Fixed ellipse theorems in metric spaces and an application of discontinuous activation function in neural networks

In this article, we establish new fixed ellipse theorems that will provide a geometrical perspective to the classical metric fixed point theory. Also, we provide novel fixed ellipse consequences as the expanding of some existing and known fixed circle theorems in literature to ellipse case, and give non-trivial illuminating examples, which show the impressiveness of our results. On the other hand, using the given fixed ellipse, we give some solutions for the open question which is posed by Rhoades with respect to the presence of contractive mappings that permissive discontinuity at the fixed point. Furthermore, we give an application to usage respecting activation functions handled at neural networks via the discontinuity and the obtained results of fixed ellipse.