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 binterval metric space.Further, an improved binterval metric variant of a quasi-contraction characterizing the completeness of a binterval 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 binterval metric space.As an application of the binterval metric, a near-fixed interval ellipse and its unique equivalence -class are introduced to study the geometry of non-unique nearfixed points.