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Mappings contracting triangles

Ovidiu Popescu, Cristina Maria Pacurar

Abstract

The aim of the current paper is to introduce a new class of contractive mappings, which are contracting (a feature of) triangles. We prove that maps contracting triangles are continuous and give the fixed point result for such mappings. We emphasize that our main theorem encompasses many functions, with significant applicability, for which the result holds, thereby representing a notable advancement in this research domain.

Mappings contracting triangles

Abstract

The aim of the current paper is to introduce a new class of contractive mappings, which are contracting (a feature of) triangles. We prove that maps contracting triangles are continuous and give the fixed point result for such mappings. We emphasize that our main theorem encompasses many functions, with significant applicability, for which the result holds, thereby representing a notable advancement in this research domain.
Paper Structure (2 sections, 3 theorems, 42 equations, 2 figures)

This paper contains 2 sections, 3 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mappings contracting triangles

Key Result

Theorem 1.1

Let $(X,d)$, $|X|\geq 3$ be a complete metric space and let $T:X\to X$ be a mapping contracting perimeters of triangles on $X$. Then, $T$ has a fixed point if and only if $T$ does not possess periodic points of prime period $2$. The number of fixed points is at most $2$.

Figures (2)

  • Figure 1: Example 2.1
  • Figure 2: Example 2.2

Theorems & Definitions (15)

  • Definition 1.1: Petrov Petrov
  • Theorem 1.1: Petrov Petrov
  • Remark 2.1
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 2.1
  • proof
  • Remark 2.2
  • ...and 5 more