Quantifying non-compactness in fixed point theorems: a measure-theoretic and applied approach

Abstract

In this paper, we commence by establishing the theoretical foundations of fixed point theorems and their historical importance. This paper then introduces groundbreaking measures rooted in measure theory, designed to quantify non-compactness within these theorems. These measures redefine the boundaries of classical fixed point theory, unlocking new vistas of applications. Our paper presents four fundamental theorems, each extending classical fixed point results to non-compact spaces. These theorems are rigorously proven, providing a solid mathematical foundation for our framework. We delve into the nuances of each theorem, showcasing their implications and relevance in contemporary mathematics. To underscore the practicality of our approach, we offer a diverse array of applications. From optimizing traffic flow in urban environments to modeling intricate ecological systems supported by many related examples, our framework provides innovative solutions to complex problems. These applications are accompanied by concrete examples and numerical simulations, illustrating the tangible benefits of our methodology. Throughout the paper, the synergy between measure theory and fixed point theorems is a central theme. We explore how measure-theoretic concepts enrich our comprehension of these theorems and offer fresh perspectives on their utility across various domains.