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Fixed-Point Theorems in $b$-Metric Spaces via a Novel Simulation Function

Anuradha Gupta, Rahul Mansotra

TL;DR

The paper addresses fixed-point results in generalized $b$-metric spaces by introducing a novel $\mathbb{A}_{\mathbb{R}}$-simulation function framework. It builds on prior simulation-function theory to formulate $\mathfrak{J}_{_{\mathbb{A}_{\mathbb{R}}}}$-contractions and proves a fixed-point theorem ensuring existence and uniqueness of a fixed point under these contractions on complete $b$-metric spaces. The approach combines sequence arguments to establish Cauchy behavior and leverages the contraction inequality to force convergence, with several explicit finite-space examples demonstrating a concrete fixed point (often $3$) and illustrating the constructions. The work extends classical metric-fixed-point principles to a broader $b$-metric setting and offers a practical, verifiable framework for nonlinear contractions in generalized spaces.

Abstract

This paper introduces a new type of simulation function within the framework of $b$-metric spaces, leading to the derivation of fixed-point results in this general setting. We explore the theoretical implications of these results and demonstrate their utility through a concrete example.

Fixed-Point Theorems in $b$-Metric Spaces via a Novel Simulation Function

TL;DR

The paper addresses fixed-point results in generalized -metric spaces by introducing a novel -simulation function framework. It builds on prior simulation-function theory to formulate -contractions and proves a fixed-point theorem ensuring existence and uniqueness of a fixed point under these contractions on complete -metric spaces. The approach combines sequence arguments to establish Cauchy behavior and leverages the contraction inequality to force convergence, with several explicit finite-space examples demonstrating a concrete fixed point (often ) and illustrating the constructions. The work extends classical metric-fixed-point principles to a broader -metric setting and offers a practical, verifiable framework for nonlinear contractions in generalized spaces.

Abstract

This paper introduces a new type of simulation function within the framework of -metric spaces, leading to the derivation of fixed-point results in this general setting. We explore the theoretical implications of these results and demonstrate their utility through a concrete example.
Paper Structure (2 sections, 3 theorems, 41 equations)

This paper contains 2 sections, 3 theorems, 41 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Fixed Point Theorems Using $\mathfrak{J}_{_{\mathbb{A}_{\mathbb{R}}}}$-Contractions

Key Result

Theorem 1.3

Every $\vartheta$-contraction on a complete metric space has a unique fixed point.

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Theorem 2.5
  • ...and 6 more