On the Geometry of Fixed Points and Discontinuity

Recently, there has been a considerable effort to obtain new solutions to the Rhoades' open problem on the existence of contractive mappings that admit discontinuity at the fixed point. An extended version of this problem is also stated using a geometric approach. In this paper, we obtain new solutions to this extended version of the Rhoades' open problem. A related problem, the fixed-circle problem (resp. fixed-disc problem) is also studied. Both of these problems are related to the geometric properties of the fixed point set of a self-mapping on a metric space. Furthermore, a new result about metric completeness and a short discussion on the activation functions used in the study of neural networks are given. By providing necessary examples, we show that our obtained results are effective.