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The fixed-circle problem is a recent problem about the study of geometric properties of the fixed point set of a self-mapping on metric (resp. generalized metric) spaces. The fixed-disc problem occurs as a natural consequence of this problem. Our aim in this paper, is to investigate new classes of self-mappings which satisfy new specific type of contraction on a metric space. We see that the fixed point set of any member of these classes contains a circle (or a disc) called the fixed circle (resp. fixed disc) of the corresponding self-mapping. For this purpose, we introduce the notions of an $F_{c}$-contractive mapping and an $F_{c}$-expanding mapping. Activation functions with fixed circles (resp. fixed discs) are often seen in the study of neural networks. This shows the effectiveness of our fixed-circle (resp. fixed-disc) results. In this context, our theoretical results contribute to future studies on neural networks., Comment: 15 pages, 1 figure

In this paper we introduce the concept of the rectangular metric like spaces, along with its topology and we prove some fixed point theorems under different contraction principles. We introduce the concept of modified metric-like space as well and prove some topological and convergence properties under the symmetric convergence. Some examples are given to illustrate the proven results and enrich the new introduced metric type spaces.

We generalize Lewis's result about a central series associated with the vanishing off subgroup. We write $V_{1}=V(G)$ for the vanishing off subgroup of $G$, and $V_{i}=[V_{i-1},G]$ for the terms in this central series. Lewis proved that there exists a positive integer $n$ such that if $V_{3} < G_{3}$, then $|G:V_{1}|=|G':V_{2}|^{2}=p^{2n}$. Let $D_{3}/V_{3} = C_{G/V_{3}}(G'/V_{3})$. He also showed that if $V_{3} < G_{3}$, then either $|G:D_{3}|=p^{n}$ or $D_{3}=V_{1}$. We show that if $V_{i}

Özgür, Nihal Yılmaz (Balikesir Author), In this paper, we present a new notion which is called an extended Mb-metric space as a generalization of an M-b-metric space. We investigate some basic and topological properties of this new space. Furthermore, an extended M-b-metric space is a new generalization of an M-metric space and partial metric space. So it is important to study fixed-point theorems for non-M -metric (or non-partial metric) functions on an extended M-b-metric space. Also we generalize some known results in literature.