Perimetric Contraction on n-gon and Related Fixed Point Results

Abstract

This paper introduces a novel extension of the classical Banach Contraction Principle, focusing on "perimetric contractions" in n-gon. Unlike traditional contractions that deal with the distances between pairs of points, perimetric contractions are concerned with the contraction of the entire perimeter of an n-gon, considering the distances between consecutive points along the boundary. This new perspective enables the development of fixed-point results in higher-dimensional metric spaces. The core objective is to establish a fixed-point theorem for mappings that contract the perimeters of n-gon, providing a generalization of Banach's original theorem. The paper demonstrates that such mappings are continuous and presents conditions under which fixed points exist and are unique. Additionally, the relationships between perimetric contractions and conventional contraction mappings are examined, thus expanding the applicability of fixed-point theorems in more complex settings.