Extensions of Banach Contraction Principle in Ordered Metric Spaces and Their Computational Implications

Abstract

A significant emphasis is placed on the evolution of fixed-point theory in the context of generalized metric spaces. The thesis delineates the structural characteristics and mathematical motivations behind alternative metric formulations, including b-metric, G-metric, and 2-metric spaces. These generalizations, which respectively relax or redefine the classical triangle inequality or the notion of pairwise distances, are introduced to accommodate a broader spectrum of mathematical and real-world systems. For instance, b-metric spaces, characterized by a constant multiplicative factor in the triangle inequality, provide a flexible framework for modeling systems with inherently imperfect or noisy distance metrics. Similarly, G-metric spaces, defined over triples of points, offer a more nuanced representation of spatial relationships, particularly in topological and functional analyses. The thesis not only reviews the formal definitions and axiomatic properties of these spaces but also presents newly formulated fixed-point theorems within each context, alongside illustrative examples and computational applications. In addition to theoretical generalizations, the thesis contributes to methodological innovations through the development and analysis of iterative algorithms tailored to fixed-point approximation. These include numerical schemes based on successive approximations, gradient iterations, and hybrid methods. The convergence behavior, rate of approximation, and computational complexity of these algorithms are analyzed rigorously, thereby emphasizing their practical viability in fields such as optimization, signal processing, and data science. The research also investigates non-classical mappings—namely expansive, non-expansive, and Lipschitz continuous mappings—and examines the conditions under which fixed-points can be guaranteed. These explorations challenge the constraints of classical fixed-point theory and open avenues for dealing with chaotic or diverging systems.