Fixed point theory and some applications in modular metric spaces

This doctoral thesis consists of seven chapters. It includes modular metric spaces, which generalize ordinary metric spaces, and fixed point theorems in these spaces. First, the generalized versions of some particular conditions defined in ordinary metric spaces are described in modular metric spaces. For these generalized conditions defined in each chapter, fixed point theorems are first stated and then proven in modular metric spaces. Illustrative examples are considered to reinforce the results of the theorems proven in these chapters. As applications, the existence and uniqueness of the solutions of first-order nonlinear differential and integral equations given in general are proven by using the results of fixed point theorems for different types of mappings in the mentioned spaces. The introduction section explains the concept of fixed points, and a summary of the literature on this subject is given. Brief information about the fixed point theorem and the first applications of this theorem is given. In the second part, the concepts of metric and metric spaces are first explained, and brief information about the fixed point theorems based on the Banach contraction principle in these spaces is given. Then, short descriptions of the C-condition and F-contraction conditions defined in ordinary metric spaces and some particular functions utilized in this thesis are provided. Information about the fixed point theorems and the studies carried out for the mappings that satisfy the mentioned conditions in these spaces is given. The third section starts by giving a summary of the literature on modular metric space. Before the modular metric space, the concepts of modular and modular space are introduced. It is mentioned when and by whom these concepts were introduced. Then, the definition of the modular metric, introduced by Chistyakov, is given. Based on this definition, the structure of modular metric spaces is explained. It is mentioned that the fixed point theorem was expressed and proved for the first time in modular metric spaces. After this development, many scientists began to work on fixed point theorems for different types of mappings in modular metric spaces. It is stated that many results regarding the fixed point theorem emerged in these spaces. Some specific modular metric spaces are introduced. In addition, some important definitions such as limit, convergence, completeness, and being Cauchy series, in modular metric spaces are given. In addition, the definition of the contraction condition in modular metric spaces is given, and some fixed point theorems existing in the literature for this condition are mentioned. In the fourth part, the generalized C-condition given in metric spaces is extended to modular metric spaces and defined in these spaces for the first time. First of all, the fixed point theorems in modular metric spaces for mappings that satisfy this condition are expressed and proven. Based on the results obtained here, the existence and uniqueness of the solution of the anti-periodic boundary value problem and the initial value problem for first-order nonlinear differential equations are demonstrated. Therefore, separated modular metric spaces are defined for these problems. By using the fixed point theorems proven at the beginning of this section, the existence and uniqueness of the solutions for the problems mentioned in these defined spaces are demonstrated. Moreover, a non-metric modular metric is considered. It is shown that these theorems are valid in the difference space between modular metric spaces and ordinary metric spaces. In the fifth section, the sort connection ≼ is mentioned. By using this definition, partially ordered modular metric spaces are defined. Expressions and proofs of the fixed-point theorems in partially ordered modular metric spaces are presented for the generalized C-conditions defined in the previous section within modular metric spaces. In addition, some illustrative examples of partially ordered modular metric spaces are given to reinforce the results obtained from these theorems. It is proven that the solutions of the mappings given in these examples exist and are unique. Additionally, by using the intersection of the mapping's graph and the line y=x, it is shown that the fixed point is unique. As an application, an integral equation in a specific partially ordered modular metric space is considered. Finally, the existence and uniqueness of the solution of this integral equation in the mentioned modular metric space are proven by using the theorems proven in this section. In the sixth part, the generalized type of the F-contraction condition defined in ordinary metric spaces is defined in modular metric spaces by using α-admissible mappings. This generalized F-contraction condition, defined for the first time in this study, is denoted as the F_α-contraction condition. Two different F_α-contraction conditions, type 1 and type 2, are defined and separate fixed point theorems are expressed and proven for each of them. First of all, the existence of a fixed point is proven for the mappings that satisfy certain conditions. An example is given, and it is shown that the fixed point of the mapping in the example exists but is not unique. In addition, the existence of a fixed point of a mapping under different conditions is expressed and proven. Many results are given, as well. Later, by changing some of the conditions in the theorems proven here, it is stated and proven that the fixed point is unique. An example for this theorem is given. It is shown that the fixed point of the mapping in the example exists and is unique. Additionally, many results are expressed regarding the uniqueness of the fixed point. Finally, the existence of a solution of an integral equation given in a general form in the defined modular metric space is demonstrated. As an application of this, it is shown that the solution of the given integral equation exists in the specifically considered modular metric space. Furthermore, this existing solution is stated. In the final section, a summary of this thesis is given. Significant results regarding fixed point theorems in modular spaces are presented. Within the scope of the thesis, the extended versions of certain conditions existing in ordinary metric spaces are defined in modular metric spaces. Fixed point theorems are stated and proven for the mappings that satisfy these conditions in the considered modular metric spaces. In addition, separate examples are given to reinforce the results of the theorems proven here. Based on these proofs, each chapter contains significant conclusions and outcomes. The generalized conditions defined in modular metric spaces in the fourth, fifth, and sixth chapters of this thesis are defined for the first time in this thesis work. The fixed-point theorems expressed and proven in these chapters are the original results of this thesis work. As a result, this thesis study contains original results that do not exist in the literature, and so it contributes to the literature with new results.