Fixed hyperbola and fixed Apollonius circle theorems in G-metric space

The origin of fixed-point theory dates back to the end of the 19th century. It is based on the work of many famous mathematicians such as Fredholm, Cauchy, Lipschitz, Picard, Liouville and Peano especially on the existence and uniqueness of solutions of differential equations. Banach's formulation is considered the beginning in the Metric Fixed-point theory. Fixed-point theory has applications in many areas. In particular, we can change the structure of the geometric shapes by changing the metrics and the formulas used. Therefore, metric change represents the impressions of a new world to us. Our aim in this thesis is completely on this structure. Inspired by the studies of Nihal Özgür and Nihal Ta¸s in 2017, where the geometry of fixed points is examined in the case of more than one fixed point, and also by the doctoral thesis written by G. Zaim Erçınar under the supervision of Ö. Geli¸sgen and T. Ermi¸s, theorems that give the existence and uniqueness conditions for the two geometric shapes defined in the G-metric space, the hyperbola and the Apollonius circle, to be fixed under a transformation, which Mustafa and Sims introduced in 2016, were produced. Thus we added a new vision to area, and also we have expanded the application areas. This thesis consists of seven chapters. In the first chapter, the subject and purpose of the thesis are explained and the thesis is introduced. In the second chapter, brief information is given about the studies conducted in the literature within the scope of the thesis subject. In the third chapter, the basic definitions, theorems and concepts to be used in the thesis are briefly reminded. In the fourth and fifth chapters, the concepts of fixed hyperbola and fixed Apollonius circle in G-metric spaces are defined and introduced, and the conditions for existence and uniqueness of fixed hyperbola and fixed Apollonius circle are given, and the applications of these concepts are mentioned. In the sixth chapter, the findings obtained in the thesis are expressed. In the seventh chapter, the results obtained in the thesis are expressed and suggestions are given for what can be done in future studies on this subject.