Fixed point theorems in digital topology

In this thesis, fixed point theorems in the context of digital topology are examined. Digital topology enables the application of classical topological concepts to finite and discrete structures, making it possible to analyze digital images. This study first introduces fundamental concepts such as digital adjacency relations, digital intervals, connectedness, digital homeomorphism, and homotopy. Then, digital metric spaces are presented, and topological measures such as Hausdorff distance, Euler characteristic, and Lusternik–Schnirelmann category are explained. The main focus of this thesis is to establish digital analogues of fixed point theorems of Banach, Kannan, Chatterjea, Reich, and Zamfirescu types under contraction conditions in digital metric spaces. The properties of digital contraction mappings and their behavior on κ-connected and disconnected structures are examined in detail with examples. This study aims to systematically demonstrate the validity of fixed point theorems in digital metric spaces and to strengthen the theoretical foundation in this area.