  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.
