Fixed ellipse and fixed Cassini oval theorems in G-metric space

This study, comprised of seven chapters, explores the concept of G-metric spaces by defining ellipses and Cassini curves previously defined in other spaces as G-ellipses and G-Cassini curves within G-metric spaces. Additionally, fixed point theorems concerning the solution of the fundamental fixed point problem and some contraction conditions are discussed, outlining their equivalents in G-metric spaces. Conditions necessary for the existence and uniqueness of transformations preserving any G-ellipse and G-Cassini curve are also identified. The first chapter briefly discusses the historical development stages of the fixed point theorem and outlines the objectives of this study. The second chapter covers the axiomatic structures of certain metric spaces and their generalizations in the literature. The third chapter presents fundamental definitions, examples, and theorems related to G-metric spaces examined throughout the thesis. The fourth chapter introduces G-ellipses, discussing the conditions for the existence and uniqueness of fixed G-ellipses, supported by examples and applications. The fifth chapter introduces G-Cassini curves and discusses the conditions for the existence and uniqueness of fixed G-Cassini curves, also supported by examples and applications. The sixth and seventh chapters summarize the findings of this study and provide suggestions for further research.