$\varphi $-fixed points of self-mappings on metric spaces with a geometric viewpoint

TL;DR

This paper investigates the geometric properties of $\

Contribution

It provides solutions to an open problem by linking zeros of a function $\

Findings

Zeros of $\

Fixed circles and discs are contained in the zeros of $\

Abstract

A recent open problem was stated on the geometric properties of $\varphi $-fixed points of self-mappings of a metric space in the non-unique fixed point cases. In this paper, we deal with the solutions of this open problem and present some solutions via the help of appropriate auxiliary numbers and geometric conditions. We see that a zero of a given function $\varphi $ can produce 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) is also contained in the set of zeros of the function $\varphi $.