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In this paper, as a geometric approach to the fixed-point theory, we provenew fixed-figure results using the notion of $k$-ellipse on a metric space. Forthis purpose, we are inspired by the Caristi type contraction, Kannan typecontraction, Chatterjea type contraction and \'{C}iri\'{c} type contraction.After that, we give some existence and uniqueness theorems of a fixed$k$-ellipse. We also support our obtained results with illustrative examples.Finally, we present a new application to the $S$-Shaped Rectified LinearActivation Unit ($SReLU$) to show the importance of our theoretical results.

New Fixed Figure Results with the Notion of $k$-Ellipse