Equivalence Class Characterizations of Near Fixed Points and Interval Ellipses in Coupled Metric and G-Metric Spaces With Applications

In this paper, we suggest a rigorous equivalence class description of interval ellipses and near fixed points in coupled metric and G-metric spaces with applications. In both metric and G-metric frameworks, we prove that the equivalence classes formed by the division of near fixed points naturally reflect their convergence to underlying fixed points. We demonstrate that tolerance zones of iterative convergence for interval ellipses are encoded by equivalence classes, enabling a geometric interpretation of coupled fixed point behavior. By applying these findings to coupled structures, we show how pairs of near fixed points are arranged into synchronized convergence classes using equivalence relations. We make three contributions: (i) we present formulations of equivalence classes for interval ellipses and near fixed points in metric and G-metric spaces; (ii) we establish convergence theorems connecting these classes to the stability and uniqueness of fixed points; and (iii) we provide examples and applications to stability analysis, machine learning, control systems, and coupled systems. This characterization offers a versatile toolkit for examining iterative processes in computational and nonlinear contexts, in addition to strengthening the structural knowledge of fixed point theory.