Systems of linear equations in generalized b−metric spaces

This paper introduces a pioneering concept in the realm of metric spaces, specifically focusing on a novel category termed controlled generalized b-metric spaces (CGbMS). The study delves into the investigation of fixed points within CGbMS for self-mappings that exhibit both linear and non-linear contraction characteristics. The analysis establishes the existence and uniqueness of such fixed points, contributing valuable insights into the properties of these spaces. Moreover, the paper extends its impact by exploring diverse applications and implementations derived from the established results. One notable application is the application of these findings in solving systems of linear equations. The comprehensive examination of these applications not only underscores the practical significance of the proposed concept but also offers a broader understanding of its potential utility in various mathematical contexts. In summary, this research not only introduces and rigorously defines the concept of controlled generalized b-metric spaces but also provides a robust theoretical foundation by establishing the existence and uniqueness of fixed points. The exploration of applications, with a focus on solving linear equations, further highlights the practical implications and versatility of the proposed framework within the broader mathematical landscape.