Fixed point theory and application in modular metric spaces

In this thesis, it is aimed to find the existence and uniqueness of the common fixed point of various contraction mappings defined in modular 𝑏−metric spaces and modular 𝑏−metric spaces endow with Branciari distance function. In addition, the applicability of the obtained results to various fields of mathematics such as graph theory, homotopy, integral equations is also shown. Basically, this study consists of two main parts. In the first chapter, the development of fixed point theory, the concept of metric space and its properties, some basic definitions and theorems, the concept of fixed point, extensions of Banach fixed point theorem, graph theory and some generalized metric space structures are mentioned. In the second part, by using various auxiliary functions, the new concepts such as Suzuki Σ−contraction, 𝒢(Φ,ϑ,Ξ)−contraction, generalized 𝐹−contraction involving quadratics terms, generalized Suzuki-Berinde 𝑍̂Ψ∗Θ−contraction and generalized simulation Θ̂℮𝛽−contraction mappings are introduced. The common fixed point theorems obtained from these new concepts have been proved. These results are supported by various examples and mathematical applications. Also, the modular 𝑏−metric space equipped with the Branciari distance function and some properties of these spaces have been brought to the literature and the fixed point theorem on this space has been proved.