Perimetric Contraction on Polygons and Related Fixed Point Theorems

TL;DR

This paper introduces perimetric contractions on polygons, generalizing perimeter-based mappings, and establishes fixed point theorems with conditions for uniqueness, extending classical results like Banach's theorem.

Contribution

It defines new classes of perimeter-based contractions on polygons and proves fixed point theorems, including Kannan type contractions, with conditions for uniqueness.

Findings

Established fixed point theorems for perimetric contractions.

Extended Banach's fixed point theorem to polygon-based mappings.

Provided examples validating the main results.

Abstract

In the present paper, a new type of mappings called perimetric contractions on $k$-polygons is introduced. These contractions can be viewed as a generalization of mappings that contracts perimeters of triangles. A fixed point theorem for this type of mappings in a complete metric space is established. Achieving a fixed point necessitates the avoidance of periodic points of prime period $2,3,\cdots, k-1$. The class of contraction mappings is encompassed by perimeter-based mappings, leading to the recovery of Banach's fixed point theorem as a direct outcome from our main result. A sufficient condition to guarantee the uniqueness of the fixed point is also provided. Moreover, we introduce the Kannan type perimetric contractions on $k$-polygons, establishing a fixed point theorem and a sufficient uniqueness condition. The relationship between these contractions, generalized Kannan type mappings, and mappings contracting the perimeters on $k$-polygons is investigated. Several examples are illustrated to support the validity of our main results.