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

Abstract

In this paper, we deal with the geometric properties of non-unique $\varphi $%
-fixed points. A $\varphi $-fixed point of a self-mapping $\mathcal{T}$ of a metric space $X$ is a recently introduced notion. An element $x\in X$ is
said to be a $\varphi $-fixed point of the self-mapping $\mathcal{T} :X\rightarrow X$, where $\varphi :X\rightarrow \left[ 0,\infty \right) $ is
a given function, if $x$ is a fixed point of $\mathcal{T}$ and $\varphi
\left( x\right) =0$. A recent open problem was stated on the geometric
properties of $\varphi $-fixed points as the investigation of the existence
of a $\varphi $-fixed circle and of a $\varphi $-fixed disc. We study on
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 $\mathcal{T}$ on a metric space.
Moreover, this circle (resp. fixed disc) is also contained in the set of
zeros of the function $\varphi $.