Investigation of near fixed points, near fixed interval ellipse and its equivalence classes

The objective of the manuscript is to employ the Hardy-Roger contraction to determine the near fixed point and its unique equivalence class in the context of the b − interval metric space. Further, an improved b − interval metric variant of a quasi-contraction characterizing the completeness of a b − interval metric space is exhibited. Various illustrations have been provided to show the existence of a near fixed point and its distinct equivalence class for both continuous and discontinuous maps developed in the b − interval metric space. As an application of the b − interval metric, a near-fixed interval ellipse and its unique equivalence ε −class are introduced to study the geometry of non-unique nearfixed points.