Geometry of curves which are fixed by some transformations

In this thesis, which consists of five chapters, existence and uniqueness theorems are presented for fixed -ellipse and fixed -Cassini oval in metric spaces. The obtained theoretical results are supported by various examples and applications to activation functions. The first chapter of this thesis constitutes the introduction section. The second chapter includes the fundamental definitions and various illustrative examples that support the thesis study. In the third chapter, the concept of -ellipse on metric spaces is introduced with geometric examples, and various existence and uniqueness theorems for the fixed -ellipse concept are obtained using two different auxiliary functions. Additionally, a condition excluding the identity transformation is provided from the derived theorems. The theoretical results are further strengthened with an application to the S-Shaped Rectified Linear Activation Unit activation function. In the fourth chapter, the concept of -Cassini oval is presented with geometric examples on metric spaces, and various existence and uniqueness theorems for its fixed points are stated and proven. The obtained theoretical results are supported with examples, and theorems excluding the identity transformation are provided. Finally, an application to the Leaky Rectified Linear Unit activation function is presented. The fifth chapter is the conclusion and recommendations section, where a brief summary of the thesis is provided, and some open problems are suggested.