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The fixed-circle problem is a recent problem about the study of geometricproperties of the fixed point set of a self-mapping on metric (resp.generalized metric) spaces. The fixed-disc problem occurs as a naturalconsequence of this problem. Our aim in this paper, is to investigate newclasses of self-mappings which satisfy new specific type of contraction on ametric space. We see that the fixed point set of any member of these classescontains a circle (or a disc) called the fixed circle (resp. fixed disc) of thecorresponding self-mapping. For this purpose, we introduce the notions of an$F_{c}$-contractive mapping and an $F_{c}$-expanding mapping. Activationfunctions with fixed circles (resp. fixed discs) are often seen in the study ofneural networks. This shows the effectiveness of our fixed-circle (resp.fixed-disc) results. In this context, our theoretical results contribute tofuture studies on neural networks.

New fixed-circle results related to Fc-contractive and Fc-expanding mappings on metric spaces