Exploring fixed point results in fuzzy $\mathscr{F}$-metric spaces with an application to satellite web coupling problem

TL;DR

The paper develops and analyzes compactness, boundedness, and total boundedness for fuzzy $\mathscr{F}$-metric spaces and proves a fixed-point theorem for fuzzy $\psi$-contractions in $\mathscr{F}$-complete spaces, establishing uniqueness under a generalized contraction framework. It demonstrates practical applicability by solving a nonlinear boundary-value problem via an integral formulation for a satellite web coupling, using a $\mathscr{F}$-metric with a contraction encoded through a $\psi$-function. The results extend fuzzy metric theory, providing rigorous existence-uniqueness results and constructive examples, including a geometric illustration of contractions and several motivating applications in nonlinear analysis and dynamic systems. The work offers new tools for uncertainty modeling and applied analysis, with potential extensions to broader generalized spaces and related computational problems.

Abstract

This article explores several fundamental aspects of fuzzy $\mathscr{F}$-metric spaces and their applications in mathematical analysis. We investigate some essential properties concerning compactness and total boundedness in fuzzy $\mathscr{F}$-metric spaces. Within this framework, we present a fixed point theorem and demonstrate its utility by applying it to the satellite web coupling problem. To support the theoretical findings, illustrative examples and a graphical representation of the contraction condition are also provided.

Figures (1)

• Figure 1: (a) $M ( Tx, Ty, t )$ vs $\psi ( M (x, y, t ))$ at $t = 2$ (b) $M ( Tx, Ty, t )$ vs $\psi ( M (x, y, t ))$ for $| x -y | = 10$

Theorems & Definitions (44)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Lemma 2.4
• Remark 2.5
• Example 2.6
• Definition 2.7
• Example 2.8
• Proposition 2.9
• Proposition 2.10