  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 $.
