On Fixed Circles in $C^{\ast }$-Algebra Valued $S$-Metric Spaces and Application to Exponential Linear Unit Function

In this article, we introduce the concept of a fixed circle in a $C^{\ast }$-algebra valued $S$-metric space and explore some interesting existence and uniqueness theorems for self-mappings that have fixed circles in various directions. With a geometric viewpoint, we delve into the properties of these self-mappings and provide a deeper understanding of their mathematical foundations and applications. Additionally, we present several illustrative examples to verify the accuracy of our findings and to concretely demonstrate the applicability of the concept. These examples serve to validate the theoretical results related to fixed circles and their extendability. We also investigate the interplay between the algebraic structure of the $C^{\ast }$-algebra and the geometric properties of the fixed circles, highlighting how these interactions contribute to the richness of the theory. Finally, we apply the obtained fixed-circle results to activation functions used in neural networks, providing a meaningful example of how these mathematical structures can be utilized in practice. In doing so, we offer a broader perspective on the potential applications and practical implications of these theoretical insights, particularly in fields such as machine learning and nonlinear analysis, where the understanding of such mappings can lead to advancements in both theory and application.