On generalized distance functions in Fixed Point Theory

The development of various generalizations of metric spaces has significantly enriched the study of fixed point theory, enabling the extension of classical results to broader and more flexible frameworks. In this talk, we examine several of these generalized metrics, with particular attention to the bipolar metric, suprametric, and perturbed metric. We investigate their topological properties, including completeness, compactness, and continuity structures, and discuss how these properties interact with fixed point principles.

A key aspect of our study is the introduction and analysis of remetrization techniques—methods of redefining distance functions while preserving completeness of the underlying space. Approaches can provide deeper insights into the behavior of iterative schemes and broaden the applicability of fixed point theorems. The necessary and sufficient assumptions of fixed point theorems will be compared in generalized metric space and induced metric space.

We also explore the stability properties of fixed point problems within generalized and remetrized spaces, highlighting differences and similarities in the convergence behavior of Picard iterative sequences. Comparative analysis reveals how these generalizations influence not only the existence and uniqueness of fixed points but also the efficiency and robustness of numerical methods used to approximate them. The presented results offer a unified perspective both on fixed point results and the scope of their applications, in that way combining both pure and applied mathematical contexts.