Fixed Points of Mappings Contracting Perimeters of Polygons: A Geometric Generalization of Banach’s Principle

Abstract

In this paper, we investigate fixed points of mappings that contract the perimeters of polygons. Our study is motivated by the idea that the perimeter, as a global geometric measure, provides a more natural and flexible framework than the individual edge lengths when analyzing contraction properties in metric spaces. We extend Petrov’s fixed point theorem from triangles to
polygons with an arbitrary number of vertices and establish conditions under which such mappings admit unique fixed points. The methodology relies on generalizations of contraction mappings and properties of metric spaces. Our main contributions include a new perimeter-based contraction principle, a demonstration of its application in proving Banach’s contraction theorem, and examples that validate the theoretical results. This generalization enriches the existing literature on fixed point theory and opens avenues for further applications in geometric analysis and related areas.