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

TL;DR

This paper introduces new fixed-figure results based on the concept of $k$-ellipse in metric spaces, extending fixed-point theory with geometric insights and applications to neural network activation functions.

Contribution

It develops novel fixed-figure theorems using $k$-ellipse and contraction types, providing existence, uniqueness, and practical applications.

Findings

Established new fixed $k$-ellipse theorems with existence and uniqueness.

Provided illustrative examples demonstrating the results.

Applied the theory to $S$-Shaped ReLU activation in neural networks.

Abstract

In this paper, as a geometric approach to the fixed-point theory, we prove new fixed-figure results using the notion of $k$-ellipse on a metric space. For this purpose, we are inspired by the Caristi type contraction, Kannan type contraction, 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 Linear Activation Unit ($SReLU$) to show the importance of our theoretical results.