φ-fixed points of self-mappings on metric spaces with a geometric viewpoint

In this paper, we investigate the geometric properties of non-unique φ-fixed points. The concept of a φ-fixed point of a self-mapping T on a metric space X has been introduced recently. An element x ∈ X is called a φ-fixed point of the self-mapping T : X → X, where φ : X → [0,∞) is a given function, if x is a fixed point of T and φ(x) = 0. A recent open problem concerns the geometric properties of φ-fixed points, particularly the existence of a φ-fixed circle and a φ-fixed disc. In this study, we address this problem and present several solutions by employing suitable auxiliary numbers and geometric conditions. We demonstrate that a zero of a given function φ can generate a fixed circle (resp. fixed disc) contained in the fixed point set of a self-mapping T on a metric space. Moreover, this circle (resp. fixed disc) also lies within the set of zeros of the function φ.