The solutions of some nonlinear q-difference equations systems

This thesis mainly consists of five chapters. The first part of the thesis consists of two subsections. In the first subsection, the purpose and focus of the study is stated. In the second subsection, the studies carried out in the literature on the subjects that constitute the thesis study are mentioned and it is also aimed to examine the place of the studies in the literature in order to emphasize the history of the subjects and why they are important. On the other hand, this section is intended to be a resource for examining the studies that have been done as well as for a better understanding of the studies that can be done. The second part of the thesis consists of four subsections. In the first subsection, basic definitions and theorems about 𝑞-calculus, known as unlimited calculus, are given. Additionally, in some places the subject has been tried to be explained with examples. These concepts form the basis for the third chapter, which constitutes the first original part of the study. Therefore, in this subsection, firstly, the 𝑞-analogue of any natural number 𝑛 and the 𝑞 - analogue of the factorial of any natural number 𝑛 are defined and explained with examples. In addition, the definitions of the 𝑞-analogues of the exponential function 𝑒𝑡, which will be used frequently in the first original part of the work, and some connections between these definitions are given. Finally, 𝑞-derivative, 𝑞-integral, and some basic theorems are included. In the second subsection, it is aimed to include the basic definitions, theorems and concepts related to (𝑝,𝑞)-calculus, which is a generalization of 𝑞-calculus and needed for the fourth chapter, which constitutes the second original part of this thesis study. Therefore, first of all, the (𝑝,𝑞)-analogue and (𝑝,𝑞)-factorial of any natural number 𝑛 are defined. The definition of (𝑝,𝑞)-derivative is explained by giving an example. Additionally, it is emphasized with an example that there is no general derivation for the chain rule in (𝑝,𝑞)-calculus. Finally, the definition of (𝑝,𝑞)-integral, which will be used frequently in the second original part of the study, and some related basic theorems are given. In the third subsection, the definition of the concept of time scale, which will be encountered in the fourth chapter of the thesis, is given and explained with examples. In the last subsection, before talking about fixed point theory, definitions of metric space, Cauchy sequence, convergence, completeness, continuity, uniform continuity, sequential continuity, equicontinuity, compact set, relatively compact set, normed space and Banach space are given. The connection between metric space and normed space is emphasized. Fundamental theorems such as the Arzela-Ascoli theorem and Lebesgue limited convergence theorem are included. Then, the definition of fixed point is explained with examples and the concept of contraction transformation is introduced. Finally, Schaefer fixed point theorem, Krasnoselskii fixed point theorem and Banach fixed point theorem are discussed. In addition, the initial condition of the initial value problem, which will be defined in the third section, consists of a matrice, and the inequality 𝐶>0 must be defined if 𝐶 is any matrice. However, unlike two real numbers, two matrices cannot be directly compared. Therefore, it is defined the following cone in 𝒦𝑛 𝒦𝑛+={𝐶=(𝑐𝑖𝑗)∈𝒦𝑛:𝑐𝑖𝑗≥0,∀𝑖,𝑗=1,𝑛̅̅̅̅̅} where 𝒦𝑛 is the set of matrices of type 𝑛×𝑛. Then, a partial ordering relation is defined on this 𝒦𝑛+ cone consisting of matrices. The third part of the study constitutes the first original part of the thesis and investigates the existence and uniqueness of solutions to the first-order initial value problem on 𝑞-calculus. For this reason, first of all, the problem is defined in q-calculus. Then, an auxiliary theorem is proven, stating that the system of equations defined using the initial conditions and the properties of the 𝑞-integral is equivalent to the solution of a system of integral equations. This equivalent solution is rewritten by constructing an appropriate Green's function. In the next stage, in order to prove the existence and uniqueness of the solutions of the given equation, a 𝑈 operator was defined as 𝑈(𝑥,𝑦)=(𝑥,𝑦) and the problem was transformed into a fixed point problem. As a result, the proof of three main theorems is given using Schaefer fixed point theorem, Krasnoselskii fixed point theorem and Banach fixed point theorem. The fourth part of the study consists of two subsections and the solutions of a second-order (𝑝,𝑞)-difference equation on (𝑝,𝑞)-calculus, which is a generalization of 𝑞-calculus, are examined with different methods. In the first part of the fourth chapter, it is considered the following second order (𝑝,𝑞)-difference equation with non-local and (𝑝,𝑞)-integral boundary conditions { 𝐷𝑝,𝑞2𝑥(𝑡)=𝜑(𝑡,𝑥𝜎(𝑞𝑡)), 𝑡∈[0,𝑇𝑝2.𝑞2]𝕋0𝑥(0)=𝑥0+𝑘(𝑥), 𝑥(𝑇)=𝛿∫𝑥(𝑠)𝑑𝑝,𝑞𝑠𝑇0 where 𝑥0∈ℝ𝑛 and 𝑇∈𝕋 is a fixed constant. Solutions of this second-order (𝑝,𝑞)-difference equation are investigated using Banach fixed point theorem and an example is given. In the second part of the fourth chapter, the equation defined aboveis transformed into a second-order (𝑝,𝑞)-difference equation of the form 𝐷𝑝,𝑞2𝑥(𝑡)+𝜌(𝑡).𝑥𝜎(𝑞𝑡)=0, without boundary conditions, by taking the 𝜑 function as 𝜑(𝑡,𝑥𝜎(𝑞𝑡))=−𝜌(𝑡).𝑥𝜎(𝑞𝑡). Then, the second-order derivative of the 𝑥(𝑡) function and the first order derivative of the 𝑥(𝑞𝑡) function are found. An Euler-Cauchy-like (𝑝,𝑞)-difference equation is reached as 𝑞𝑡.𝜎(𝑡).𝐷𝑝,𝑞2𝑥(𝑡)+𝑎𝑡.𝐷𝑝,𝑞𝑥(𝑞𝑡)+𝑏.𝑥(𝑞2𝑡)=0 when the derivatives are substituted. Therefore, the solution metheod of the Euler-Cauchy-like 𝑞-difference equation is applied to the new equation obtained. Additionally, the oscillation of the solutions is examined. Finally, some results available in the literature on 𝑞-calculus have been generalized to (𝑝,𝑞)-calculus. In the fifth chapter of the thesis, the results obtained and the generalizations made are included and suggestions are presented for further studies.