. Examples of mappings contracting total pairwise distance on points and having different properties are constructed.
Paper Structure (2 sections, 9 theorems, 49 equations, 1 figure)

This paper contains 2 sections, 9 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main results

Key Result

Proposition 1.2

Suppose that in Definition d1 inequality (e1) holds for any $n$ points $x_1, x_2, \ldots, x_n\in X$ with $|\{x_1, x_2, \ldots, x_n\}|=k$, where $2\leqslant k\leqslant n-1$. Then $T$ is a mapping contracting total pairwise distance on $k$ points.

Figures (1)

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

Theorems & Definitions (21)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • ...and 11 more