Novel fixed point outcomes in compact generalized metric spaces

Abstract

This work presents new advancements in fixed point theory within the framework of G-metric spaces, initially introduced by Mustafa and Sims. Contrary to prior findings that often reduced such results to their standard metric counterparts, our approach yields genuinely intrinsic and non-reducible theorems. We establish extended versions of the Banach, Kannan, and Reich fixed point theorems, leveraging the assumption of compactness in the G-metric setting. Our methodology eschews reliance on equivalence to standard metrics, instead furnishing stronger conclusions inherent to the G-metric structure. Furthermore, we explore applications involving mappings that contract the perimeters of triangles—a geometric condition with significant implications for nonlinear analysis. Included examples demonstrate the necessity of our hypotheses and delineate scenarios where existing results fail. These contributions propel the theory of generalized metric structures and their practical use.