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Full-Text Articles in Physical Sciences and Mathematics
A New Solution To The Discontinuity Problem On Metric Spaces, Ufuk Çeli̇k, Ni̇hal Özgür
A New Solution To The Discontinuity Problem On Metric Spaces, Ufuk Çeli̇k, Ni̇hal Özgür
Turkish Journal of Mathematics
We study on the Rhoades' question concerning the discontinuity problem at fixed point for a self-mapping T of a metric space. We obtain a new solution to this question. Our result generalizes some recent theorems existing in the literature and implies the uniqueness of the fixed point. However, there are also cases where the fixed point set of a self-mapping contains more than 1 element. Therefore, by a geometric point of view, we extend the Rhoades' question to the case where the fixed point set is a circle. We also give a solution to this extended version.
Bilateral-Type Solutions To The Fixed-Circle Problem With Rectified Linear Units Application, Ni̇hal Taş
Bilateral-Type Solutions To The Fixed-Circle Problem With Rectified Linear Units Application, Ni̇hal Taş
Turkish Journal of Mathematics
In this paper, we prove new fixed-circle (resp. fixed-disc) results using the bilateral type contractions on a metric space. To do this, we modify some known contractive conditions called the Jaggi-type bilateral contraction and the Dass-Gupta type bilateral contraction. We give some examples to show the validity of our obtained results. Also, we construct an application to rectified linear units activation functions used in the neural networks. This application shows the importance of studying "fixed-circle problem".