On unique and non-unique fixed point in parametric N_b−metric spaces with application

We propose 𝒮𝒜, η−𝒮𝒜, η−𝒮 𝒜_min, and 𝒮𝒜_η,δ,ζ−contractions and notions of η−admissibility type b and η_b−regularity in parametric N_b-metric spaces to determine a unique fixed point, a unique fixed circle, and a greatest fixed disc. Further, we investigate the geometry of non-unique fixed points of a self mapping and demonstrate by illustrative examples that a circle or a disc in parametric N_b−metric space is not necessarily the same as a circle or a disc in a Euclidean space. Obtained outcomes are extensions, unifications, improvements, and generalizations of some of the well-known previous results. We provide non-trivial illustrations to exhibit the importance of our explorations. Towards the end, we resolve the system of linear equations to demonstrate the significance of our contractions in parametric N_b−metric space.