Existence of best proximity points in convex metric spaces

In this thesis, the concepts of noncyclic contraction and noncyclic relatively nonexpansive mappings are examined within the framework of strictly convex metric spaces, and a detailed investigation is conducted regarding the existence of best proximity points. The thesis consists of five chapters. In the first chapter, studies related to the existence of best proximity points are discussed, followed by the preliminary information section in the second chapter, where fundamental definitions and theorems are provided. In the third chapter, the properties of convex metric spaces are emphasized, and certain fixed-point theorems for nonexpansive mappings are investigated. In the fourth chapter, the classes of noncyclic contractions and noncyclic relatively nonexpansive mappings are addressed, and the existence of best proximity points for such mappings is examined in detail. Finally, in the last chapter, the results are discussed, and suggestions for future studies are provided.