Fixed point theorem for mappings contracting perimeters of triangles

TL;DR

This paper introduces a new class of mappings in metric spaces that contract the perimeters of triangles, proves a fixed-point theorem for them, and shows they are continuous, extending classical fixed-point results.

Contribution

It defines triangle perimeter contracting mappings, proves their continuity, and establishes a fixed-point theorem generalizing Banach's theorem with a novel example.

Findings

Mappings contract perimeters of triangles and are continuous

Fixed-point theorem established for these mappings

Classical Banach fixed-point theorem derived as a corollary

Abstract

We consider a new type of mappings in metric spaces which can be characterized as mappings contracting perimeters of triangles. It is shown that such mappings are continuous. The fixed-point theorem for such mappings is proved and the classical Banach fixed-point theorem is obtained like a simple corollary. An example of a mapping contractive perimeters of triangles which is not a contraction mapping is constructed.